ALL PERMUTATIONS OF INFINITY

[Graham Cooper]

How many ways can you re-order <f, a, t> ?
<f a t>
<f t a>
<a f t>
<a t f>
<t f a>
<t a f>

3! = 3 Factorial = 3X2X1 = 6

How many different ways can you order <1,2,3,4....> ?

oo X oo-1 X oo-2 X .... X 3 X 2 X 1
It's not well defined!

But you can use all possible computable orders of the infinite
sequence of (emulated) computer programs.
e.g.
TM1 --> Real1
TM2 --> Real2
TM3 --> Real3
...

use a different Universal Turing Machine and
TM1 -> Real2
TM2 -> Real1
TM3 -> Real3
...

Since there is an infinite sequence of Universal Turing Machines
increasing in size as you list all computer programs, every possible
permutation of the set of Emulated Turing Machines is computable.

_____________________________________________

Turing's Computer specification, the Turing Machine

TAPE
...00010110101010000...
^TP

STATE1 -> READ0 -> WRITE-1 -> TP-LEFT -> NEXT STATE
-> READ1 -> WRITE-1 -> TP-RIGHT -> NEXT STATE

STATE2
...


A method for representing all possible TURING MACHINES that indexes
them
TM1, TM2, TM3..

The first 8 TMs are all size 1 with only 1 internal state the starting
state S, where each state has 2 branches: INPUT-0 and INPUT-1

TM-1
S-00L->S
S-10L->S

TM-2
S-00L->S
S-10R->S
...
TM-8
S-11R->S
S-01R->S

--------

Next comes TM-9, the first 2 State TM.

TM-9
S-00L->1
S-10L->1
1-00L->S
1-10L->S
..

There should be another 16 or 32 of those!, then onto 3 state TMs and
so on.

This is 1 method to count all possible computer programs.

____________________


The first Universal Turing Machine is about 5,495 bits long, around
200 states.

Its roughly the 6^200^200th Turing Machine, but let's call it TM-5000

TM-5000 doesn't just input a number, it decodes the input number into
2 numbers!

TM-5000(InputTape)
<=>
UTM1(a,b)

Not only does it do that, a is the number of ANY-TURING MACHINE!

Even Turing Machines BIGGER than 5000!

TM-5000(111111111111...1)
<=>
UTM1(6000,111)
<=>
TM-6000(111)

Using TM-5000 we can emulate any other Turing Machine, including all
Universal Turing Machines!

The original sequence of Turing Machines goes:

TM1
TM2
TM3
...
TM5000 UTM1
TM5001
...
TM7777 UTM2
TM7778
...
TM12000 UTM3
...
TM13333 UTM4
...

and so on!

___________

EACH UTM emulates the SAME INFINITE SET of computer programs
TM1, TM2, TM3...
but in infinitely many different orders!

It may turn out UTM1 and UTM2 have slightly different order of
computer functions!

TM-5000( X, Y ) = TM-7777( X, Y ) | X>2
TM-5000( 1, Y ) = TM-7777( 2, Y )
TM-5000( 2, Y ) = TM-7777( 1, Y )

So by emulating all programs using TM-5000 you get
<TM1, TM2, TM3, TM4, ...>
and using TM-7777 you get
<TM2, TM1, TM3, TM4, ...>

The infinite set of Universal Turing Machines provides infinitely many
orderings of the infinite set of programs.

Graham Cooper (BInfTech)
KINGS BEACH QUEENSLAND

[K_h]

"Graham Cooper" wrote in message
news:302a8508-af13-4fa2...@t2g2000pbg.googlegroups.com...

>
> How many different ways can you order <1,2,3,4....> ?
>

2^aleph_0 ways, assuming the axiom of choice and defining the factorial of
the cardinality of a set S to be equal to the cardinality of the set of all
bijections of S to itself.

+

[Graham Cooper]

On May 21, 10:45 am, "K_h" <KHol...@SX729.com> wrote:
> "Graham Cooper"  wrote in message
>
>

> > How many different ways can you order <1,2,3,4....> ?
>
> 2^aleph_0 ways, assuming the axiom of choice and defining the factorial of
> the cardinality of a set S to be equal to the cardinality of the set of all
> bijections of S to itself.
>
>

The presented algorithm can list all computable permutations of N.

All 1X2X3X4X5X6X7... of them!

The set of binary string 2^oo=2X2X2X2X2...
is much smaller.

Herc

[Torben Ægidius Mogensen]

Graham Cooper <graham...@gmail.com> writes:

> How many ways can you re-order <f, a, t> ?
> <f a t>
> <f t a>
> <a f t>
> <a t f>
> <t f a>
> <t a f>
>
> 3! = 3 Factorial = 3X2X1 = 6
>
> How many different ways can you order <1,2,3,4....> ?
>
> oo X oo-1 X oo-2 X .... X 3 X 2 X 1
> It's not well defined!

Not if you write it like this, but the number of permutations of a set
is well defined even for infinite sets -- just like the number of all
subsets of a set is defined even for infinite sets.

For finite and infinite sets, two sets are considered to have the same
number of elements if there exists a bijection between them. For
example, the set of even naturals has the same size as the set of all
naturals (N), because f(n)=2*n is a bijection between naturals and even
naturals. For the same reason, the set of all sets of naturals (P(N)
or, equivalently, 2^N) has the same size as the set of real numbers (R):
There is a bijection between sets of naturals and reals.

There is a bijection between permutations of N and subsets of N, so the
set of permutations has the same size as the set of subsets.

Torben

[Torben Ægidius Mogensen]

You can not assume that a property that holds for finite numbers also
holds for infinite numbers. For example, n+1 > n for all (finite)
naturals, but oo+1 = oo (which I'm sure you agree).

So you can not conclude that n! > 2^n for naturals n>2 implies that c! >
2^c for infinite cardinals c. In fact, for infinite cardinals c, c! =
2^c (if c! is defined as the number of permutations of c and 2^c is
defined as the number of subsets of c).

Torben

[Graham Cooper]

On May 22, 2:46 am, torb...@diku.dk (Torben Ægidius Mogensen) wrote:

How are you calculating O(2^alepth_0) for an algorithm PS(N) you claim
doesn't exist?

You must have ASSUMED the amount of subsets of N is 2X2X2... = |R|

You're the ones who can't find algorithms to Halt(), Prv(), PS() not
me!

Perm(N) clearly has complexity O(1X2X3X4X5...)

1X2X3X4X5X6... > 2X2X2X2X2X2...

UTM^2 > 2^aleph_0

I can COMPUTE the enumeration of 1X2X3X4X5X6... computable terms
(Permutations of N)

Herc

[Graham Cooper]

On May 22, 2:46 am, torb...@diku.dk (Torben Ægidius Mogensen) wrote:

> For the same reason, the set of all sets of naturals (P(N)
> or, equivalently, 2^N) has the same size as the set of real numbers (R):
> There is a bijection between sets of naturals and reals.
>
> There is a bijection between permutations of N and subsets of N, so the
> set of permutations has the same size as the set of subsets.
>
>         Torben

Thanks for the NON-SEQUITUR of the Century.

So there's a BIJECTION between O(2^n) and O(n!) sized sets now?

SETSIZE 1 {a} = 1 Permutation
SETSIZE 2 {a,b} = 2! Permutations
SETSIZE 3 {a,b,c} = 3! Permutations
...

SETSIZE |N| {1,2,3..} = 2^|N| Permutations

How convenient!

Herc
--
http://tinyurl.com/BiggestNumber

[Torben Ægidius Mogensen]

Graham Cooper <graham...@gmail.com> writes:

> On May 22, 2:46 am, torb...@diku.dk (Torben Ægidius Mogensen) wrote:
>> For the same reason, the set of all sets of naturals (P(N)
>> or, equivalently, 2^N) has the same size as the set of real numbers (R):
>> There is a bijection between sets of naturals and reals.
>>
>> There is a bijection between permutations of N and subsets of N, so the
>> set of permutations has the same size as the set of subsets.
>>
>>         Torben
>
> Thanks for the NON-SEQUITUR of the Century.
>
> So there's a BIJECTION between O(2^n) and O(n!) sized sets now?

Only for infinite n.

> SETSIZE 1 {a} = 1 Permutation
> SETSIZE 2 {a,b} = 2! Permutations
> SETSIZE 3 {a,b,c} = 3! Permutations
> ...

True so far.

> SETSIZE |N| {1,2,3..} = 2^|N| Permutations

But this is a non sequitur. A property that holds for all finite values
does not necessarily hold for infinite values. Or, more geneally, a
property that holds for all elements of a sequence of values does not
necessarily hold for the limit of the sequence -- whether the limit is
finite or not.

> How convenient!

Not at all. It would be so much more convenient if a property that
holds for elements of a sequence would also hold in the limit. But,
alas, we are not so lucky.

Torben

[Torben Ægidius Mogensen]

> How are you calculating O(2^alepth_0) for an algorithm PS(N) you claim
> doesn't exist?

I didn't say I could _calculate_ the value, just that I could reason
about it. This is similar to reasoning about the exact value of pi even
though you can never calculate it.

> You must have ASSUMED the amount of subsets of N is 2X2X2... = |R|

Not at all. This is, in fact, fairly easy to prove: We use the theorem
that there exists a bijection between two sets A and B if there exists
an injection from A to B and an injection from B to A. So all we need
to do to prove that |P(N)| = |R| is to find injections both ways.

First, we define a mapping from R to P(N). This works as follows:

Write the real number x as a binary string s b_n ... b_1 . d_1 d_2 ...
where s is the sign, b_i are the bits of the integral part and d_j are
the bits of the fractional part.

For reals that have two binary representations (such as 1 which can be
represented both as 1.000... and 0.111...), we choose the one without an
infinite sequence of 1 bits.

From this we construct a set of naturals:

- if s is positive, 0 is in the set.
- if b_i is 1, 2i is in the set.
- if d_j is 1, 2j+1 is in the set.

Clearly, different real number produce different sets, so this mapping
is an injection.

Going the other way, we have a set which is either finite or infinite.
For a finite set {n_1,n_2,...,n_k}, we produce the number 2 + 2^n_1 +
2^n_2 + ... + 2^n_k. So we get a number greater than or equal to 2.

For an infinite set {n_1,n_2,....}, we produce the number 2^(-n_1) +
2^{-n_2} + ..., which is a number between 0 (exclusive) and 1
(inclusive).

Clearly, every set maps to a different real number, so this mapping is
an injection.

Since we have injections both ways, the sets are equipotent (i.e.,
having the same size).

> You're the ones who can't find algorithms to Halt(), Prv(), PS() not
> me!

With all due respect, neither can you.

> Perm(N) clearly has complexity O(1X2X3X4X5...)
>
> 1X2X3X4X5X6... > 2X2X2X2X2X2...

I can in a similar way "prove" that 1X2X3X4X5X6... < 2X2X2X2X2X2...:

Each integer n is smaller than 2^k for some k. So by grouping a
sufficient number of 2s in the product to the right, we get:

1 x 2 x 3 x 4 x 5 x 6 ... < 2 x (2x2) x (2x2) x (2x2x2) x (2x2x2) ...

So we have a product to the left where all factors are smaller than the
factors in the product to the right. So, clearly, the product on the
left is smaller. And by associativity of multiplication, 2 x (2x2) x
(2x2) x (2x2x2) x (2x2x2) ... = 2x2x2x2x2x2x2x2x2x2x2..., so we have
1x2x3x4x5x6... < 2x2x2x2x2x2.... QED.

Torben

[Graham Cooper]

On May 22, 7:53 pm, torb...@diku.dk (Torben gidius Mogensen) wrote:
> Graham Cooper <grahamcoop...@gmail.com> writes:

Yes Yes I know all that, you threw a red herring that there is a
bijection between permutations and powersets,

That's like saying there's a bijection between {a,b,c} and {a,b,c,d,e}

Are you saying there is a Permutation Missing from all permutations
of

< TM1, TM2, TM3, TM4, ...>

such that no Universal Turing Machine can emulate all TMs in that
*missing* sequence?

What is it?

Say Penrose 1st UTM taking up 5495 bits can emulate
TM1, TM2, TM3, ...

Which permutation of those TMs can NO UTM emulate?
TM4, TM7, TM1, ... ?

There exists a UTM that will produce that sequence!


Herc

[Graham Cooper]

On May 22, 7:53 pm, torb...@diku.dk (Torben gidius Mogensen) wrote:
> Graham Cooper <grahamcoop...@gmail.com> writes:

Yes Yes I know all that, you threw a red herring that there is a
bijection between permutations and powersets,

That's like saying there's a bijection between {a,b,c} and {a,b,c,d,e}

Are you saying there is a Permutation Missing from all permutations
of

< TM1, TM2, TM3, TM4, ...>

[WM]

On 21 Mai, 18:46, torb...@diku.dk (Torben Ægidius Mogensen) wrote:

> Graham Cooper <grahamcoop...@gmail.com> writes:
> > How many ways can you re-order <f, a, t> ?
> > <f a t>
> > <f t a>
> > <a f t>
> > <a t f>
> > <t f a>
> > <t a f>
>
> > 3! = 3 Factorial = 3X2X1 = 6
>
> > How many different ways can you order <1,2,3,4....> ?
>
> > oo X oo-1 X oo-2 X .... X 3 X 2 X 1
> > It's not well defined!
>
> Not if you write it like this, but the number of permutations of a set
> is well defined even for infinite sets -- just like the number of all
> subsets of a set is defined even for infinite sets.

Take a well-ordered set Q of all rational numbers (simply an
enumeration). Does the well-ordered permutation exist, that well-
orders Q by magnitude of its elements? If not, is this permutation
taken into account, nevertheless, when counting all permutations of Q?

Regards, WM

[Graham Cooper]

Say 2/3 and 3/4 were adjacent, then it's trivial to find a rational
inbetween.

8/12 9/16 //find common denominator
16/24 18/36 //*2

17/36 //increment numerator

--------

As the UTM's get bigger and bigger the starting TM increases.
Referenced to UTM1.

Early UTMS first TM#.

UTM1= <TM1, TM2, TM3...>
UTM2= <TM1, TM2, TM20, TM25...>
UTM3= <TM3, TM1, TM2, TM4...>

...

UTM10000000= <TM500, TM20, TM400, TM1...>

Each sequence position will cover every Natural Infinitely many times.

But it would be difficult to specify what the pattern is. Every trick
in every recipe-book would make it as erratic as the explosive growth
of BB().

TM-SIZE MAX-1s OUTPUT
BB(2) 6
BB(3) 38
BB(4) 3,932,964
BB(5) 1.7 x 10^352
BB(6) 1.9 x 10^4933
...


Herc

[Graham Cooper]

On May 22, 9:53 pm, WM <mueck...@rz.fh-augsburg.de> wrote:

[Graham Cooper]

On May 23, 12:34 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> Say 2/3 and 3/4 were adjacent, then it's trivial to find a rational
> inbetween.
>
> 8/12 9/16    //find common denominator
> 16/24 18/36  //*2
>
> 17/36        //increment numerator
>

8/12 9/12    //find common denominator, i.e 12
16/24 18/24  //*2

17/24        //increment numerator


Herc

[Uirgil]

In article
<ee9bc91c-bc2f-42db...@b1g2000vbb.googlegroups.com>,
WM <muec...@rz.fh-augsburg.de> wrote:

> On 21 Mai, 18:46, torb...@diku.dk (Torben Ćgidius Mogensen) wrote:
> > Graham Cooper <grahamcoop...@gmail.com> writes:
> > > How many ways can you re-order <f, a, t> ?
> > > <f a t>
> > > <f t a>
> > > <a f t>
> > > <a t f>
> > > <t f a>
> > > <t a f>
> >
> > > 3! = 3 Factorial = 3X2X1 = 6
> >
> > > How many different ways can you order <1,2,3,4....> ?
> >
> > > oo X oo-1 X oo-2 X .... X 3 X 2 X 1
> > > It's not well defined!
> >
> > Not if you write it like this, but the number of permutations of a set
> > is well defined even for infinite sets -- just like the number of all
> > subsets of a set is defined even for infinite sets.
>
> Take a well-ordered set Q of all rational numbers (simply an
> enumeration). Does the well-ordered permutation exist, that well-
> orders Q by magnitude of its elements?

Obviously not, since ordering by magnitude does not provide a well
ordering of Q.

It is not even a total ordering of Q until you determine which of q and
-q is to have the greater magnitude.

> If not, is this permutation
> taken into account, nevertheless, when counting all permutations of Q?

Since the ordering as described is not an ordering, no such
"permutation" exists, not even to be counted.

[WM]

On 22 Mai, 22:43, Uirgil <uir...@uirgil.ur> wrote:
> In article
> <ee9bc91c-bc2f-42db-91de-20da98d73...@b1g2000vbb.googlegroups.com>,
>
>
>
>
>
>  WM <mueck...@rz.fh-augsburg.de> wrote:

> > On 21 Mai, 18:46, torb...@diku.dk (Torben gidius Mogensen) wrote:
> > > Graham Cooper <grahamcoop...@gmail.com> writes:
> > > > How many ways can you re-order <f, a, t> ?
> > > > <f a t>
> > > > <f t a>
> > > > <a f t>
> > > > <a t f>
> > > > <t f a>
> > > > <t a f>
>
> > > > 3! = 3 Factorial = 3X2X1 = 6
>
> > > > How many different ways can you order <1,2,3,4....> ?
>
> > > > oo X oo-1 X oo-2 X .... X 3 X 2 X 1
> > > > It's not well defined!
>
> > > Not if you write it like this, but the number of permutations of a set
> > > is well defined even for infinite sets -- just like the number of all
> > > subsets of a set is defined even for infinite sets.
>
> > Take a well-ordered set Q of all rational numbers (simply an
> > enumeration). Does the well-ordered permutation exist, that well-
> > orders Q by magnitude of its elements?
>
> Obviously not, since ordering by magnitude does not provide a well
> ordering of Q.
>
> It is not even a total ordering of Q until you determine which of q and
> -q is to have the greater magnitude.

That's right. So reduce the set such that Q contains only the positive
rationals.

>
> > If not, is this permutation
> > taken into account, nevertheless, when counting all permutations of Q?
>
> Since the ordering as described is not an ordering, no such

> "permutation" exists, not even to be counted.-

A well-ordering of the naturals can be the same as the usual order by
magnitude.
The positive rationals have not yet been well-ordered by magnitude.
But if all permutations exist, then this one should also exist, no?
Why not?

Regards, WM

[WM]

On 22 Mai, 16:34, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> Say 2/3 and 3/4 were adjacent, then it's trivial to find a rational
> inbetween.

Of course. You are right. My point is that there is no well-ordering
of all rational numbers at all. If such a well-ordering existed, for
instance the Cantor-enumeration, then this could be reordered by
aleph_0 transpositions such that a well-ordering and simultaneously an
ordering by magnitude was accomplished. Contradiction.

Regards, WM

[Don Stockbauer]

1. Potential

2. Actual

[Torben Ægidius Mogensen]

Graham Cooper <graham...@gmail.com> writes:

> On May 22, 7:53 pm, torb...@diku.dk (Torben gidius Mogensen) wrote:
>> Graham Cooper <grahamcoop...@gmail.com> writes:

>> > You must have ASSUMED the amount of subsets of N is 2X2X2... = |R|
>>
>> Not at all.  This is, in fact, fairly easy to prove: We use the theorem
>> that there exists a bijection between two sets A and B if there exists
>> an injection from A to B and an injection from B to A.  So all we need
>> to do to prove that |P(N)| = |R| is to find injections both ways.
>>

>> [...]

>>
>> Since we have injections both ways, the sets are equipotent (i.e.,
>> having the same size).
>

> Yes Yes I know all that, you threw a red herring that there is a
> bijection between permutations and powersets,

Whay is that a red herring?

> That's like saying there's a bijection between {a,b,c} and {a,b,c,d,e}

There isn't. Do you even know what a bijection is?

Torben

[Graham Cooper]

You empty your pockets 1 item for every item in mine?

Herc

[Don Stockbauer]

That's like when you have 2 diseases and you have to get two shots for
them??????

[Uirgil]

In article
<430e6d7f-a73f-48fd...@eh4g2000vbb.googlegroups.com>,

But it is the rationals ,not the naturals, that are under consideration.

> The positive rationals have not yet been well-ordered by magnitude.
> But if all permutations exist, then this one should also exist, no?

NO!

> Why not?

A permutation of the rationals does not have any effect on their
magnitudes, so that their dense ordering after such a permutation is
still not well ordering.

While it is possible to define an ordering of the rationals that is a
well-orderingof the rationals, a "permutation" of them doesn't do it.

[WM]

On 24 Mai, 08:17, Uirgil <uir...@uirgil.ur> wrote:
> In article

> <430e6d7f-a73f-48fd-808c-95280b4c0...@eh4g2000vbb.googlegroups.com>,

Take any well ordering of the rationals:
q_1, q_2, q_3, q_4, ...
Subdivide the set into pairs and re-order every pair such that the
smaller one comes first:
(q_1', q_2'), (q_3', q_4'), ...
Then subdivide the set into pairs again, leaving out q_1', and re-
order every pair such that the smaller one comes first:
q_1', (q_2'', q_3''), (q_4'', ...
Then subdivide the set into pairs again, including q_1', and re-order
every pair such that the smaller one comes first - and continue in
this way. After aleph_0 transpositions you get a well-order which is
simultaneously an order by magnitude.

No convergence? In set theory there is no convergence (for example
when enumerating the rationals). Nevertheless the limit (aleph_0
rationals) exists. Same holds for my example.

There are uncountably many transpositions of the rationals. Should
just this simple single one be missing???

Regards, WM

[Graham Cooper]

Could be! If SCI.MATH wants to open a Casino with $aleph_1 in the
bank, I'll double my bets on the Casino table all night using my
modest $1+1+1..

Herc

TM-SIZE MAX-1s OUTPUT
-------------------------

BB(2) 6
BB(3) 38
BB(4) 3,932,964
BB(5) 1.7 x 10^352
BB(6) 1.9 x 10^4933
...

BB(199) COOPERS NUMBER
BB(200) UNIVERSAL TURING MACHINE SIZE
includes PorkyPig Jnr's Number = CN+1

[Alan Smaill]

Write the proof out properly, you become famous, maths moves on.

You would need some axioms, though ...


> Regards, WM

--
Alan Smaill

[WM]

On 24 Mai, 16:18, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:

> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 22 Mai, 16:34, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> >> Say 2/3 and 3/4 were adjacent, then it's trivial to find a rational
> >> inbetween.
>
> > Of course. You are right. My point is that there is no well-ordering
> > of all rational numbers at all. If such a well-ordering existed, for
> > instance the Cantor-enumeration, then this could be reordered by
> > aleph_0 transpositions such that a well-ordering and simultaneously an
> > ordering by magnitude was accomplished. Contradiction.
>
> Write the proof out properly, you become famous, maths moves on.

Please don't confuse proper and ZF.

Here you see a proper proof showing that Cantor's way of counting is
wrong.

Take any well ordering of the rationals:
q_1, q_2, q_3, q_4, ...
Subdivide the set into pairs and re-order every pair such that the
smaller one comes first:
(q_1', q_2'), (q_3', q_4'), ...
Then subdivide the set into pairs again, leaving out q_1', and re-
order every pair such that the smaller one comes first:
q_1', (q_2'', q_3''), (q_4'', ...
Then subdivide the set into pairs again, including q_1', and re-order
every pair such that the smaller one comes first - and continue in
this way. After aleph_0 transpositions you get a well-order which is
simultaneously an order by magnitude.

No convergence? In set theory there is no convergence (for example
when enumerating the rationals). Nevertheless the limit (aleph_0
rationals) exists. Same holds for my example.

And here you see why this way is wrong:

It is astonishing that mathematicians are satisfied with Cantor's
"enumeration" of the rational numbers. If you count an infinite set
like Q you never count a share of more than lim{n-->oo} 2^-n = 0
because every natural number is followed by infinitely many others.
How can you obtain anything sensible from the result? ("All natural
numbers" means all elements of a set of which you cannot have all
elements. Infinite means never, not "arrived at after all".)
Same is true for infinite "words". They cannot be used for
communicating information. Therefore the following reversion of
implication, usually applied in set theory, is wrong:
A finite formula defines an infinite sequence. <==> An infinite
sequence defines a finite formula.
We can never obtain a finite formula like 1/9 or sqrt2 from an
infinite sequence unless we know the last term (which is impossible by
definition). An infinite sequence (like Cantor's diagonal) never
defines a number. Up to every digit it defines an interval out of
countably many. In order to know the limit, you need information about
the infinitely many digits to follow. That requires a finite formula.


Regards, WM

[Curt Welch]

WM <muec...@rz.fh-augsburg.de> wrote:
> On 24 Mai, 08:17, Uirgil <uir...@uirgil.ur> wrote:
> > In article
> > <430e6d7f-a73f-48fd-808c-95280b4c0...@eh4g2000vbb.googlegroups.com>,
> >
> >
> >
> >
> >
> > =A0WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 22 Mai, 22:43, Uirgil <uir...@uirgil.ur> wrote:
> > > > In article
> > > > <ee9bc91c-bc2f-42db-91de-20da98d73...@b1g2000vbb.googlegroups.com>,
> >
> > > > =A0WM <mueck...@rz.fh-augsburg.de> wrote:
> > > > > On 21 Mai, 18:46, torb...@diku.dk (Torben gidius Mogensen) wrote:
> > > > > > Graham Cooper <grahamcoop...@gmail.com> writes:
> > > > > > > How many ways can you re-order <f, a, t> ?
> > > > > > > <f a t>
> > > > > > > <f t a>
> > > > > > > <a f t>
> > > > > > > <a t f>
> > > > > > > <t f a>
> > > > > > > <t a f>
> >

> > > > > > > 3! =3D 3 Factorial =3D 3X2X1 =3D 6

> >
> > > > > > > How many different ways can you order <1,2,3,4....> ?
> >
> > > > > > > oo X oo-1 X oo-2 X .... X 3 X 2 X 1
> > > > > > > It's not well defined!
> >
> > > > > > Not if you write it like this, but the number of permutations

> > > > > > of =

> a set
> > > > > > is well defined even for infinite sets -- just like the number

> > > > > > of=

> all
> > > > > > subsets of a set is defined even for infinite sets.
> >
> > > > > Take a well-ordered set Q of all rational numbers (simply an
> > > > > enumeration). Does the well-ordered permutation exist, that well-
> > > > > orders Q by magnitude of its elements?
> >
> > > > Obviously not, since ordering by magnitude does not provide a well
> > > > ordering of Q.
> >
> > > > It is not even a total ordering of Q until you determine which of q

> > > > a=

> nd
> > > > -q is to have the greater magnitude.
> >
> > > That's right. So reduce the set such that Q contains only the
> > > positive rationals.
> >
> > > > > If not, is this permutation
> > > > > taken into account, nevertheless, when counting all permutations

> > > > > of=

> Q?
> >
> > > > Since the ordering as described is not an ordering, no such
> > > > "permutation" exists, not even to be counted.-
> >
> > > A well-ordering of the naturals can be the same as the usual order by
> > > magnitude.
> >
> > But it is the rationals ,not the naturals, that are under
> > consideration.
> >
> > > The positive rationals have not yet been well-ordered by magnitude.
> > > But if all permutations exist, then this one should also exist, no?
> >
> > NO!
> >
> > > Why not?
> >
> > A permutation of the rationals does not have any effect on their
> > magnitudes, so that their dense ordering after such a permutation is

> > still not =A0well ordering.

>
> Take any well ordering of the rationals:
> q_1, q_2, q_3, q_4, ...
> Subdivide the set into pairs and re-order every pair such that the
> smaller one comes first:
> (q_1', q_2'), (q_3', q_4'), ...
> Then subdivide the set into pairs again, leaving out q_1', and re-
> order every pair such that the smaller one comes first:
> q_1', (q_2'', q_3''), (q_4'', ...
> Then subdivide the set into pairs again, including q_1', and re-order
> every pair such that the smaller one comes first - and continue in
> this way. After aleph_0 transpositions you get a well-order which is
> simultaneously an order by magnitude.

Why is it valid to say "After aleph_0 transpositions"? How can there be an
"after" of a process that by definition never terminates?

> No convergence? In set theory there is no convergence (for example
> when enumerating the rationals). Nevertheless the limit (aleph_0
> rationals) exists.

What exactly "exists" that that sentence is referring to?

> Same holds for my example.
>
> There are uncountably many transpositions of the rationals. Should
> just this simple single one be missing???
>
> Regards, WM

--
Curt Welch http://CurtWelch.Com/
cu...@kcwc.com http://NewsReader.Com/

[WM]

On 24 Mai, 16:37, c...@kcwc.com (Curt Welch) wrote:
>
> > Take any well ordering of the rationals:
> > q_1, q_2, q_3, q_4, ...
> > Subdivide the set into pairs and re-order every pair such that the
> > smaller one comes first:
> > (q_1', q_2'), (q_3', q_4'), ...
> > Then subdivide the set into pairs again, leaving out q_1', and re-
> > order every pair such that the smaller one comes first:
> > q_1', (q_2'', q_3''), (q_4'', ...
> > Then subdivide the set into pairs again, including q_1', and re-order
> > every pair such that the smaller one comes first - and continue in
> > this way. After aleph_0 transpositions you get a well-order which is
> > simultaneously an order by magnitude.
>
> Why is it valid to say "After aleph_0 transpositions"?  How can there be an
> "after" of a process that by definition never terminates?

This has been introduced by Cantor: omega is followed by omega + 1.
And aleph_0 you only the beginning of the hierarchy of infinities.
Cantor's set theory needs the completed infinity (Cantor's own words)
because the usual infinite is only potential, i.e., there is no end.
But in that case there is also no complete Cantor-list and no means to
decide whether or not an entry is contained therein.

>
> > No convergence? In set theory there is no convergence (for example
> > when enumerating the rationals). Nevertheless the limit (aleph_0
> > rationals) exists.
>
> What exactly "exists" that that sentence is referring to?

The complete enumeration of all rational numbers, such that no
rational number is missing.

In fact, Cantor has only proven that for every rational number there
is a natural number indexing the former. But then he has turned this
into the meaning of indexing "all" rationals.

Only by this reversion he could insist that an irrational number is
tantamount to its infinite sequence of digits, and, another reversion,
that an infinite sequence of digits can define a number. The latter
claim is false. There is no infinite definition defining anything.

Regards, WM

[Curt Welch]

WM <muec...@rz.fh-augsburg.de> wrote:
> On 24 Mai, 16:37, c...@kcwc.com (Curt Welch) wrote:
> >
> > > Take any well ordering of the rationals:
> > > q_1, q_2, q_3, q_4, ...
> > > Subdivide the set into pairs and re-order every pair such that the
> > > smaller one comes first:
> > > (q_1', q_2'), (q_3', q_4'), ...
> > > Then subdivide the set into pairs again, leaving out q_1', and re-
> > > order every pair such that the smaller one comes first:
> > > q_1', (q_2'', q_3''), (q_4'', ...
> > > Then subdivide the set into pairs again, including q_1', and re-order
> > > every pair such that the smaller one comes first - and continue in
> > > this way. After aleph_0 transpositions you get a well-order which is
> > > simultaneously an order by magnitude.
> >

> > Why is it valid to say "After aleph_0 transpositions"? =A0How can there
> > b=

> e an
> > "after" of a process that by definition never terminates?
>
> This has been introduced by Cantor: omega is followed by omega + 1.
> And aleph_0 you only the beginning of the hierarchy of infinities.
> Cantor's set theory needs the completed infinity (Cantor's own words)
> because the usual infinite is only potential, i.e., there is no end.
> But in that case there is also no complete Cantor-list and no means to
> decide whether or not an entry is contained therein.

Ok. So by what logic does he justify "adding the completed infinity"?

How does it become valid to define something that can't exist, as existing?

By declaring "infinity exists" as an axiom, it seems to me one would be
introducing a contradiction into their set of axioms that would only lead
to problems - like a nonsensical hierarchy of infinities to start with, but
many more I'm sure show up as well.

Has there been found any application in this universe where the act of
pretending infinity exists becomes useful?

> > > No convergence? In set theory there is no convergence (for example
> > > when enumerating the rationals). Nevertheless the limit (aleph_0
> > > rationals) exists.
> >
> > What exactly "exists" that that sentence is referring to?
>
> The complete enumeration of all rational numbers, such that no
> rational number is missing.

But that can't exist. What is the advantage of pretending it can when it
clearly can't?

> In fact, Cantor has only proven that for every rational number there
> is a natural number indexing the former. But then he has turned this
> into the meaning of indexing "all" rationals.

Right. We can define a process that can index any given rational. However,
the list of all rationals and their indexes can never exist.

> Only by this reversion he could insist that an irrational number is
> tantamount to its infinite sequence of digits, and, another reversion,
> that an infinite sequence of digits can define a number. The latter
> claim is false. There is no infinite definition defining anything.

Right. But there are processes that never end, and we can use any of these
processes as a definition. Such as the process that produces the digits of
pi. We could call the digits produced by that process the definition of
pi. The infinite list of digits of pi can never exist, but the process
can, and we can call the process the definition of the infinite list of the
digits of pi. One must just be careful not to make the stake of leaping
from the existence of the process, to the existence of all the digits
however. It seems to me that this mistake is commonly made in mathematics
- often (as far as I can tell) with no awareness that the mistake is being
made.

[Graham Cooper]

On May 25, 4:33 am, c...@kcwc.com (Curt Welch) wrote:

You mean Irrationals. Rationals have a finite formula.

1/1
1/2
2/1
1/3
2/2
3/1
1/4
2/3
3/4
4/4

Formulas, functions, predicates, .. exist using the existential
quantifier.

Also PROOF(Formula) either Exist or Not in Theories.

By using a SET SPECIFICATION AXIOM, Set's are inferred to exist.

X e Y <-> EXISTS(function-n) function-n(X)

where function-n is some lexicographical index of the Theory.

For arguments sake you could call this SYNTAX the point at which the
SET EXISTS, not just the function or 'proof of existence'.

Y = { x | fn(x) }

since fn exists, so does Y, by the SPECIFICATION AXIOM.

Herc

--
On May 21, 9:16 am, Virgil <vir...@ligriv.com> wrote:
Actually I believe that there are more that aleph_0 irrationals
in any real interval of positive length. |R|>oo

On May 17, 6:13 am, Virgil <vir...@ligriv.com> wrote:
RE:"There are more Reals than Naturals." |R|>oo
If that means that there is a surjection from the reals to the
naturals but no surjection from the naturals to the reals ,Yes!

[WM]

On 24 Mai, 20:33, c...@kcwc.com (Curt Welch) wrote:

> > This has been introduced by Cantor: omega is followed by omega + 1.
> > And aleph_0 you only the beginning of the hierarchy of infinities.
> > Cantor's set theory needs the completed infinity (Cantor's own words)
> > because the usual infinite is only potential, i.e., there is no end.
> > But in that case there is also no complete Cantor-list and no means to
> > decide whether or not an entry is contained therein.
>
> Ok.  So by what logic does he justify "adding the completed infinity"?

Not by logic. That is impossible. First he found that Augustinus
claimed God knew all numbers.

> How does it become valid to define something that can't exist, as existing?

Set theorists introduced an axiom saying that there exists an infinite
set. This means in set theory; every element exists.

>

>
> By declaring "infinity exists" as an axiom, it seems to me one would be
> introducing a contradiction into their set of axioms that would only lead
> to problems - like a nonsensical hierarchy of infinities to start with, but
> many more I'm sure show up as well.

Indeed, that is the case.

>
> Has there been found any application in this universe where the act of
> pretending infinity exists becomes useful?

No a single one. (In fact some have been claimed. I have listed them
in
http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-.pdf
Nos 1053 - 1060, partially in English. But all that is rubbish.)

The only "application": Set theory is claimed to be necessary in order
to put mathematics on a solid basis. That's even greater rubbish.

>
> > > > No convergence? In set theory there is no convergence (for example
> > > > when enumerating the rationals). Nevertheless the limit (aleph_0
> > > > rationals) exists.
>
> > > What exactly "exists" that that sentence is referring to?
>
> > The complete enumeration of all rational numbers, such that no
> > rational number is missing.
>
> But that can't exist.  What is the advantage of pretending it can when it
> clearly can't?

Honestly, I don't know.

>
> > In fact, Cantor has only proven that for every rational number there
> > is a natural number indexing the former. But then he has turned this
> > into the meaning of indexing "all" rationals.
>
> Right. We can define a process that can index any given rational.  However,
> the list of all rationals and their indexes can never exist.
>
> > Only by this reversion he could insist that an irrational number is
> > tantamount to its infinite sequence of digits, and, another reversion,
> > that an infinite sequence of digits can define a number. The latter
> > claim is false. There is no infinite definition defining anything.
>
> Right.  But there are processes that never end, and we can use any of these
> processes as a definition.  Such as the process that produces the digits of
> pi.  We could call the digits produced by that process the definition of
> pi.  The infinite list of digits of pi can never exist, but the process
> can, and we can call the process the definition of the infinite list of the
> digits of pi.

You have an absolute correct view of the things.

>  One must just be careful not to make the stake of leaping
> from the existence of the process, to the existence of all the digits
> however.  It seems to me that this mistake is commonly made in mathematics
> - often (as far as I can tell) with no awareness that the mistake is being
> made.

So it is.

Regards, WM

[Shmuel Metz]

In <20120524143314.779$a...@newsreader.com>, on 05/24/2012

at 06:33 PM, cu...@kcwc.com (Curt Welch) said:

>By declaring "infinity exists" as an axiom, it seems to me one
>would be introducing a contradiction into their set of axioms

Mathematics is not about how it seems to you. Produce a contradiction
if you can; until then no rational person will take your belief
seriously.

>nonsensical hierarchy of infinities

The fact that you don't like it doesn't make it nonsensical.

>Has there been found any application in this universe where the
>act of pretending infinity exists becomes useful?

Has there been found any application in this universe where the act of

pretending numbers exist becomes useful? Like numbers, cardinals are
abstractions. Theories using those abstractions have found practical
use.

>But that can't exist.

Asserting that they can't exist doesn't mean that they don't.

>But there are processes that never end, and we can use any of
>these processes as a definition.

Mathematics is not founded on a concept of processes.

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org

[WM]

On 24 Mai, 22:35, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <20120524143314.779...@newsreader.com>, on 05/24/2012

>    at 06:33 PM, c...@kcwc.com (Curt Welch) said:
>
> >By declaring "infinity exists" as an axiom, it seems to me one
> >would be introducing a contradiction into their set of axioms
>
> Mathematics is not about how it seems to you. Produce a contradiction
> if you can; until then no rational person will take your belief
> seriously.

The most prominent contradiction is the claim that numbers, that
cannot be determined, can be well-ordered. The set of elements that
can be well-ordered is a subset of the set of elements that can be
identified.

>
> >nonsensical hierarchy of infinities
>
> The fact that you don't like it doesn't make it nonsensical.

I don't dislike it. I have proven a contradiction with mathematics.

This is it: Cover every rational q_n of the real axis (-oo, oo) with
an interval I_n of measure 10^-n. All remaining irrationals then must
be separated from each other by at least one interval I_n. So the
irrationals can only occupy the borders of intervals. That is a well-
ordered and even countable set. Contradiction, because there should
uncountably many irrationals be uncovered.

>
> >Has there been found any application in this universe where the
> >act of pretending infinity exists becomes useful?
>
> Has there been found any application in this universe where the act of
> pretending numbers exist becomes useful? Like numbers, cardinals are
> abstractions. Theories using those abstractions have found practical
> use.
>
> >But that can't exist.
>
> Asserting that they can't exist doesn't mean that they don't.

It can't exists since finished infinity does not exist in reality.

>
> >But there are processes that never end, and we can use any of
> >these processes as a definition.
>
> Mathematics is not founded on a concept of processes.

You are in error. Compare Newton, Euler and even Cantor: Enumerating
some set is a process. Don't confuse mathematics with set theory,
please.

Regards, WM

[Uirgil]

In article
<bf5a91a9-a32d-4c4a...@eh4g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 22 Mai, 16:34, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > Say 2/3 and 3/4 were adjacent, then it's trivial to find a rational
> > inbetween.
>
> Of course. You are right. My point is that there is no well-ordering
> of all rational numbers at all.

To claim that the rationals cannot be well ordered when it has so
frequently been done, including by mathematicians of considerable
greater talent than WM, is the height of arrogance by an incompetent.

> If such a well-ordering existed, for
> instance the Cantor-enumeration, then this could be reordered by
> aleph_0 transpositions such that a well-ordering and simultaneously an
> ordering by magnitude was accomplished. Contradiction.

Claimed but, as usual, not proven, and, as usual, unprovable.

Note that a transposition of element positions has no effect on the
order type of an ordered set.

[Graham Cooper]

On May 25, 6:56 am, Uirgil <uir...@uirgil.ur> wrote:
> In article

> <bf5a91a9-a32d-4c4a-a18a-476319409...@eh4g2000vbb.googlegroups.com>,

>
>  WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 22 Mai, 16:34, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > Say 2/3 and 3/4 were adjacent, then it's trivial to find a rational
> > > inbetween.
>
> > Of course. You are right. My point is that there is no well-ordering
> > of all rational numbers at all.
>
> To claim that the rationals cannot be well ordered when it has so
> frequently been done, including by mathematicians of considerable
> greater talent than WM, is the height of arrogance by an incompetent.
>

You can prove anything with a contradiction in your system.


SCI.MATH |- ->oo
SCI.MATH |- oo->


Once you say |X|>oo is a fact
Then of course there will be an explosion of Mutually-Assured-Formulas
infecting all the ordinary logical truths.

[Uirgil]

In article
<af76b6df-6b5b-4a4f...@ec4g2000vbb.googlegroups.com>,

Not so! If it were true then WM should be able to tell us what the first
member of that final ordering must be, since it must be less that every
other rational.

Each of WM's reorderings only exists after some last permutation of
elements of Q, so without a LAST permutation for it to follow, your
alleged ordering does not exist!

>
> There are uncountably many transpositions of the rationals. Should
> just this simple single one be missing???

Yes!

[Uirgil]

In article
<66417374-027a-4749...@m3g2000vbl.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 24 Mai, 16:37, c...@kcwc.com (Curt Welch) wrote:
> >
> > > Take any well ordering of the rationals:
> > > q_1, q_2, q_3, q_4, ...
> > > Subdivide the set into pairs and re-order every pair such that the
> > > smaller one comes first:
> > > (q_1', q_2'), (q_3', q_4'), ...
> > > Then subdivide the set into pairs again, leaving out q_1', and re-
> > > order every pair such that the smaller one comes first:
> > > q_1', (q_2'', q_3''), (q_4'', ...
> > > Then subdivide the set into pairs again, including q_1', and re-order
> > > every pair such that the smaller one comes first - and continue in
> > > this way. After aleph_0 transpositions you get a well-order which is
> > > simultaneously an order by magnitude.
> >
> > Why is it valid to say "After aleph_0 transpositions"?  How can there be an
> > "after" of a process that by definition never terminates?
>
> This has been introduced by Cantor: omega is followed by omega + 1.
> And aleph_0 you only the beginning of the hierarchy of infinities.
> Cantor's set theory needs the completed infinity (Cantor's own words)
> because the usual infinite is only potential, i.e., there is no end.
> But in that case there is also no complete Cantor-list and no means to
> decide whether or not an entry is contained therein.

Cantor's Aleph_0 is incompete only in the sense that it does not have a
last member but is complete in the sense of having all of its members.

> >
> > > No convergence? In set theory there is no convergence (for example
> > > when enumerating the rationals). Nevertheless the limit (aleph_0
> > > rationals) exists.

How can it 'exist' until it has produced a rational that is smaller than
every other rational. Clearly until the process has produced such a
minimal rational it cannot have reached its limit.

> >
> > What exactly "exists" that that sentence is referring to?
>
> The complete enumeration of all rational numbers, such that no
> rational number is missing.

And at the same time, WM claims that in the "limit" it will have
produced a rational smaller than any other rational.

>
> In fact, Cantor has only proven that for every rational number there
> is a natural number indexing the former. But then he has turned this
> into the meaning of indexing "all" rationals.
>
> Only by this reversion he could insist that an irrational number is
> tantamount to its infinite sequence of digits, and, another reversion,
> that an infinite sequence of digits can define a number. The latter
> claim is false. There is no infinite definition defining anything.

Then in WM's world, 0.999(9) =/= 1. And he joins the nuts!

[Graham Cooper]

On May 25, 6:42 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 24 Mai, 22:35, Shmuel (Seymour J.) Metz
>
> <spamt...@library.lspace.org.invalid> wrote:
> > In <20120524143314.779...@newsreader.com>, on 05/24/2012
> >    at 06:33 PM, c...@kcwc.com (Curt Welch) said:
>
> > >By declaring "infinity exists" as an axiom, it seems to me one
> > >would be introducing a contradiction into their set of axioms
>
> > Mathematics is not about how it seems to you. Produce a contradiction
> > if you can; until then no rational person will take your belief
> > seriously.

Pi/10, _/2/10, e/10, ... are ALL missing from my LIST OF REALS!

REAL1 = 0.0123456789...
REAL2 = 0.9876541210
REAL3 = 0.2469097531
REAL4 = 0.1357889426
REAL5 = 0.3012674859
REAL6 = 0.4201365578
REAL7 = 0.8940123567
REAL8 = 0.7394210622
REAL9 = 0.5732802095
REAL10 = 0.6585331103
REAL11 = 0.0679768344
...



AD(pos) = changedigit(LIST(n2n(pos),pos))

3141592653.. is missing
1414213562.. is missing
2718281828.. is missing

n2n(1) = 3 (Row 3)
n2n(2) = 5 (Row 5)
..

0123 [4] 56789...
987654 [1] 210
[2] 469097531
13578 [8] 9426
3 [0] 12674859
4201365 [5] 78
894 [0] 123567
739421062 [2]
57 [3] 2802095
6585331103
06797683 [4] 4
...

the possible diagonal [2][0][3][0][4][8][1][5][4][2]...
gives AD 3141592653...

PI IS MISSING!

EXTRAPOLATE THAT LIST TO oo WIDE AND oo LONG
AND PI IS STILL MISSING!

>
> The most prominent contradiction is the claim that numbers, that
> cannot be determined, can be well-ordered. The set of elements that
> can be well-ordered is a subset of the set of elements that can be
> identified.
>
>
>
> > >nonsensical hierarchy of infinities
>
> > The fact that you don't like it doesn't make it nonsensical.
>
> I don't dislike it. I have proven a contradiction with mathematics.
>
> This is it: Cover every rational q_n of the real axis (-oo, oo) with
> an interval I_n of measure 10^-n. All remaining irrationals then must
> be separated from each other by at least one interval I_n. So the
> irrationals can only occupy the borders of intervals. That is a well-
> ordered and even countable set. Contradiction, because there should
> uncountably many irrationals be uncovered.
>
>

That segments the number line into a total order.

Although it holds for all n, q_n

the LIMIT at n->oo is how reals are defined (by SCI.MATH)

i.e. a real is the convergence of an infinite sequence of rationals.

3
3.1
3.14
...


Herc

[Uirgil]

In article
<a47ff2f6-2172-4076...@s5g2000vbc.googlegroups.com>,

If there were any "convergence" to this process, it would necessarily
produce a rational that was strictly smaller than every other rational,
which should be a hint that the process does not ever produce a limit.

Also, each new ordering can only exist after a particular finite
sequence of reordering of some pairs. i.e. after a finite number of
reordering operations, so each ordering requires an immediately previous
last reordering, but with aleph_0 of them ther is no last one, thus no
limit.

As usual. WM's inability to think things through as deeply as
mathematics sometimes requires requires has betrayed him.

[Curt Welch]

Shmuel (Seymour J.) Metz <spam...@library.lspace.org.invalid> wrote:
> In <20120524143314.779$a...@newsreader.com>, on 05/24/2012
> at 06:33 PM, cu...@kcwc.com (Curt Welch) said:
>
> >By declaring "infinity exists" as an axiom, it seems to me one
> >would be introducing a contradiction into their set of axioms
>
> Mathematics is not about how it seems to you. Produce a contradiction
> if you can; until then no rational person will take your belief
> seriously.
>
> >nonsensical hierarchy of infinities
>
> The fact that you don't like it doesn't make it nonsensical.
>
> >Has there been found any application in this universe where the
> >act of pretending infinity exists becomes useful?
>
> Has there been found any application in this universe where the act of
> pretending numbers exist becomes useful?

There is no need to pretend a number exists. Counts of objects do exist.
No pretending is needed.

> Like numbers, cardinals are
> abstractions.

Abstractions exist. No pretending needed. To suggest they don't is to
make an error of duality - the error that the thoughts aren't physical -
that they don't have physical existence. An abstraction is just the
behavior of hardware that can recognize a class of sensory patterns. This
hardware exists both in the human brain, and in our machines. When we talk
about an abstraction, say the abstraction of a circle, we are talking about
some for of pattern recognizing hardware that can identify a circle.

In mathematics, we advance from the simple abstraction of a circle being a
shape, to some formal language based definition, such as "the set of all
points that lie on a plan which are equal distant from the the center point
of the circle". But even when we do and moved away from a visual pattern
of circle, we are still talking about pattern matching hardware that can
correctly recognize a formal language definition of "circle". Either way,
the abstraction is just a behavior of a pattern matching machine - which
must be real and physical to "exist".

If we removed all humans from the earth, and there were no machines left
that could recognize the "circle" pattern, then we would have removed all
forms of the "circle" abstraction from the planet. The circle abstraction
at that point would not exist on the planet - (even though many circles
would still exist). This is because there's a clear distinction between
the circles and the circle abstractions.

> Theories using those abstractions have found practical
> use.

Yes, machines which can count, have found endless practical application.

That was not my question. My question was whether anyone has yet found a
practical application for pretending machines that can finish counting to
infinity can "exist" has any practical application.

> >But that can't exist.
>
> Asserting that they can't exist doesn't mean that they don't.

That's right. Show my an example of one existing please.

Humans operate on probabilities. Nothing for us is certain. There are no
absolutes. All we have, are events which are highly likely, and highly
unlikely.

It's highly unlikely that gravity will work differently starting tomorrow
for example. But it's not an absolute fact that it will work the same
tomorrow.

What we do however, is _pretend_ such high probability events are
absolutes, because they are so close to being absolutes, it would be a
waste of time and computational energy in most applications, to do anything
other than plan our future based on the assumption gravity will keep
working the same.

In the language of logic, we pretend absolutes can exist. We pretend
"true" and "false" are valid absolute axioms. We pretend they exist as
axioms, and we pretend the results of our logical analyse is likewise, an
absolute that grows from the axioms. Doing this is highly useful when
working with ideas that are extremely close to being absolutes - like the
idea of gravity working the same say for us tomorrow as we believe it has
for the entire history of humans science. Doing this type of pretending
allows us to greatly simply problems based on events close to absolutes.
It's practically very useful to us, even though it's a lie.

We don't need to prove a negative. We assume a negative is false, until
there is data to show us otherwise. This is a practical fact of how we are
forced to operate.

There could be troll under my desk right now. But I don't need to prove
the suggest false. I just choose to act that it's false, based on the fact
I have no data to suggest it could be true. Just like the idea that
gravity is highly likely to work the same way tomorrow, is not an absolute,
the idea that there is no troll under my desk is also not an absolute. He
could be there.

But it's valid for me to _ACT_ as if it was an absolute fact that he is not
there. All things we consider to be absolutes are in fact not. We only
act as if they are beause they are so close to being absolute truths, or
absolute non-truths, that it is completely unproductive to act as if they
were not absolute.

So, when I write, as I did above:

> > But that can't exist.

What does that mean? It means that the existence of a counting machine
that can finishing counting to infinity, is so unlikely, based on all the
data we have access to, that it would be a waste of our time to act is if
it were not an absolute.

I need to pull any punches and covers my ass by saying, "we don't know if
they could exist". It's 100% valid to just say, "they can't exist" due to
the fact there is no evidence to suggest they can.

So, in order for you to show my behavior invalid, you need to provide the
evidence (or at least some small hint of a suggestion about how and where
it might be hidden from us), that such a machine can exist in this
universe. You have to provide evidence strong enough to suggest we should
no long assume this is an absolute.

If you can not provide such evidence, then the correct beahvior, is to say,
"it can't exist" (or a little more toned down, "I doesn't exist".

> >But there are processes that never end, and we can use any of
> >these processes as a definition.
>
> Mathematics is not founded on a concept of processes.

Then what it is founded on?

If it exists, it MUST be founded on something. Mathematics certainly
exists, so tell me, what is it founded on?

The problem here, is that people use their brain to create, and to perform,
mathematics. But for most the history of mathematics, they had no clue
what their brain was, or what it was doing. The machine they used, to do
all their work, was a mystery to them. They had no clue how it worked.
They had no clue what "it" was. They didn't even know it was their brain
doing the work. Many thought it was a magical "soul" given to them by some
"God" which likewise, had one of these magical "souls" (but a better one).

But the idea of trying to figure out what mathematics was founded on was an
interest to many - so they tried to "guess". They started making crap up
to fill in the unknown to see if they discover, or create, the axioms of
mathematics. Nothing wrong with trying to do that.

But, what they seemed to have overlooked, was the fact that they don't get
to just "make up" the foundations of mathematics. The foundations of
mathematics, comes from the machine that does the mathematics. And the
foundations of that machine, come from the foundations of the universe it
exists in.

So, if you want to know the foundations of mathematics, you must uncover,
and use, the true foundations of the machine that will be doing the
mathematics.

Mathematics, can't "exist" if there is no machine that can do the work.

We are free to speculate about a different foundation, but if we choose a
foundation which is impossible for us to build a machine to follow, then we
will be hard pressed to do much with that foundation, since it becomes
impossible to build, or test, machines that can do mathematics according to
this "impossible" foundation.

What strikes me as highly problematic about the current mathematics, is
that most of what I'm saying seems to be still be ignored by the field (as
far as I can see). They ignore it, because they are playing the old
(totally invalid) game of "thoughts are not physical, and have no physical
limitations". That's just not true. Thoughts have the same limitations as
our computers. They can't, for example, count to infinity.

They also seem to make endless map territory mistakes. Words are "maps"
which make reference to some "territory". Some territories can exist, and
some simply can not. But we can still draw maps for things that can't
exist. We can talk about things that can't exist in our universe. Talking
about them, however, does not make them true or possible.

When we talk about numbers, as I talked about above, we are talking about
machines that can count - or count objects detected as patterns in sensory
data. The "abstraction" of a number, is real hardware, it's not "a
non-physical thought". Using language, we can very easily talk about a
machine that can finish counting to infinity. But in doing that, we have
created a "real" map, (those words), about a "territory" which does not,
and can not, exist, in this u universe.

We step over a very significant line once we start to draw maps, of
territories that can't exist. But mathematicians, seem to be unaware of
this line. They act as if it's not there at all - that it has nothing to
do with their work. And that's where everything falls apart for them.

They then act as if, the territory was as "real" as the "map". They act
that anything they can draw a map for (aka talk about), "exists" as much as
the "map" does. They mistake the existence of the map, with the existence
of the territory. They act that since they can say "the infinite set
exists", then that's all they need to do to prove that "it does exist".

But them then use concepts about what their real machine can do, and
pretend that these operations are valid for this territory which can't
exist. And once they do that, they have wondered into a land of shear and
utter nonsense that as far as I can tell, has no value to anyone, or
anything.

It's as useful as starting a project by saying something like, "Assume all
elephants can fly - but they have just never done it for a human to see",
now lets figure out what the world is really like if this were true. And
then they go off exploring the possibilities of what the world is like,
based on the idea that all elephants can fly. They have to fill in all
sorts of gaps about how the elephants can fly, even though there is nothing
in their physiology to justify the ability, etc, etc. We could spend 1000
years, and give endless people PhDs on flying elephant speculation
research. But what would it's value be? None that I can see.

I see all this work in mathematics about hierarchies of infinity being just
as pointless - a total and utter waste of time because it's exploring a
world that does not, and can not, exist for us. It's just science fiction.

The rest of mathematics is not science fiction. It's very real - it's the
world of what computers (and brains) can do.

What frustrates me, is that the mathematics have wondered outside of the
fence of reality in this regard, and seem to have no understanding that
they have done it (or that they should care that have done it). They don't
seem to understand, they are studying flying elephants, and seem to think
their work on this nonsense is as valuable as the rest of mathematics. As
far as I can tell, the only value to society in the work, is that it has
the value of being entertaining to mathematicians.

[Graham Cooper]

On May 25, 8:31 am, c...@kcwc.com (Curt Welch) wrote:
> Shmuel (Seymour J.) Metz <spamt...@library.lspace.org.invalid> wrote:
>

> > In <20120524143314.779...@newsreader.com>, on 05/24/2012

WRONG! for the 100th time!

Godel's "incompleteness" and Heisenberg's "uncertainty" are nothing to
do with absolute truth.

You can't SEE an electron perfectly because the photon knocks it out
of the experimental domain.

You can KNOW 100% of the electron's position, and have no clue what
the momentum is.

You can KNOW 100% of the electron's momentum, and have no clue what
it's position is.

You can KNOW 50% of position and 50% of momentum!

There are no AXIOMS to ascertain the universal fact that Russell's Set
cannot exist.

~E(R)xeR<->xex

this is an ABSOLUTE FACT!

> All we have, are events which are highly likely, and highly
> unlikely.
>
> It's highly unlikely that gravity will work differently starting tomorrow
> for example.  But it's not an absolute fact that it will work the same
> tomorrow.
>
> What we do however, is _pretend_ such high probability events are
> absolutes, because they are so close to being absolutes, it would be a
> waste of time and computational energy in most applications, to do anything
> other than plan our future based on the assumption gravity will keep
> working the same.
>
> In the language of logic, we pretend absolutes can exist.  We pretend
> "true" and "false" are valid absolute axioms.  We pretend they exist as

They DO EXIST as solidly as HOT and COLD exist.

Here is the INEVITABLE SEQUENCE.

What is Truth?

IN THE BEGINNING..
there was the physical world
from that came evolution
from that came event associated positive and negative awareness
then also came language
from that came proposition associated true and false properties
from that came predicate() associated true and false properties
from that came logic
from that and the physical world came computers


SEE THREAD : MATHEMATICS IS.. on defining Maths and Logic with
Tautologies.

Herc
--

[Graham Cooper]

On May 25, 8:47 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> ~E(R)xeR<->xex
>
> this is an ABSOLUTE FACT!
>

Damn you Noise Demon!

~E(R)xeR<->x~ex

Herc

[LudovicoVan]

"Curt Welch" <cu...@kcwc.com> wrote in message
news:20120524143314.779$a...@newsreader.com...
> WM <muec...@rz.fh-augsburg.de> wrote:
<snip>

> Has there been found any application in this universe where the act of
> pretending infinity exists becomes useful?

Exists in what sense? I'd think limits are the prototypal example of
mathematical usage of completed infinities. Would you guys object to the
usage (and usefulness!) of limits and the existence of divergent limits
(i.e. to the fact that limits exist and have values), or, for some reason,
think that limits must necessarily introduce inconsistencies? In fact, the
limit of a sequence is *not* a sequence (not any of its elements), so IMHO
we are *already* talking about completed infinities here: for instance,
lim_{n->oo} (1/n) = 0, despite 0 is not the value of any of the elements of
the sequence (1/n), so that, to use ordinals, we *should* indeed say that
(1/w) = 0.

And *that*, to me, would be the essential contradiction: defining reals as
limits of sequences (so point-like numbers) and still pretending that 'pi',
or in fact any irrational, is a sequence of digits in the standard sense is
only an instance of it. The very diagonal argument becomes the joke of
asserting that an inductive (countable) infinity is never complete. Not to
mention some well known logical absurdities like the vase eventually empty
in the balls and vase problem, despite at every step one puts more balls in
then one removes (so that the derivation is patently false!), etc.

To sum it up: while I still have to see any solid/direct justification to
the assertion that "completed" infinities (meant as sequences extended with
their limit points, or equivalent definitions) bring contradictions (which
is, if you like, a challenge to the finitists), I'd venture that the
contradictions we are encountering (with reals that are and are not
sequences, with vanishing balls, with complete-incomplete lists, etc.)
rather have to do with an improper/ambiguous/self-contradictory mix of
complete and incomplete infinities in the same definitions or arguments,
maybe specifically with the fact that, wherever we are allowed to take
limits, we should also be using the corresponding limit ordinals.

-LV

[LudovicoVan]

Uirgil" <uir...@uirgil.ur> wrote in message
news:uirgil-AE5B47....@bignews.usenetmonster.com...

> Cantor's Aleph_0 is incompete only in the sense that it does not have a
> last member but is complete in the sense of having all of its members.

Speak of the devil...

-LV

[Curt Welch]

Graham Cooper <graham...@gmail.com> wrote:

> On May 25, 4:33=A0am, c...@kcwc.com (Curt Welch) wrote:
> > WM <mueck...@rz.fh-augsburg.de> wrote:
> > > On 24 Mai, 16:37, c...@kcwc.com (Curt Welch) wrote:
> >
> > > > > Take any well ordering of the rationals:
> > > > > q_1, q_2, q_3, q_4, ...
> > > > > Subdivide the set into pairs and re-order every pair such that
> > > > > the smaller one comes first:
> > > > > (q_1', q_2'), (q_3', q_4'), ...
> > > > > Then subdivide the set into pairs again, leaving out q_1', and
> > > > > re- order every pair such that the smaller one comes first:
> > > > > q_1', (q_2'', q_3''), (q_4'', ...
> > > > > Then subdivide the set into pairs again, including q_1', and

> > > > > re-ord=

> er
> > > > > every pair such that the smaller one comes first - and continue
> > > > > in this way. After aleph_0 transpositions you get a well-order

> > > > > which i=

> s
> > > > > simultaneously an order by magnitude.
> >

> > > > Why is it valid to say "After aleph_0 transpositions"? =3DA0How can
> > > > t=
> here
> > > > b=3D

> > > e an
> > > > "after" of a process that by definition never terminates?
> >
> > > This has been introduced by Cantor: omega is followed by omega + 1.
> > > And aleph_0 you only the beginning of the hierarchy of infinities.
> > > Cantor's set theory needs the completed infinity (Cantor's own words)
> > > because the usual infinite is only potential, i.e., there is no end.
> > > But in that case there is also no complete Cantor-list and no means
> > > to decide whether or not an entry is contained therein.
> >

> > Ok. =A0So by what logic does he justify "adding the completed

> > infinity"?
> >
> > How does it become valid to define something that can't exist, as

> > existin=

> g?
> >
> > By declaring "infinity exists" as an axiom, it seems to me one would be
> > introducing a contradiction into their set of axioms that would only
> > lead to problems - like a nonsensical hierarchy of infinities to start

> > with, b=

> ut
> > many more I'm sure show up as well.
> >
> > Has there been found any application in this universe where the act of
> > pretending infinity exists becomes useful?
> >
> > > > > No convergence? In set theory there is no convergence (for
> > > > > example when enumerating the rationals). Nevertheless the limit
> > > > > (aleph_0 rationals) exists.
> >
> > > > What exactly "exists" that that sentence is referring to?
> >
> > > The complete enumeration of all rational numbers, such that no
> > > rational number is missing.
> >

> > But that can't exist. =A0What is the advantage of pretending it can
> > when =

> it
> > clearly can't?
> >
> > > In fact, Cantor has only proven that for every rational number there
> > > is a natural number indexing the former. But then he has turned this
> > > into the meaning of indexing "all" rationals.
> >
> > Right. We can define a process that can index any given rational.

> > =A0Howe=

> ver,
> > the list of all rationals and their indexes can never exist.
> >
>
> You mean Irrationals. Rationals have a finite formula.
>
> 1/1
> 1/2
> 2/1
> 1/3
> 2/2
> 3/1
> 1/4
> 2/3
> 3/4
> 4/4
>
> Formulas, functions, predicates, .. exist using the existential
> quantifier.
>
> Also PROOF(Formula) either Exist or Not in Theories.
>
> By using a SET SPECIFICATION AXIOM, Set's are inferred to exist.

Naming something with language does not make it exist.

I doubt it's possible to define a reality where language use defines
existence without creating an endless list of logical contradictions in the
same reality.

The first obvious problem is that if you could write "X exists" to make X
exist, then I think you are forced to also include "X does not exist" to
define non-existence of X. So then you write "This sentence does not
exist", and instantly you have shown the reality is flawed at the core.
The sentence can't exist if the sentence says it doesn't exist when an
axiom of the reality is that any sentence created in the reality defines
existence in the same reality.

It's a fatal as trying to include two axioms in a system where one says 1
!= 2, and the second defines 1 == 2. What's the point (or use) of the
system of axioms if they contradict each other?

The axiom "language use defines existence", is just as fatal. The axiom is
self contradicting.

> X e Y <-> EXISTS(function-n) function-n(X)
>
> where function-n is some lexicographical index of the Theory.
>
> For arguments sake you could call this SYNTAX the point at which the
> SET EXISTS, not just the function or 'proof of existence'.
>

> Y =3D { x | fn(x) }

>
> since fn exists, so does Y, by the SPECIFICATION AXIOM.

Such an axiom seems unworkable at the core. If you try to build a system
from such an axiom, the system will filled to the brim with self
contradictions.

What can work, is to just ignore whether the set exists, and just
acknowledge that the language to define it exists and stop at that. The
set doesn't need to exist. But them you must be very careful and
understand every time you make reference to "the set" you are not
referencing a real set you are only referencing what does exist - the
language that defines it.

If you take care to get the references correct like than, then the system
need not have any contradictions.

At last that is how I see it.

> Herc

[Curt Welch]

WM <muec...@rz.fh-augsburg.de> wrote:
> On 24 Mai, 20:33, c...@kcwc.com (Curt Welch) wrote:
>
> > > This has been introduced by Cantor: omega is followed by omega + 1.
> > > And aleph_0 you only the beginning of the hierarchy of infinities.
> > > Cantor's set theory needs the completed infinity (Cantor's own words)
> > > because the usual infinite is only potential, i.e., there is no end.
> > > But in that case there is also no complete Cantor-list and no means
> > > to decide whether or not an entry is contained therein.
> >

> > Ok. =A0So by what logic does he justify "adding the completed

> > infinity"?
>
> Not by logic. That is impossible. First he found that Augustinus
> claimed God knew all numbers.
>
> > How does it become valid to define something that can't exist, as

> > existin=

> g?
>
> Set theorists introduced an axiom saying that there exists an infinite
> set. This means in set theory; every element exists.

Yeah, I sort of understood they did that. But it seems to me, that once
you do that, you have stepped outside of reality and created endless self
contradictions in your system that basically turns it all into useless
nonsense.

> > By declaring "infinity exists" as an axiom, it seems to me one would be
> > introducing a contradiction into their set of axioms that would only
> > lead to problems - like a nonsensical hierarchy of infinities to start

> > with, b=

> ut
> > many more I'm sure show up as well.
>
> Indeed, that is the case.
> >
> > Has there been found any application in this universe where the act of
> > pretending infinity exists becomes useful?
>
> No a single one. (In fact some have been claimed. I have listed them
> in
> http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-.pdf
> Nos 1053 - 1060, partially in English. But all that is rubbish.)
>
> The only "application": Set theory is claimed to be necessary in order
> to put mathematics on a solid basis. That's even greater rubbish.

It looks to me like it destroys any hope of giving it a sold base. :)

> > > > > No convergence? In set theory there is no convergence (for
> > > > > example when enumerating the rationals). Nevertheless the limit
> > > > > (aleph_0 rationals) exists.
> >
> > > > What exactly "exists" that that sentence is referring to?
> >
> > > The complete enumeration of all rational numbers, such that no
> > > rational number is missing.
> >

> > But that can't exist. =A0What is the advantage of pretending it can
> > when =

> it
> > clearly can't?
>
> Honestly, I don't know.
> >
> > > In fact, Cantor has only proven that for every rational number there
> > > is a natural number indexing the former. But then he has turned this
> > > into the meaning of indexing "all" rationals.
> >
> > Right. We can define a process that can index any given rational.

> > =A0Howe=

> ver,
> > the list of all rationals and their indexes can never exist.
> >
> > > Only by this reversion he could insist that an irrational number is
> > > tantamount to its infinite sequence of digits, and, another
> > > reversion, that an infinite sequence of digits can define a number.
> > > The latter claim is false. There is no infinite definition defining
> > > anything.
> >

> > Right. =A0But there are processes that never end, and we can use any of
> > t=
> hese
> > processes as a definition. =A0Such as the process that produces the
> > digit=
> s of
> > pi. =A0We could call the digits produced by that process the definition
> > o=
> f
> > pi. =A0The infinite list of digits of pi can never exist, but the

> > process can, and we can call the process the definition of the infinite

> > list of t=

> he
> > digits of pi.
>
> You have an absolute correct view of the things.
>

> > =A0One must just be careful not to make the stake of leaping

> > from the existence of the process, to the existence of all the digits

> > however. =A0It seems to me that this mistake is commonly made in
> > mathemat=

> ics
> > - often (as far as I can tell) with no awareness that the mistake is

> > bein=

> g
> > made.
>
> So it is.
>
> Regards, WM

Looks like we are thinking the same things....

[Graham Cooper]

On May 25, 11:26 am, c...@kcwc.com (Curt Welch) wrote:

Yes Y = {x|f(x)} is Naive Set Theory.

> > For arguments sake you could call this SYNTAX the point at which the
> > SET EXISTS, not just the function or 'proof of existence'.

PROOFS (of the existence of N) EXIST in LOGIC,
the same way WORDS exist in BOOKS.

apart from that, your only high ground is abstractions don't exist.

Herc

[Curt Welch]

Graham Cooper <graham...@gmail.com> wrote:

> On May 25, 6:42=A0am, WM <mueck...@rz.fh-augsburg.de> wrote:
> > On 24 Mai, 22:35, Shmuel (Seymour J.) Metz
> >
> > <spamt...@library.lspace.org.invalid> wrote:
> > > In <20120524143314.779...@newsreader.com>, on 05/24/2012

> > > =A0 =A0at 06:33 PM, c...@kcwc.com (Curt Welch) said:
> >
> > > >By declaring "infinity exists" as an axiom, it seems to me one
> > > >would be introducing a contradiction into their set of axioms
> >
> > > Mathematics is not about how it seems to you. Produce a contradiction
> > > if you can; until then no rational person will take your belief
> > > seriously.
>
> Pi/10, _/2/10, e/10, ... are ALL missing from my LIST OF REALS!
>

> REAL1 =3D 0.0123456789...
> REAL2 =3D 0.9876541210
> REAL3 =3D 0.2469097531
> REAL4 =3D 0.1357889426
> REAL5 =3D 0.3012674859
> REAL6 =3D 0.4201365578
> REAL7 =3D 0.8940123567
> REAL8 =3D 0.7394210622
> REAL9 =3D 0.5732802095
> REAL10 =3D 0.6585331103
> REAL11 =3D 0.0679768344
> ...
>
> AD(pos) =3D changedigit(LIST(n2n(pos),pos))

>
> 3141592653.. is missing
> 1414213562.. is missing
> 2718281828.. is missing
>

> n2n(1) =3D 3 (Row 3)
> n2n(2) =3D 5 (Row 5)

You can't define existence by the convergence of an infinite process.
That's just absurd.

Some infinite processes converge _towards_ a thing that does exist. 1,
1/2, 1/3, 1/4, 1/5, etc converges to zero. But it does not create or even
define the number zero.

By saying an infinite processes converges to zero, we are saying that the
longer the infinite process runs, the closer it can get to being _replaced_
by the finite process (that represents the number zero).

The infinite process did not define the finite process. The finite process
already existed, and we can demonstrate the infinite process gets closer
and closer to the finite one the longer it runs. But since it can't
complete, it will never be the same as the finite process.

An infinite process that does not converge towards any finite process does
not magically become a finite process (aka, can complete).

If you try to define rationals as what the processes converges to, then we
have the issue of 0.9999.. being a very different process than 1.0000....
but yet they must be said to coverage on the same real number. So now we
have an infinite number of processes producing different sequences, which
must be claimed to be "the same real". What a disaster. The instant you
start trying to pretend the infinite process can "finish", everything turns
into a huge nasty mess.

If instead, we just understand that we have processes that can produce
sequences (which never finish), and that the process can be defined, but
never run to completion, and limit our proofs and statements to be be about
what can exist (a partially generated list), then nothing gets messy.

The diagonal "proof" is no longer a proof if we do this for example - it
just becomes nonsense.

> Herc

[Graham Cooper]

On May 25, 11:56 am, c...@kcwc.com (Curt Welch) wrote:

I tend to agree, in Computer Science most of the maths constructs are
specifications.

e.g. there are 2X2X2X2... non-terminating binary strings.
2^aleph0 or 2^oo

A definition is more a how-to function spec.

But the Software Development Cycle is out of scope in Academia.

Herc

[Uirgil]

In article
<3eaf7e42-e8b8-42a0...@b26g2000vbt.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 24 Mai, 20:33, c...@kcwc.com (Curt Welch) wrote:
>
> > > This has been introduced by Cantor: omega is followed by omega + 1.
> > > And aleph_0 you only the beginning of the hierarchy of infinities.
> > > Cantor's set theory needs the completed infinity (Cantor's own words)
> > > because the usual infinite is only potential, i.e., there is no end.
> > > But in that case there is also no complete Cantor-list and no means to
> > > decide whether or not an entry is contained therein.
> >
> > Ok.  So by what logic does he justify "adding the completed infinity"?
>
> Not by logic. That is impossible. First he found that Augustinus
> claimed God knew all numbers.
>
> > How does it become valid to define something that can't exist, as existing?
>
> Set theorists introduced an axiom saying that there exists an infinite
> set. This means in set theory; every element exists.

And no one has yet proved that that axiom has created any contradictions.

> >
>
> >
> > By declaring "infinity exists" as an axiom, it seems to me one would be
> > introducing a contradiction into their set of axioms that would only lead
> > to problems - like a nonsensical hierarchy of infinities to start with, but
> > many more I'm sure show up as well.
>
> Indeed, that is the case.

WHere is you proof that that is the case?

> >
> > Has there been found any application in this universe where the act of
> > pretending infinity exists becomes useful?

Classical Calculus requires infinite sets, and has been remarkably
useful.

>
> No a single one. (In fact some have been claimed. I have listed them
> in
> http://www.hs-augsburg.de/~mueckenh/KB/KB%201001-.pdf

From that pdf,
1003 Das Kalenderblatt 120303
(2) Axiom of choice, II; multiplicative axiom: The Cartesian product of
nonempty sets is always nonempty.

If you deny this, as WM would have everyone do, you are requiring that
there exist a cartesian product of non-empty sets that IS empty, which
I, for one, regard as far more offensive than the axiom of choice.
nonempty.

>
> The only "application": Set theory is claimed to be necessary in order
> to put mathematics on a solid basis. That's even greater rubbish.

It is either set theory or category theory, and in category theory there
is no easy way to deny infiniteness.

> >
> > > > > No convergence? In set theory there is no convergence (for example
> > > > > when enumerating the rationals). Nevertheless the limit (aleph_0
> > > > > rationals) exists.
> >
> > > > What exactly "exists" that that sentence is referring to?
> >
> > > The complete enumeration of all rational numbers, such that no
> > > rational number is missing.
> >
> > But that can't exist.  What is the advantage of pretending it can when it
> > clearly can't?

Since it has been done repeatedly, I have even done it myself, both via
a surjection from N to Q and an injection from Q to N, what is the point
of claiming it cannot be done.

>
> Honestly, I don't know.
> >
> > > In fact, Cantor has only proven that for every rational number there
> > > is a natural number indexing the former. But then he has turned this
> > > into the meaning of indexing "all" rationals.

"For every" implies "for all"

> >
> > Right. We can define a process that can index any given rational.  However,
> > the list of all rationals and their indexes can never exist.

Functions surjecting N onto Q and injecting Q into N can exist by virtue
of being unambiguously defined. And they have been unambiguously defined.

[Uirgil]

In article <20120524213225.934$b...@newsreader.com>,
cu...@kcwc.com (Curt Welch) wrote:

> WM <muec...@rz.fh-augsburg.de> wrote:

>
> > Set theorists introduced an axiom saying that there exists an infinite
> > set. This means in set theory; every element exists.
>
> Yeah, I sort of understood they did that. But it seems to me, that once
> you do that, you have stepped outside of reality and created endless self
> contradictions in your system that basically turns it all into useless
> nonsense.

Trying to justify calculus when one presumes all sets must be finite is
a bitch.

[Uirgil]

In article
<d69a8fbb-4680-45b6...@w13g2000vbc.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 24 Mai, 22:35, Shmuel (Seymour J.) Metz
> <spamt...@library.lspace.org.invalid> wrote:
> > In <20120524143314.779...@newsreader.com>, on 05/24/2012
> >    at 06:33 PM, c...@kcwc.com (Curt Welch) said:
> >
> > >By declaring "infinity exists" as an axiom, it seems to me one
> > >would be introducing a contradiction into their set of axioms
> >
> > Mathematics is not about how it seems to you. Produce a contradiction
> > if you can; until then no rational person will take your belief
> > seriously.
>
> The most prominent contradiction is the claim that numbers, that
> cannot be determined, can be well-ordered. The set of elements that
> can be well-ordered is a subset of the set of elements that can be
> identified.

As usual, claimed but not proven!

Things are not true just because WM believes them to be.

[K_h]

"Curt Welch" wrote in message news:20120524143314.779$a...@newsreader.com...

>
> By declaring "infinity exists" as an axiom, it seems to me one would be
> introducing a contradiction into their set of axioms that would only lead
> to problems - like a nonsensical hierarchy of infinities to start with,
> but
> many more I'm sure show up as well.

The axiom of infinity does not introduce any known contradictions.

> Has there been found any application in this universe where the act of
> pretending infinity exists becomes useful?

Infinity does exist in the sense of mathematical Platonism. But to answer
your question, infinity is necessary for calculus and the whole of analysis.
Integrals and derivatives don't make much sense without the real number
line.

> > The complete enumeration of all rational numbers, such that no
> > rational number is missing.
>
> But that can't exist. What is the advantage of pretending it can when it
> clearly can't?

They do exist. It is easy to show that a bijection exists between N and Q.

> Right. We can define a process that can index any given rational.
> However,
> the list of all rationals and their indexes can never exist.

No

> Right. But there are processes that never end, and we can use any of
> these
> processes as a definition. Such as the process that produces the digits
> of
> pi. We could call the digits produced by that process the definition of
> pi. The infinite list of digits of pi can never exist, but the process

All of the digits do exist; they are all totally fixed regardless of the
fact that they are not physically enumerated in the world.

> can, and we can call the process the definition of the infinite list of
> the
> digits of pi. One must just be careful not to make the stake of leaping
> from the existence of the process, to the existence of all the digits
> however. It seems to me that this mistake is commonly made in mathematics

To get to the end of an unending process, just go through every step of the
process. It is called a limit and it exists if the process converges. So,
for example, take whatever algorithm you use to calculate every numeral of
pi and the limit of that process is the value pi.

+

[muec...@rz.fh-augsburg.de]

Am Freitag, 25. Mai 2012 07:43:19 UTC+2 schrieb K_h:
> "Curt Welch" wrote in message news:20120524143314.779$a...@newsreader.com...
> >
> > By declaring "infinity exists" as an axiom, it seems to me one would be
> > introducing a contradiction into their set of axioms that would only lead
> > to problems - like a nonsensical hierarchy of infinities to start with,
> > but
> > many more I'm sure show up as well.
>
> The axiom of infinity does not introduce any known contradictions.

Your claim merely shows a great amount of ignorance.
The axiom of infinite introduces contradictions. For instance, the set of well-orderable elements is a subset of the set of indentifyable elements. The latter has cardinality aleph_0, the former is uncountable. Contradiciton.

Regards, WM

[netzweltler]

On 24 Mai, 20:33, c...@kcwc.com (Curt Welch) wrote:
>

> Has there been found any application in this universe where the act of
> pretending infinity exists becomes useful?
>

> --
> Curt Welch                                            http://CurtWelch.Com/

> c...@kcwc.com                                        http://NewsReader.Com/

I can display every natural number n as a line of slope n in the
coordinate system. w can be displayed as a vertical line then. Does
infinity exist thus?

--
netzweltler

[Alan Smaill]

WM <muec...@rz.fh-augsburg.de> writes:

> On 24 Mai, 20:33, c...@kcwc.com (Curt Welch) wrote:
>
>> > This has been introduced by Cantor: omega is followed by omega + 1.
>> > And aleph_0 you only the beginning of the hierarchy of infinities.
>> > Cantor's set theory needs the completed infinity (Cantor's own words)
>> > because the usual infinite is only potential, i.e., there is no end.
>> > But in that case there is also no complete Cantor-list and no means to
>> > decide whether or not an entry is contained therein.
>>
>> Ok.  So by what logic does he justify "adding the completed infinity"?
>
> Not by logic. That is impossible. First he found that Augustinus
> claimed God knew all numbers.
>
>> How does it become valid to define something that can't exist, as existing?
>
> Set theorists introduced an axiom saying that there exists an infinite
> set. This means in set theory; every element exists.
>>
>
>>
>> By declaring "infinity exists" as an axiom, it seems to me one would be
>> introducing a contradiction into their set of axioms that would only lead
>> to problems - like a nonsensical hierarchy of infinities to start with, but
>> many more I'm sure show up as well.
>
> Indeed, that is the case.

So, you do in fact think that set theory is inconsistent.

>
> Regards, WM

--
Alan Smaill

[Graham Cooper]

On May 25, 5:59 pm, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>
> >> By declaring "infinity exists" as an axiom, it seems to me one would be
> >> introducing a contradiction into their set of axioms that would only lead
> >> to problems - like a nonsensical hierarchy of infinities to start with, but
> >> many more I'm sure show up as well.
>
> > Indeed, that is the case.
>
> So, you do in fact think that set theory is inconsistent.
>

you cannot show the inconsistency to the system that is inconsistent
as anything is now possible according to them.

For instance, none of you have answered a straight question in the
last 2 years.

Do you believe EXIST(Z) Z>oo ?
HINT: |N|=oo
HINT: |R|=Z

SCI.MATH |- ->oo
SCI.MATH |- oo->

SCI.MATH |- LIM x->oo 2^x =/= 2X2X2X..
SCI.MATH |- ALL(d) AD(permute(list))[d]=[0|1]
SCI.MATH |- ALEPH_1>oo <-> {n|n~ef(n)}=/={n|n~en}=RS
SCI.MATH |- A(f):N->PS(N) A(g):N->PS(N) {n|n~ef'(n)}={n|n~eg'(n)}
SCI.MATH |- !E(U)<->!E(RS)<->E(GS)<->E(t)t~eT<->E(p)p~eHALT<->E(r)r=/
=Rn
SCI.MATH |- ALL(t) SCI.MATH |- t
SCI.MATH |- CONS(SCI.MATH)

Herc

--
1 X ^ NOT(X)
2 G = NOT(PRV(G))
3 S > INF
4 R = {X | NOT(X e X)}
5 IF HALT() GOTO 5
6 ALL(F) MAX(F)
=
THE 6 DEAD ENDS IN MATHEMATICS
but only 4 are recognised contradictions

[Jesse F. Hughes]

The set of well-orderable elements! Oh my!

--
"He isn't capable of actually defining his terms, or axiomatizing
them, or deriving consequences from them. The kindest course of action
is to humor him[...]Just pat him on the head and say 'Tony, aren't you
the cutest little mathematician!'" -- Daryl McCullough on Tony Orlow.

[WM]

On 25 Mai, 09:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
>
> So, you do in fact think that set theory is inconsistent.

It is not an opinion, it is a proof.

The most prominent contradiction is the claim that numbers, that
cannot be determined, can be well-ordered. The set of elements that
can be well-ordered is a subset of the set of elements that can be
identified.

Try to convince a freshman or anintelligent outsider, that my argument
is wrong. I have hundreds of students convinced that my arguments are
right. Of course they had not been spoilt by matheology yet. The
additcts of finished infinity cannot be saved.

Here is a very simple proof:

Cover every rational q_n of the real axis (-oo, oo) with
an interval I_n of measure 10^-n. All remaining irrationals then must
be separated from each other by at least one interval I_n. So the
irrationals can only occupy the borders of intervals. That is a well-
ordered and even countable set. Contradiction, because there should
uncountably many irrationals be uncovered.

Regards, WM

[Shmuel Metz]

In <20120524183145.077$1...@newsreader.com>, on 05/24/2012

at 10:31 PM, cu...@kcwc.com (Curt Welch) said:

>There is no need to pretend a number exists.

Whoosh! Humor is such a subjective thing.

>Counts of objects do exist.

Finger folding and marks on paper are not numbers.


>Abstractions exist.

Ah, so you concede the point.

>That was not my question.

You didn't have a question; you had a statement masquerading as a

question.

>My question was whether anyone has yet found a
>practical application for pretending machines that can finish
>counting to infinity can "exist" has any practical application.

As long as you refuse to ask an honest question youi can expect
ridicule. Begging the question doesn't cut it. You've already admitted
that abstractions exist and have not demonstrated that there is any
pretending involved.

>Show my an example of one existing please.

Not until you justify your claim.

>We don't need to prove a negative. We assume a negative is false,
>until there is data to show us otherwise.

So if the negative is "Curt is not dishonest", we should assume that
it is false?

>What does that mean? It means that the existence of a counting
>machine that can finishing counting to infinity,

Then it has no relevance, because you are the one that wants such a
machine. It has nothing to do with whether either "infinity"[1] or
infinite sets exist.

>Then what it is founded on?

Reaoning from axioms.

>The foundations of mathematics, comes from the machine that does
>the mathematics.

Is it your position that neither the human brain nor written text are
part of the Universe?

>What strikes me as highly problematic about the current
>mathematics, is that most of what I'm saying seems to be still be
>ignored by the field

Well, if you told a conductor that concerti don't exist, how long
would it take you to convince him and his colleagues? Would they take
you seriously if you claimed that a kettledrum was a string
instrument?

>They ignore it, because

It reflects a misunderstanding of what Mathematics is about, not
because of the bogus reasons you give.

>When we talk about numbers, as I talked about above, we are talking
>about machines that can count

Demonstrably false; *you* are talking about such machines, *we* are
not.

[1] A word that has narrow[2] meanings in Mathematics quite
unrelated to the way that amateurs sling it around here.

[2] Google for "compactification" and for "series"

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org

[Shmuel Metz]

In <20120524213225.934$b...@newsreader.com>, on 05/25/2012

at 01:32 AM, cu...@kcwc.com (Curt Welch) said:

>But it seems to me, that once you do that, you have stepped
>outside of reality and created endless self contradictions in
>your system

How it seems to you is irrelevant. Violating your aesthetics and
prejudices does not prove a self contradiction. Showing that it is
incompatible with some other system that you prefer is not good
enough. You won't be taken seriously until you produce an actual proof
of a contradiction.

[Shmuel Metz]

In <20120524215637.243$Z...@newsreader.com>, on 05/25/2012

at 01:56 AM, cu...@kcwc.com (Curt Welch) said:

>You can't define existence by the convergence of an infinite process.
>That's just absurd.

Graham Cooper doesn't understand the definition of the reals. However,
one common definition is that a real is an equivalence class of Cauchy
sequences in the rationals, and one theorem that applies to the reals
however defined is that every Cauchy sequence converges. Note that a
sequence is *not* a process, and there is no process involved.

>Some infinite processes converge _towards_ a thing that does exist.

Define process and convergence of processes.

>If you try to define rationals as what the processes converges to,
>then we have the issue of 0.9999.. being a very different process
>than 1.0000....

Only if you have a really bizarre definition of convergence. IAC, it
has nothing to do with convergence of sequences, which is what is
relevant.

>So now we have an infinite number of processes producing
>different sequences, which must be claimed to be "the same real".

The limit of a sequence is not the sequence.

>What a disaster.

If a_1 = 1 for all i, b_1 = 0 and b_i = 1 for all i>0, does it bother
you that a and b both converge to 1? If so, why?

>The diagonal "proof" is no longer a proof if we do this for example

Wrong again, because it does not involve processes at all.

[Shmuel Metz]

In <e1c57183-a5f1-4d2f...@googlegroups.com>, on
05/24/2012

at 10:49 PM, muec...@rz.fh-augsburg.de said:

>Your claim merely shows a great amount of ignorance.

PKB.

[WM]

On 25 Mai, 16:20, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <20120524215637.243...@newsreader.com>, on 05/25/2012

>    at 01:56 AM, c...@kcwc.com (Curt Welch) said:
>
> >Some infinite processes converge _towards_ a thing that does exist.
>
> Define process and convergence of processes.

Thinking, for instance, is a process. Sometimes it converges to a
result. Mathematics is a subset of thinking. Everything that we know
of happens in a reality and proceeds in time. And nobody (except
photons) has accomplished yet to stop time from marching on.

Regards, WM

[K_h]

"Curt Welch" wrote in message news:20120524183145.077$1...@newsreader.com...

>
> Abstractions exist. No pretending needed. To suggest they don't is to
> make an error of duality - the error that the thoughts aren't physical -
> that they don't have physical existence. An abstraction is just the

Why are you bringing the mind-body problem into this thread??

> the abstraction is just a behavior of a pattern matching machine - which
> must be real and physical to "exist".

No. All digits of pi exist even though they are not physically enumerated.

> Humans operate on probabilities. Nothing for us is certain. There are no

What about death and taxes?

> In the language of logic, we pretend absolutes can exist. We pretend

It is an absolute fact that 12 + 13 = 25. It is absolutely true that the
second root of 2 is 1.414213...

> There could be troll under my desk right now. But I don't need to prove

There isn't.

> gravity is highly likely to work the same way tomorrow, is not an
> absolute,

Why do you even think that gravity might behave differently?

> So, in order for you to show my behavior invalid, you need to provide the
> evidence (or at least some small hint of a suggestion about how and where
> it might be hidden from us), that such a machine can exist in this
> universe. You have to provide evidence strong enough to suggest we should
> no long assume this is an absolute.

The evidence can be found in physics. The quanta are in a superposition of
an infinite number of all possible histories, until state reduction happens.

> to just "make up" the foundations of mathematics. The foundations of
> mathematics, comes from the machine that does the mathematics. And the
> foundations of that machine, come from the foundations of the universe it
> exists in.

The foundations of mathematics does not come from a machine. The human mind
has discovered mathematical truth and set theory grounds that truth in
self-evident axioms that all reasonable people agree to.

> Mathematics, can't "exist" if there is no machine that can do the work.

No. The mathematical truths pre-exist machines.

> What strikes me as highly problematic about the current mathematics, is
> that most of what I'm saying seems to be still be ignored by the field (as
> far as I can see). They ignore it, because they are playing the old
> (totally invalid) game of "thoughts are not physical, and have no physical
> limitations". That's just not true. Thoughts have the same limitations
> as
> our computers. They can't, for example, count to infinity.

Nobody has solved the mind-body mystery. So the possible physicality of
"thoughts" is currently an open question. Roger Penrose has some ideas
about it as do others.

> of the territory. They act that since they can say "the infinite set
> exists", then that's all they need to do to prove that "it does exist".

It is a self-evident truth that the naturals exist just like all the digits
of pi exist. They are not physically instantiated, obviously, but they
don't have to be.

> It's as useful as starting a project by saying something like, "Assume all
> elephants can fly - but they have just never done it for a human to see",

Stop trying to denigrate mathematics.

+

[WM]

On 25 Mai, 21:17, "K_h" <KHol...@SX729.com> wrote:
> "Curt Welch"  wrote in messagenews:20120524183145.077

> The foundations of mathematics does not come from a machine.  The human mind
> has discovered mathematical truth and set theory grounds that truth in
> self-evident axioms that all reasonable people agree to.

Finished infinity and time after never is reasonable?

>
> > Mathematics, can't "exist" if there is no machine that can do the work.
>
> No.  The mathematical truths pre-exist machines.

Of course you have not the least hint or proof for your assumption.
You are a believer in God and His mathematics.
It is not useful to discuss with such people. It is a shame to be
called reasonable by such people.

Regards, WM

[WM]

On 25 Mai, 16:06, Shmuel (Seymour J.) Metz
<spamt...@library.lspace.org.invalid> wrote:
> In <20120524213225.934...@newsreader.com>, on 05/25/2012

>    at 01:32 AM, c...@kcwc.com (Curt Welch) said:
>
> >But it seems to me, that once you do that, you have stepped
> >outside of reality and created endless self contradictions in
> >your system
>
> How it seems to you is irrelevant. Violating your aesthetics and
> prejudices does not prove a self contradiction. Showing that it is
> incompatible with some other system that you prefer is not good
> enough. You won't be taken seriously until you produce an actual proof
> of a contradiction.

Here is a proof of a contradiction.

Ordering objects is impossible without identifying them as distinct.
Reason: Unless you can identify objects as distinct, you cannot be
sure to have more than one (or even to have any).
ZFC "proves" that the set of real numbers is uncountable and can be
well-ordered. Mathematics proves that only countably many objects can
be distinguished and identified.

This is a contradiction in any theory that does not preclude the
existence of contradictions and does not commit its followers to
forcefully deny the existence of any contradiction.

Regards, WM

[Graham Cooper]

On May 26, 12:06 am, Shmuel (Seymour J.) Metz wrote:
> In <20120524213225.934...@newsreader.com>, on 05/25/2012

>    at 01:32 AM, c...@kcwc.com (Curt Welch) said:
>
> >But it seems to me, that once you do that, you have stepped
> >outside of reality and created endless self contradictions in
> >your system
>
> How it seems to you is irrelevant. Violating your aesthetics and
> prejudices does not prove a self contradiction. Showing that it is
> incompatible with some other system that you prefer is not good
> enough. You won't be taken seriously until you produce an actual proof
> of a contradiction.
>

We are claiming your system is TOO BIG! We are claiming there is NO
NEED for a superset of N in size. You are dismissing countable models
as incomplete, based on your web of top down justifications.

SCI.MATH |- !E(U)<->!E(RS)<->E(GS)<->E(t)t~eT<->E(p)p~eHALT

<->E(r)r=/=fn<-E(s)s=/=PS(N)<->~E(U)

This is your cyclic argument, all the same antidiagonal 'proof'!

I asked SCI.MATH "WHAT IS A SET WITHOUT A FORMULA?" All you came up
with was "This sentence is missing from U"

It is extremely important to understand that the classical treatment
of language interpretation parameterizes the universal quantifier over
collections. Moreover, the existential quantifier is interpreted only
with
respect to its status as a derivative concept relative to the
universal quantifier. These are the underlying assumptions of
construction that allow the self-inconsistency of a specific syntactic
form to be extended to a
metaphysical assertion of reality.

Think about the problem differently. If a set is a collection taken
as an object, how are we to understand a collection as
indecomposable? What object can be dually interpreted as a
collection?

It is possible to interpret the universal quantifier with respect to a
reflexive order relation. The language primitive must be a strict
transitive order, and, an axiom explicitly assuming "almost
universality" must relate a primitive membership predicate to that
order relation. A congruence relation
derived from these two primitives (incorrectly referred to as identity
in first-order logic with identity) provides for a reflexive case
juxtaposed against the strict order relation. This exclusive
disjunction distinguishes between terms referring to proper parts (the
strict order relation corresponds
to the mereological "proper part" predicate) and terms referring to
the class universe (the set of all sets).
If you look at the class universe as some sort of primitive topology,
the proper parts are topologies with the subspace topology. The axiom
schema of separation ensures that these topologies are totally
separated. The power set axiom ensures that all separations
correspond to proper subspaces of the class universe. Hereditary
definition ensures that all proper subspaces of the universe
correspond to dichotomies (labelings of term referents into two
categories) by virtue of characteristic functions. That is, all of
the proper
subspaces of the class universe are totally disconnected.

With this formulation, the language can support reference to a
greatest class. The assertion of "almost universality" ensures that
"proper classes" different from the class universe cannot be referred
to within the theory. So, complements of the hereditarily defined
classes are disallowed. Consequently, the class universe is a
connected topology for which every proper subspace is totally
disconnected.

That describes an indecomposable object that reflects the semantics of
a collection. Do not confuse interpretation of a particular
instantiation of self-inconsistent syntax with a metaphysical
reality.

Mr Cooper

[Uirgil]

In article
<1eed6342-d3f3-4c22...@z19g2000vbe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Mai, 21:17, "K_h" <KHol...@SX729.com> wrote:
> > "Curt Welch"  wrote in messagenews:20120524183145.077
>
> > The foundations of mathematics does not come from a machine.  The human mind
> > has discovered mathematical truth and set theory grounds that truth in
> > self-evident axioms that all reasonable people agree to.
>
> Finished infinity and time after never is reasonable?

It has proved quite useful in, for instance the development of the
infinitesimal calculus.

> >
> > > Mathematics, can't "exist" if there is no machine that can do the work.
> >
> > No.  The mathematical truths pre-exist machines.
>

> You are a believer in God and His mathematics.

Then WM must be a disbeliever in any God and His mathemtics.

[Uirgil]

In article
<9dc163c3-ed85-438a...@p27g2000vbl.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Mai, 16:20, Shmuel (Seymour J.) Metz
> <spamt...@library.lspace.org.invalid> wrote:
> > In <20120524215637.243...@newsreader.com>, on 05/25/2012
> >    at 01:56 AM, c...@kcwc.com (Curt Welch) said:
> >
> > >Some infinite processes converge _towards_ a thing that does exist.
> >
> > Define process and convergence of processes.
>
> Thinking, for instance, is a process.

Not one to which WM has as easy access as the rest of us do.

[Uirgil]

In article
<fccf2aaf-c674-4114...@q2g2000vbv.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Mai, 16:06, Shmuel (Seymour J.) Metz
> <spamt...@library.lspace.org.invalid> wrote:
> > In <20120524213225.934...@newsreader.com>, on 05/25/2012
> >    at 01:32 AM, c...@kcwc.com (Curt Welch) said:
> >
> > >But it seems to me, that once you do that, you have stepped
> > >outside of reality and created endless self contradictions in
> > >your system
> >
> > How it seems to you is irrelevant. Violating your aesthetics and
> > prejudices does not prove a self contradiction. Showing that it is
> > incompatible with some other system that you prefer is not good
> > enough. You won't be taken seriously until you produce an actual proof
> > of a contradiction.
>
> Here is a proof of a contradiction.
>
> Ordering objects is impossible without identifying them as distinct.
> Reason: Unless you can identify objects as distinct, you cannot be
> sure to have more than one (or even to have any).
> ZFC "proves" that the set of real numbers is uncountable and can be
> well-ordered.

I was under the impression that ZFC merely assumed that
well-orderability.

> Mathematics proves that only countably many objects can
> be distinguished and identified.

What you miscall mathematics merely assumes it, but never proves it
based only on assumptions that do not presume it.

>
> This is a contradiction in any theory that does not preclude the
> existence of contradictions

According to Godel, there are no such theories sufficient to produce
arithmetic.

[Uirgil]

In article
<9081c3dd-169d-42fd...@d17g2000vbv.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Mai, 09:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> >
> > So, you do in fact think that set theory is inconsistent.
>
> It is not an opinion, it is a proof.

Not by the stanards of any competent mathematician.

> Try to convince a freshman or anintelligent outsider, that my argument
> is wrong. I have hundreds of students convinced that my arguments are
> right.

The person giving out the grades can make his students say anything.

[Curt Welch]

Graham Cooper <graham...@gmail.com> wrote:

> On May 25, 8:31=A0am, c...@kcwc.com (Curt Welch) wrote:
> > Shmuel (Seymour J.) Metz <spamt...@library.lspace.org.invalid> wrote:

> You can KNOW 50% of position and 50% of momentum!
>
> There are no AXIOMS to ascertain the universal fact that Russell's Set
> cannot exist.
>
> ~E(R)xeR<->xex
>
> this is an ABSOLUTE FACT!

Then, tell me, what is a fact? What does that word mean? That is, what
does it mean in terms of physics. When is are some atoms representing (or
creating, or whatever) a fact? Where do we find "facts" in the the
behavior of atoms (without a human in the system to declare or test the
fact).

And then, what is an absolute fact?

We can use language to make up nonsense, but for the language we produce to
have any purpose or use, we must create meaning, that is defined not in
terms of axioms, but in terms of of the physical universe.

Axioms have no meaning, if their meaning is not first grounded to physical
reality. Any axiom we try to make up, is in fact, grounded in concepts
from physical reality.

One concept however that people make a mistake about, is the concept that
"absolute truths can exist".

You have not proven above that issue of whether they exist or not. You
have just stated the opinion that they do.

Generally speaking, all of mathematics is based on the belief that absolute
truths can exist. It's an implied axiom of the entire field. But it's
wrong. And as such, one must be careful in understanding the separation
between, reality, and mathematics.

The most common belief of absolutes in mathematics is the belief that
humans have absolutely perfect perception. That it's 100% impossible, for
language to misunderstood.

If we say the symbol 1 is absolutely and always different from the symbol
2, then we are also saying that perception of the symbol by all machines
that attempt to read and understand them, will be 100% perfect 100% of the
time for the rest of eternity. But that is never an absolute.

To make the statement that "1 != 2" (absolutely), we are also simplicity
making the statement that the language sequence will never be misunderstood
- because without someone's perception when they read the sentence, the
meaning of the symbols don't exist - the math doesn't exist. Only light
and dark spots on a screen exist. It doesn't become math, until a
perception system reads it and interprets it. And if there is any chance a
perception system will fail to read it correctly, then the light and dark
marks on the screen do not form in any sense an "absolute" truth.

Mathematicians choose to ignore all this. They choose to ignore the fact
that math doesn't just happen by the magical will of God (or the will of
the mathematician). "math" itself, exists only one way - as a physical
process. Math is a physical process even though due to a lack of
understanding of the mind, mathematicians have chosen to believe math
exists somehow separate from the mind and separate from any physical
limitations of the body.

But that is exactly where math goes off the rails and wanders into pure
useless nonsense.

> > All we have, are events which are highly likely, and highly
> > unlikely.
> >
> > It's highly unlikely that gravity will work differently starting
> > tomorrow for example. =A0But it's not an absolute fact that it will
> > work the same tomorrow.
> >
> > What we do however, is _pretend_ such high probability events are
> > absolutes, because they are so close to being absolutes, it would be a
> > waste of time and computational energy in most applications, to do
> > anythi=
> ng
> > other than plan our future based on the assumption gravity will keep
> > working the same.
> >
> > In the language of logic, we pretend absolutes can exist. =A0We pretend
> > "true" and "false" are valid absolute axioms. =A0We pretend they exist
> > as
>
> They DO EXIST as solidly as HOT and COLD exist.

The concept of an absolute exists. Just as the concept of pink flying
elephants exist. The elephants however don't exist, and neither do the
absolutes. It is easy to use language to define things that are nonsense.

> Here is the INEVITABLE SEQUENCE.
>
> What is Truth?
>
> IN THE BEGINNING..
> there was the physical world
> from that came evolution
> from that came event associated positive and negative awareness

Ok, but it seems that you have assumed a form of absolutes here that is the
start of all the problems. What came from evolution, was probabilistic
event responses, not "positive and negative". Positive and negative
implies absolute differences. Human behavior is only understood (and
produced) probabilistically. We can not do anything absolutely.

> then also came language
> from that came proposition associated true and false properties
> from that came predicate() associated true and false properties
> from that came logic
> from that and the physical world came computers

Yes, but even computers are not absolutes. They are not guaranteed to
produce the answer 2, every time they add 1 + 1. The sometimes make
mistakes. We build them so as to reduce the odds of that type of mistake
to such low levels that the machines remain practically useful to us, but
good enough to be practically useful is not the same as having absolute
behaviors.

Math is based on the abstraction of absolutes, which is fine, but it must
be kept in mind, when doing math, that what the abstraction represents,
does't actually exist in this universe.

For most of basic math, this difference between reality and the
abstractions used by math, doesn't really cause any issues. We can still
do math correctly, most the time, just like a computer can add numbers
correctly, most the time. But when we get down to debating the foundation
of mathematics, ignoring the difference between the reality of the machines
in this universe that "do math", and what math defines itself to be, must
the same, or else, the split with reality creates endless pointless
nonsense. It's just becomes made up fiction, and not fact.

> SEE THREAD : MATHEMATICS IS.. on defining Maths and Logic with
> Tautologies.

I only read what you (or someone) cross posts to comp.ai.philosophy. I
don't believe that thread was cross posted here.

> Herc

--
Curt Welch http://CurtWelch.Com/

cu...@kcwc.com http://NewsReader.Com/

[Graham Cooper]

On May 26, 10:51 am, c...@kcwc.com (Curt Welch) wrote:

> Graham Cooper <grahamcoop...@gmail.com> wrote:
> > On May 25, 8:31=A0am, c...@kcwc.com (Curt Welch) wrote:
> > > Shmuel (Seymour J.) Metz <spamt...@library.lspace.org.invalid> wrote:
> > You can KNOW 50% of position and 50% of momentum!
>
> > There are no AXIOMS to ascertain the universal fact that Russell's Set
> > cannot exist.
>
> > ~E(R)xeR<->xex
>
> > this is an ABSOLUTE FACT!
>
> Then, tell me, what is a fact?  What does that word mean?  That is, what
> does it mean in terms of physics.  When is are some atoms representing (or
> creating, or whatever) a fact?  Where do we find "facts" in the the
> behavior of atoms (without a human in the system to declare or test the
> fact).
>
> And then, what is an absolute fact?

a true statement without assumptions

>
> We can use language to make up nonsense, but for the language we produce to
> have any purpose or use, we must create meaning, that is defined not in
> terms of axioms, but in terms of of the physical universe.
>

which runs like clockwork, according to a small subset of mathematics

> Axioms have no meaning, if their meaning is not first grounded to physical
> reality.  Any axiom we try to make up, is in fact, grounded in concepts
> from physical reality.
>
> One concept however that people make a mistake about, is the concept that
> "absolute truths can exist".

of course they do!

you cannot dispute you are reading words right now!

>
> You have not proven above that issue of whether they exist or not.  You
> have just stated the opinion that they do.

No I thoroughly explained
1 AXIOM OF SET SPECIFICATION
2 PROOF(THEOREM) PREDICATE
3 TABLE OF TAUTOLOGIES
4 TABLE OF THEOREMS

in www.tinyurl.com/SETAXOIMS

and detailed the difference between a statement proven inside and
outside a Theory.

>
> Generally speaking, all of mathematics is based on the belief that absolute
> truths can exist.  It's an implied axiom of the entire field.  But it's
> wrong.  And as such, one must be careful in understanding the separation
> between, reality, and mathematics.

You've proven yourself to write unreliable statements if that is the
case.

>
> The most common belief of absolutes in mathematics is the belief that
> humans have absolutely perfect perception.  That it's 100% impossible, for
> language to misunderstood.
>
> If we say the symbol 1 is absolutely and always different from the symbol
> 2, then we are also saying that perception of the symbol by all machines
> that attempt to read and understand them, will be 100% perfect 100% of the
> time for the rest of eternity.  But that is never an absolute.
>

See: IFF every Natural Number was a SET in sci.logic

ALL(x) ALL(y) ( ALL(a) TM-x(a)=TM-y(a) ) -> x == y

AXIOM OF COMPUTABLE EXTENSIONALITY
(equivalent programs have the same output for every input)

> To make the statement that "1 != 2" (absolutely), we are also simplicity
> making the statement that the language sequence will never be misunderstood
> - because without someone's perception when they read the sentence, the
> meaning of the symbols don't exist - the math doesn't exist.  Only light
> and dark spots on a screen exist.  It doesn't become math, until a
> perception system reads it and interprets it.  And if there is any chance a
> perception system will fail to read it correctly, then the light and dark
> marks on the screen do not form in any sense an "absolute" truth.
>
> Mathematicians choose to ignore all this.  They choose to ignore the fact
> that math doesn't just happen by the magical will of God (or the will of
> the mathematician). "math" itself, exists only one way - as a physical
> process.  Math is a physical process even though due to a lack of
> understanding of the mind, mathematicians have chosen to believe math
> exists somehow separate from the mind and separate from any physical
> limitations of the body.
>
> But that is exactly where math goes off the rails and wanders into pure
> useless nonsense.

See my above post.

These are the underlying assumptions of construction that allow the
self-inconsistency of a specific syntactic form to be extended to a
metaphysical assertion of reality.

i.e. using FORALL(SETS) but only EXIST(MISSING_SET)
or MISSING_REAL

leads to people like Virgil stating SIZE(R)>INFINITY

No, Schrodingers Cat is not a Theory In Physics.

It's a hypothesis because the exact mathematics of quantum state
reduction is unknown.

I am strongly going to suggest that you look at orthomodular logics
and quantum logics for this in addition to Bayesian decision models
and whatever else you find. I had been looking a Paul Halmos paper on
Hilbert spaces and discovered one of the inadequacies of quantum logic
for quantum mechanics--the Heisenberg uncertainty principle cannot be
represented.

> > then also came language
> > from that came proposition associated true and false properties
> > from that came predicate() associated true and false properties
> > from that came logic
> > from that and the physical world came computers
>
> Yes, but even computers are not absolutes.  They are not guaranteed to
> produce the answer 2, every time they add 1 + 1.  The sometimes make
> mistakes.  We build them so as to reduce the odds of that type of mistake
> to such low levels that the machines remain practically useful to us, but
> good enough to be practically useful is not the same as having absolute
> behaviors.
>
> Math is based on the abstraction of absolutes, which is fine, but it must
> be kept in mind, when doing math, that what the abstraction represents,
> does't actually exist in this universe.
>
> For most of basic math, this difference between reality and the
> abstractions used by math, doesn't really cause any issues.  We can still
> do math correctly, most the time, just like a computer can add numbers
> correctly, most the time.  But when we get down to debating the foundation
> of mathematics, ignoring the difference between the reality of the machines
> in this universe that "do math", and what math defines itself to be, must
> the same, or else, the split with reality creates endless pointless
> nonsense.  It's just becomes made up fiction, and not fact.
>

ABSOLUTE is just "in all domains"

ALL (x) ..... x ....

is just ..... x ....

Removing ABSOLUTE from the lexicon is fruitless.

Herc

[Curt Welch]

Graham Cooper <graham...@gmail.com> wrote:

> On May 25, 8:47=A0am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> >
> > ~E(R)xeR<->xex
> >
> > this is an ABSOLUTE FACT!
> >
>

> Damn you Noise Demon!
>
> ~E(R)xeR<->x~ex
>
> Herc

Ha ha. Did you do that on purpose? Kinda makes my point does it not?

[Curt Welch]

You just need to understand that where you have used "infinite set" in the
past, you must substitute "process that never terminates". All of calculus
works out the same no matter which concept you use. It does not need
infinite sets, it only needs the concept of processes which approaches (but
never reaches) a limit which can be calculated.

Mathematicians have gotten sloppy and treated these two very different
concepts as if they were the same when they never were the same.

[Curt Welch]

"K_h" <KHo...@SX729.com> wrote:
> "Curt Welch" wrote in message
> >news:20120524143314.779$a...@newsreader.com...
> > By declaring "infinity exists" as an axiom, it seems to me one would be
> > introducing a contradiction into their set of axioms that would only
> > lead to problems - like a nonsensical hierarchy of infinities to start
> > with, but
> > many more I'm sure show up as well.
>
> The axiom of infinity does not introduce any known contradictions.

It contradictory with the realities of the universe (the realities of
everything that exists).

I believe they hide the contradictions by pretending time doesn't exist.

But you can't have a successor function, and a concept of "processes
completed" without having a concept of time in this universe (or the
concept of events happening in a sequence). They hide the contradictions
by pretending things like a successor function (or any function that takes
an input and produces an output), can happen in zero time. They hide the
contradiction by pretending it's possible to compare if two things are the
same, or different, in zero time. They hide the contradiction by
pretending communication is instantaneous and never has a time delay.

They form a set of axioms, which can't be implemented, but yet they act as
if it's valid to pretend they can be implemented and that they are valid
axioms when they are not.

It's this divergence of these sorts of axioms from reality, and the fact
that the bulk of people doing mathematics, seems to have no realization
they have diverged from reality, into pure fiction - just making stuff up
that can't exist, and pretending that it can to have something fun to talk
about.

> > Has there been found any application in this universe where the act of
> > pretending infinity exists becomes useful?
>
> Infinity does exist in the sense of mathematical Platonism. But to
> answer your question, infinity is necessary for calculus and the whole of
> analysis. Integrals and derivatives don't make much sense without the
> real number line.

No, that's not true. In place of infinity, you can use the concept of a
process which never terminates, and use that as your function of what a
number line is, and what limits are, and what all of calculus is.

We can define a function, like y = x^2, and talk about the pairs of numbers
we can compute from that function, without every having to pretend that we
can "create all the values of line defined by the function".

We can talk about processes which find the mid point between any two points
on the line. That process is not infinite, it's finite. We can run that
process as many times as we want without end. But again, running it N
time, for any number N, is not infinity. It's just as many times as we
want to run it.

We can talk about what the slope of the line is at x=1 because we can take
a number dx, and calculate the slope of the line between the points f(x-dx)
and f(x+dx) and see what the limit of that is as dx approaches zero. We
can calculate limits without ever having to compute an infinite number of
points.

We can do all of calculus without anyone every having to compute an
infinite number of values.

So why do you think we can't do calculus, unless someone sits down, and
first, computes an infinite number of values?

Clearly, no one has ever done that, but yet, we still have all the
algorithms of calculus right? And none of the algorithms require anyone to
compute an infinite number of steps? So clearly, calculus is based only on
finite processes that mathematicians have been able to compute.

The problem, is that mathematicians have learned to be sloppy about this,
and they have been trained to _think_ in terms of an infinite number of
points "existing" when they never have exited, or needed to exist.

We can calculate 100 points on the line y=x^2, and talk about how there is
no limit to how many more points we can calculate. So we can understand
that there is no end to the number of points that could be calculated.
But, we take that, and change it around just a little, into pretending that
someone could actually calculate them all, when we say "all the points on
the number line _exist_".

It's this tricky little side step from "there's no end to the number of
points we could calculate", to saying "they all exist!", which is where it
all goes off the rails when it never needed to. Mathematicians have been
trained to believe that "can be calculated" is the same thing as "exist".

That is a fair liberty to take when you limit the things to be calculated
to a finite number, but it goes off the rail, when you use what's fair game
for a finite list, and use it on an infinite number of calculations.

> > > The complete enumeration of all rational numbers, such that no
> > > rational number is missing.
> >
> > But that can't exist. What is the advantage of pretending it can when
> > it clearly can't?
>
> They do exist. It is easy to show that a bijection exists between N and
> Q.

You have been well trained to use the word "exist" in the way
mathematicians have been trained to use it. But this is my very point, if
you don't limit "exists" to mean what does, or can exist in this universe,
you have taken a real word (exist), and turned it into total fiction.

It's like starting with a real word, like "human", and then using it to
mean not only real humans, but also these fictional characters that can fly
(like superman). When we right fiction, about a human that can fly, we
know we have stepped out of reality, and are just writing fiction for the
fun of it.

Mathematicians do this same thing, they start with reality, then at some
point, they step outside of reality, and start writing fiction - such as
when the introduced the axiom of infinity. It's the same thing as
introducing the axiom of superman like human flight. But for the most part,
no one, including the mathematicians, seemed to realize they had done that.
They walked over the line from reality, to fiction, with absolutely no clue
they had done it.

Then all sorts of problems start to show up, as they mixed realty with
fiction, that they couldn't really explain.

Like if you add superman flight to humans, we would be forced to ask
ourselves, why would superman walk anywhere? He might pretend to walk,
just to hide the fact he was superman. But in all the comics, superman
spends lots of time walking, even when he's superman. It's like why would
a fish walk on the bottom of the ocean? They answer is, they generally
don't. When you mix a reality that works, with fiction, all sorts of
nonsense falls out. You find yourself caught in your "lie" when all sorts
of holes show up.

This seems to me to be what happens to mathematics, once you mix in the
"lie" of the axiom of infinity.

> > Right. We can define a process that can index any given rational.
> > However,
> > the list of all rationals and their indexes can never exist.
>
> No
>
> > Right. But there are processes that never end, and we can use any of
> > these
> > processes as a definition. Such as the process that produces the
> > digits of
> > pi. We could call the digits produced by that process the definition
> > of pi. The infinite list of digits of pi can never exist, but the
> > process
>
> All of the digits do exist; they are all totally fixed regardless of the
> fact that they are not physically enumerated in the world.

Right, we are talking about the problem of how in math, the word "exist",
is used to mean "if a single number can be calculated, we are allowed to
say it exists". That is the problem. That is a lie which is common in
math but which is invalid in this universe.

It's a lie that comes from the fact that mathematicians had no clue how the
brain and thought worked, so they just made shit up that seemed "safe" to
be the foundation of mathematics - but what they thought was safe, has
turned out to be totally 100% bogus. It's just like making up a story
about the god Zeus being the cause of lightning and thunder. It's
harmless, until the point it stops being harmless. Which normally means,
it's harmless as long as it's not important what causes lightning. But
once understanding the true cause of lightning becomes important, it's no
longer harmless - its dangerous - like when someone refuses to install a
lightning rod because he believes Zeus will not be tricked by such a thing,
and that if Zeus decides to burn down your barn with lightning, a metal rod
is not going to stop Zeus from doing it.

Using the word "exist" so carelessly is in the same way, harmless, up until
the point it stops being harmless. It stops being harmless, the minute you
start messing around with the axiom of infinity.

> > can, and we can call the process the definition of the infinite list of
> > the
> > digits of pi. One must just be careful not to make the stake of
> > leaping from the existence of the process, to the existence of all the
> > digits however. It seems to me that this mistake is commonly made in
> > mathematics
>
> To get to the end of an unending process, just go through every step of
> the process. It is called a limit and it exists if the process
> converges. So, for example, take whatever algorithm you use to calculate
> every numeral of pi and the limit of that process is the value pi.

The value of PI does not exist. It can not exist. This is where your
careless use of the word "exist" changes from harmless, to dangerous (and
even to stupid when the person doesn't understand why the careless and
sloppy use of the word "exist" is harmful).

[WM]

On 25 Mai, 22:36, Uirgil <uir...@uirgil.ur> wrote:
> > Ordering objects is impossible without identifying them as distinct.
> > Reason: Unless you can identify objects as distinct, you cannot be
> > sure to have more than one (or even to have any).
> > ZFC "proves" that the set of real numbers is uncountable and can be
> > well-ordered.
>
> I was under the impression that ZFC merely assumed that
> well-orderability.

Zermelo has "proved" well-orderability, given the axiom of choice.
This axiom, however, is applied to lay the groundwork for mathematics.
And that is applied to shooting rockets to the moon as well as to
calculating the revenues of shareholders.

If it were only "assumed", then also the "foundations of mathematics"
were only _assumed_ to hold. How could such a fragile building supply
a the foundations of our daily life?

You see, it is very comforting that ZFC is only assumed to be sensible
and in reality has nothing to do with mathematics.

Regards, WM

[Curt Welch]

netzweltler <reinhard...@arcor.de> wrote:
> On 24 Mai, 20:33, c...@kcwc.com (Curt Welch) wrote:
> >
> > Has there been found any application in this universe where the act of
> > pretending infinity exists becomes useful?
> >
> > --

> > Curt Welch =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0
> > =
> =A0 =A0 =A0 =A0 =A0 =A0 =A0http://CurtWelch.Com/
> > c...@kcwc.com =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0 =A0
> > =A0=
> =A0 =A0 =A0 =A0 =A0http://NewsReader.Com/

>
> I can display every natural number n as a line of slope n in the
> coordinate system. w can be displayed as a vertical line then. Does
> infinity exist thus?

Nah. The slope is the output of a process of dividing two measurements -
the rise divided by the run. When the run becomes zero, the process either
becomes undefined, or never terminates (you get to pick). No where does it
become "infinity".

We use the word "infinity" to mean just that - a process that never
terminates. These processes most certainly do exist. An infinite number
of objects, do not exist. An infinite number of values output by one of
these processes will never exist. You must be clear as to which of these
two things you are talking about - but often in mathematics, these two very
different things, are conflated with very odd results happening as a
result.

The concept of a process that never terminates is very useful in
mathematics. It's the foundation of calculus for example. And to make
things simpler to talk about, we can just use the word "infinity" to mean
"process that never terminates". And when we use the word like that, then
that type of "infinity" can and does exist.

But when we then forget what short-hand we have been using, and try to act
as if "infinity" means "the never ending processes has finished", then odd
crap starts to happen.

We make this error in the diagonal argument for example. We wonder whether
a mapping from natural numbers to reals is possible, so we attack the
question backwards, by first assuming it does exist and we talk about as
the table. then we build the anti-diagonal, and make the claim the
antidiagonal can't be in any row of the table, so the mapping is not
complete. It all sounds logical, IFF, you make the error of assuming the
table exists in the first place! Which it can not. Arguments of that form
are only valid, in the case where the table of of finite size - where the
entire table exists, and can be completely scanned.

If we look at the argument again, without forgetting what we mean by
"infinity", we find this:

We build a process, which counts from 1 upwards. For each count, we add to
the process, a real number generator which adds a new number to the table,
which has not yet been added to the table. It works by checking all the
entries above it in the table, and then picking the first value not yet
seen above.

This is all we have. A definitive of a process that builds the table - not
the complete table. It's all we can have.

Now we try to make the diagonal argument by defining a process to construct
the anti-diagonal (using the normal sort of definition that is done for
this).

So now we have one process building the table, and a second process,
building the anti-diagonal.

If we look at the two processes, we find that no matter what the second
process outputs, the first process will also output the same thing, at some
point in the future, out the N digits in length, for any value of N.

So, we can prove that the anti-diagonal process is just a duplicate of the
first process. That no matter what it produces, it will only be
duplicating some row produced by the first process. So this method of
attack, fails to show that the first process has missed some important real
number.

The difference in these two arguments, is the second understands we are
dealing with never terminating processes, which act very differently from
finite lists of symbols. The diagonal argument is fine, for a finite list
of symbols, or for a process which generates a finite list of symbols, but
fails, when we try to apply it to processes which generate never-ending
lists of symbols. Never ending lists of symbols never terminates - they
have no end. Because of that, you can't not say the process never outputs
some infinite sequence. You can only ask whether it has YET to output some
finite subset of that infinite sequence.

Or, look at in this way. The two processes I defined above, are "fighting"
each other. The first, is trying to output a sequence which is not yet in
the table above, and the second, (the anti diagonal algorithm) is doing the
same thing. Every time the first algorithm produces a new row, the
anti-diagonal also does the same, but it gets to move second in this game.
By virtual of getting to move second, we guarantee that the N digits
produced SO FAR, by the second algorithm, will always be different than any
of the first N digits produced by the first algorithm SO FAR.

But we can also prove, that the first algorithm will always duplicate the
work of the second at a later point in time.

If the table was finite in length, then there would be "later point in
time" for the first algorithm to catch up to the second. The anti-diagonal
algorithm would "win" by virtual of getting to move second, once the first
algorithm had made it's last move.

But there is no last move to be made, so the second algorithm never gets to
"win" - the game of one-upmanship never ends between the two algorithms.

So the diagonal argument is only valid if the game ends at some opint - but
since it never ends, the anti-diagonal argument never "wins" over the first
argument.

The entire diagonal argument is flawed, simply because it makes the
assumption that the game does end - that the "table exists" (in it's
complete form), and then uses logic, that only applies to finite length
tables, to a problem of comparing processes that never end.

This is the subtle, but highly important difference that shows up between
infinity meaning "a process which never ends", and infinity meaning "the
process finished building the entire table and now we can run a second
process on the output of the first".

The process that generates the table exists - and grows without bounds.
But the table itself, never does get to "exist" in it's full form. We
can't use a logical argument about the table, that depend on it existing in
it's full complete form.

If we stick to only making arguments about the nature of processes, then we
find that there is only one type of infinity, and not a hierarchy of
different infinities. So it makes a big difference whether or not you
choose to step over that line from reality to fiction, and declare it ok to
assume infinity means "the process that never ends, finished all it's work
in finite time, and now, we have the infinite amount of output it produced,
held in a finite sized box which we can fully examine, or re-process, in
finite time as well". That is what the axiom of infinity ends up meaning.

--
Curt Welch http://CurtWelch.Com/

cu...@kcwc.com http://NewsReader.Com/

[Curt Welch]

Shmuel (Seymour J.) Metz <spam...@library.lspace.org.invalid> wrote:
> In <20120524213225.934$b...@newsreader.com>, on 05/25/2012
> at 01:32 AM, cu...@kcwc.com (Curt Welch) said:
>
> >But it seems to me, that once you do that, you have stepped
> >outside of reality and created endless self contradictions in
> >your system
>
> How it seems to you is irrelevant. Violating your aesthetics and
> prejudices does not prove a self contradiction. Showing that it is
> incompatible with some other system that you prefer is not good
> enough. You won't be taken seriously until you produce an actual proof
> of a contradiction.

Right.

As far as I can guess, the contradictions lie between reality and the
axioms, not between the axioms as they have been carefully defined by the
mathematicians. Math only concerns itself with happen after you make up
some axioms.

My issue is that some axioms create contractions with reality, which means
all work done with those axioms, are outside of reality - they are just
creative science fiction that could only apply in a different universe and
can have no real applications in our universe (other than for the role of
entertainment). I don't have any issues with people exploring science
fiction for entertainment, but it strikes me that when someone does that,
it would be better if they understood they were dealing with science
fiction, rather than some important aspect of reality.

I also am not convinced that the mathematicians have correctly created a
set of axioms without contradictions - but it would require far more
careful study of the work that I have any interest in to figure that out.
It's very hard to use the concepts from reality they borrow, and then add
in nonsense, and still end up with something self consistent - but it's
certainly possible they figured out how to write just that type of story.

In case it was not obvious to you, I do not read or post in sci.math. I
read and post in comp.ai.philosophy. It just so happens that some people
like to cross post to this group and when they do, I sometimes jump in and
comment.

[Curt Welch]

Shmuel (Seymour J.) Metz <spam...@library.lspace.org.invalid> wrote:

> In <20120524215637.243$Z...@newsreader.com>, on 05/25/2012
> at 01:56 AM, cu...@kcwc.com (Curt Welch) said:
>
> >You can't define existence by the convergence of an infinite process.
> >That's just absurd.
>
> Graham Cooper doesn't understand the definition of the reals. However,
> one common definition is that a real is an equivalence class of Cauchy
> sequences in the rationals, and one theorem that applies to the reals
> however defined is that every Cauchy sequence converges. Note that a
> sequence is *not* a process, and there is no process involved.

Well, that comment is an exmaple of abstracting out what you want from
reality, while at the same time, ignoring other aspects of reality
(pretending they don't exist and have nothing to do with what you are
using).

There are no sequences in reality that aren't also part of a process.
Sequences can't exist without first being created by a process, and second,
being detected and analyzed by a process. You can't just pretend the
process is not there, without escaping to a reality that is different from
this universe. Saying the process is not there, does not make it go away.

> >Some infinite processes converge _towards_ a thing that does exist.
>
> Define process and convergence of processes.

See physics for process. Everything studied by physics is the process.
It's what the universe "does"(/is). We normally abstract out some subset
of the universe and talk about the properties of that subset and refer to
that as a process.

I don't know how they defined convergence in math (I'm sure they have
multiple formal ways to do it), but I would attack it along the lines of
saying that a process that is outputting a sequence of measurements
(numbers) is said to converge towards a target if the distance between the
output value and the target grows smaller as the number of steps increases,
AND if for any arbitrary small difference DY, there is always some number
of steps N, where all the remaining distance to target values drops below
the value DY.

Note that there is no talk of the process reaching the target in this sort
of definition.

> >If you try to define rationals as what the processes converges to,
> >then we have the issue of 0.9999.. being a very different process
> >than 1.0000....
>
> Only if you have a really bizarre definition of convergence. IAC, it
> has nothing to do with convergence of sequences, which is what is
> relevant.
>
> >So now we have an infinite number of processes producing
> >different sequences, which must be claimed to be "the same real".
>
> The limit of a sequence is not the sequence.
>
> >What a disaster.
>
> If a_1 = 1 for all i, b_1 = 0 and b_i = 1 for all i>0, does it bother
> you that a and b both converge to 1? If so, why?

no.

A:
while true
print 1

B:
print 0
while true
print 1

Why would it bother me that these two programs exist? I don't understand
why you would ask this.

> >The diagonal "proof" is no longer a proof if we do this for example
>
> Wrong again, because it does not involve processes at all.

Well, the BELIEF that the proof can exist free of processes is exactly what
bothers me.

The brain is a process. If you removed all the brains from the earth,
where exactly on the earth, would the proof be located? The proof has no
existence separate from the processes that creates it, analyze it, or from
the concepts embodied in the proof.

The belief that it is process free, is exactly the problem I have with how
mathematicians have chosen to think about what they are working on.

[Curt Welch]

"K_h" <KHo...@SX729.com> wrote:
> "Curt Welch" wrote in message
> >news:20120524183145.077$1...@newsreader.com...
> > Abstractions exist. No pretending needed. To suggest they don't is to
> > make an error of duality - the error that the thoughts aren't physical
> > - that they don't have physical existence. An abstraction is just the
>
> Why are you bringing the mind-body problem into this thread??

Because this is comp.ai.philosophy and it's something I debate here all the
time.

The correct question to ask should be, why did someone cross post this
thread into comp.ai.philosophy!

If you do not want to get dragged into a debate about the mind body
problem, please remove c.a.p form the newsgroup list before you post (or
followup).

The mind body problem is the root of the error the mathematicians are
making in my view. If you choose to believe the mind is free from the
limitations of psychical reality (the foundation of the mind body problem),
then it's an easy step to think that when you are working on problems of
the mind (math), that there is no need to apply the limitations of the
physical world to your work.

This is what makes it so easy for them to slip into the idea that "thinking
about something" makes it exist. And that, "thinking about a set of
infinite size makes it valid to pretend it exists". And "we don't need a
physical process to make an idea come into existence". These are all
examples of the type of errors the mathematicians are making due to the
fact that either are dualists at heart, and believe the mind and thoughts
do not suffer the limits of physics, or, their lack of understanding of the
connection between mind and body, allows them to just carelessly ignore the
whole issue as if it were not important.

> > the abstraction is just a behavior of a pattern matching machine -
> > which must be real and physical to "exist".
>
> No. All digits of pi exist even though they are not physically
> enumerated.

Right, thoughts can happen even when there is no brain around to have the
thought? Is that what you are suggesting?

This is the problem I've talked about in may posts here now. In math, we
are trained to pretend "exist" in math means something separate from
"exists" in physics. But that's the mind body problem. There are not two
types of existence. There is only one. There is only physical existence.

For PI to "exist" it must have all it's digits calculated - which can't be
done. We can calculate it out to any level of precision we want, but we
can never calculate all of them.

> > Humans operate on probabilities. Nothing for us is certain. There are
> > no
>
> What about death and taxes?

:)

The most fundamental absolute (that I've every seen talked about or thought
of) is existence itself - the universe exists. Others would argue for their
brain being the most fundamental truth. (aka their consciousnesses). But
even these are not an absolutes. They are at best the reference signal by
which we measure the probability of all other things.

> > In the language of logic, we pretend absolutes can exist. We pretend
>
> It is an absolute fact that 12 + 13 = 25. It is absolutely true that the
> second root of 2 is 1.414213...

Yes, they are defined to be absolute truths - but just because someone
defines them to be absolutes, does not prove they are.

The fun example was the previous post where someone said like you did,
"this is an absolute truth...", then followed up sand said, "no I made an
error, that was not an absolute truth, this is...".

Truth exists as a communication event, and communication events always
happen with noise in this universe - and that noise, is exactly what
prevents any communication from being an absolute.

What we do, in day to day life, is PRETEND, that communication can be done
without noise but in this universe, that is not possible.

> > There could be troll under my desk right now. But I don't need to
> > prove
>
> There isn't.
>
> > gravity is highly likely to work the same way tomorrow, is not an
> > absolute,
>
> Why do you even think that gravity might behave differently?

I don't waste my time thinking that. But I do know, that all we have to go
on, is past experience. If something happens the same way 1000 times in a
row, we learn to expect it to happen that same way the next time. But
expecting it to happen the same way next time, is not the same thing as an
absolute truth that it will happen the same way next time.

Gravity is not an absolute truth, It's just a coin we have flipped a
million times, that came up heads every time. The odds of it being tails
on the next flip is never completely eliminated no matter how many times it
comes up heads.

> > So, in order for you to show my behavior invalid, you need to provide
> > the evidence (or at least some small hint of a suggestion about how and
> > where it might be hidden from us), that such a machine can exist in
> > this universe. You have to provide evidence strong enough to suggest
> > we should no long assume this is an absolute.
>
> The evidence can be found in physics. The quanta are in a superposition
> of an infinite number of all possible histories, until state reduction
> happens.

Or, more accurately, that is just an abstraction people currently use to
try and understand the result of some finite set of measurements.

> > to just "make up" the foundations of mathematics. The foundations of
> > mathematics, comes from the machine that does the mathematics. And the
> > foundations of that machine, come from the foundations of the universe
> > it exists in.
>
> The foundations of mathematics does not come from a machine. The human
> mind has discovered mathematical truth and set theory grounds that truth
> in self-evident axioms that all reasonable people agree to.

"all reasonable people" non of which have a fucking CLUE how their own
brain works!

They won't be really "reasonable people" until every one of them have a
completely understanding of how their own brain works.

You statement is the same as what we might have found in the past with "all
reasonable people know that the plague was an act of God".

I'm not saying there is something wrong with these people. I'm just saying
they are ignorant of factors that should be important to math, but which,
due to their ignorance they have chosen to ignore.

> > Mathematics, can't "exist" if there is no machine that can do the work.
>
> No. The mathematical truths pre-exist machines.

Yes, you have been well trained into believing such nonsense. It's
unlikely that a few posts by me on Usenet is going to have any real effect
on your life time of training by the "experts".

> > What strikes me as highly problematic about the current mathematics, is
> > that most of what I'm saying seems to be still be ignored by the field
> > (as far as I can see). They ignore it, because they are playing the
> > old (totally invalid) game of "thoughts are not physical, and have no
> > physical limitations". That's just not true. Thoughts have the same
> > limitations as
> > our computers. They can't, for example, count to infinity.
>
> Nobody has solved the mind-body mystery.

That's not true. But that is also a common belief in our society. Most
people believe the mystery has not been solved, and some believe it can't
be solved. Only a small subset of people understand it has been solved, or
more accurately, there is actually nothing there "to solve". The confusion
that leads to people believing there is a puzzle to solve, is just a simple
illusion - a perception error made by the brain that leads to the common
belief in duality.

> So the possible physicality of
> "thoughts" is currently an open question. Roger Penrose has some ideas
> about it as do others.

Yes, many have ideas, most are stupid.

> > of the territory. They act that since they can say "the infinite set
> > exists", then that's all they need to do to prove that "it does exist".
>
> It is a self-evident truth that the naturals exist just like all the
> digits of pi exist.

It's a self evident truth that God exists but it's an error none the less.

Being self evident doesn't make it true.

> They are not physically instantiated, obviously, but
> they don't have to be.

That is exactly right. We can talk about Pi, we can use the concepts of pi
to our advantage, wither the value every being instantiated.

I do a lot of blacksmithing these days, and funny enough, for Pi, I use 3.
That's all he accuracy I need to do my work! And it's common when
calculating the length of collars to use 2 for PI! This is because in
blacksmithing, when we calculate lengths, it's better to be short, than
long, because it's easy to make something longer (hit it a few more times)
than to make it shorter.

PI exists as a concept - a concept about how to do things like calculate
the length of material needed to form a circle 10" round. At no point in
time can we, or do we, attempt to calculate all the digits of pi.

> > It's as useful as starting a project by saying something like, "Assume
> > all elephants can fly - but they have just never done it for a human to
> > see",
>
> Stop trying to denigrate mathematics.

Stop trying to make me stop denigrating mathematics! :)

I'm not trying to denigrate mathematics. I'm trying to point out that the
field is missing an important opportunity to clean up it's thought process.
It needs to move beyond it's 19th and 20th century thinking.

If people don't want to hear these thoughts, they should not be posting
math debates to comp.ai.philosophy where I read and post from. :)

[Uirgil]

In article
<d9f18d6a-c9b3-41d8...@cu1g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 25 Mai, 22:36, Uirgil <uir...@uirgil.ur> wrote:
> > > Ordering objects is impossible without identifying them as distinct.
> > > Reason: Unless you can identify objects as distinct, you cannot be
> > > sure to have more than one (or even to have any).
> > > ZFC "proves" that the set of real numbers is uncountable and can be
> > > well-ordered.
> >
> > I was under the impression that ZFC merely assumed that
> > well-orderability.
>
> Zermelo has "proved" well-orderability, given the axiom of choice.
> This axiom, however, is applied to lay the groundwork for mathematics.
> And that is applied to shooting rockets to the moon as well as to
> calculating the revenues of shareholders.
>
> If it were only "assumed", then also the "foundations of mathematics"
> were only _assumed_ to hold. How could such a fragile building supply
> a the foundations of our daily life?

That puzzled Einstein, too, but according to him, it does.

>
> You see, it is very comforting that ZFC is only assumed to be sensible
> and in reality has nothing to do with mathematics.

When so large a proportion of all mathematicians agree that ZFC has
something to do with mathematics, no pipsqueak like WM can make that go
away.

[Graham Cooper]

On May 27, 5:49 am, c...@kcwc.com (Curt Welch) wrote:

> "K_h" <KHol...@SX729.com> wrote:
> > "Curt Welch"  wrote in message
> > >news:20120524183145.077$1...@newsreader.com...
> > > Abstractions exist.  No pretending needed.  To suggest they don't is to
> > > make an error of duality - the error that the thoughts aren't physical
> > > - that they don't have physical existence.  An abstraction is just the
>
> > Why are you bringing the mind-body problem into this thread??
>
> Because this is comp.ai.philosophy and it's something I debate here all the
> time.
>

You're right about use of abstractions and terminology taken for
granted, calculus' restricted use of "->oo" and "LIM...=oo" only, and
a bunch of other things.

But there is a hard problem involved behind all this!

Engineers, physicists and mathematicians alike express such things as:

Does "abc123" Exist in this system?

START -> LETS NUMS
LETS -> a TAIL
TAIL -> b FIN
FIN -> c
NUMS -> ZERO NATS
NATS -> 1 | (1+NATS)


Proof By Contradiction has a simpler cousin, Proof By Witness.

i.e. when solving an equation there must EXIST a value of the variable

GO FORWARD 50 years of Formal Mathematics...

1 <-> {1,2,3}
2 <-> {5,6,7}
3 <-> {5,4,3,2,1}
4 <-> {99,98,97}
...

Any Enumeration of a CONCISE FORMAL ORDERED MATHEMATICS LIST OF
FORMULA

has a missing Element.

{2, 4, ...} is missing from the INFINITE LIST of SUBSETS!

"2 is not in the first subset, so it can't be subset 2.."

So |POWERSET(N)| > |N|

i.e. the SIZE/CARDINALITY of all subsets of N appears BIGGER than
INFINITY.

Herc

[K_h]

"Curt Welch" wrote in message news:20120526154928.543$U...@newsreader.com...

>
> infinite size makes it valid to pretend it exists". And "we don't need a
> physical process to make an idea come into existence". These are all
> examples of the type of errors the mathematicians are making due to the

These are not errors; the axiom of infinity is really a self-evident truth.
But the mind body problem and its relationship to Platonism is a very
complex issue which does not instruct people on self-evident truths such as
the counting numbers.

> Right, thoughts can happen even when there is no brain around to have the
> thought? Is that what you are suggesting?

Well, nobody really knows. But regardless of what the answer is, axioms in
mathematics are not determined by whatever one thinks the answer to this
question is.

> This is the problem I've talked about in may posts here now. In math, we
> are trained to pretend "exist" in math means something separate from
> "exists" in physics. But that's the mind body problem. There are not two
> types of existence. There is only one. There is only physical existence.

But that is just a philosophical opinion of yours. Platonic existence and
physical existence are two different things. I don't think anybody is
claiming that perfect circles physically exist but they obviously exist
platonically.

> For PI to "exist" it must have all it's digits calculated - which can't be

No. This was already explained to you previously by myself and others in
this thread.

> > It is an absolute fact that 12 + 13 = 25. It is absolutely true that
> > the
> > second root of 2 is 1.414213...
>
> Yes, they are defined to be absolute truths - but just because someone
> defines them to be absolutes, does not prove they are.

12 + 13 = 25 is an absolute truth. Anyone who disagrees with that is out of
touch with reality.

> > The evidence can be found in physics. The quanta are in a superposition
> > of an infinite number of all possible histories, until state reduction
> > happens.
>
> Or, more accurately, that is just an abstraction people currently use to
> try and understand the result of some finite set of measurements.

No.

> "all reasonable people" non of which have a fucking CLUE how their own
> brain works!

Not true. Most mathematicians do know the basics about the nervous system,
etc..

> They won't be really "reasonable people" until every one of them have a
> completely understanding of how their own brain works.

That's absurd. Nobody has a complete understanding of the brain so you are
making the bogus claim that nobody is reasonable.

> > No. The mathematical truths pre-exist machines.
>
> Yes, you have been well trained into believing such nonsense. It's
> unlikely that a few posts by me on Usenet is going to have any real effect
> on your life time of training by the "experts".

It is not nonsense. What you are claiming is nonsense. Nobody in their
right mind disbelieves 12+13=25.

> > Nobody has solved the mind-body mystery.
>
> That's not true. But that is also a common belief in our society. Most

No. Even the brain experts are trying to solve this problem and they
acknowledge that is is unsolved. Discover magazine called it the `holy
grain of neuroscience' and it hasn't been found yet.

> It's a self evident truth that God exists but it's an error none the less.
>
> Being self evident doesn't make it true.

Wrong. It is not self-evident that God exists but self-evident things are
true.

> I'm not trying to denigrate mathematics. I'm trying to point out that the
> field is missing an important opportunity to clean up it's thought
> process.
> It needs to move beyond it's 19th and 20th century thinking.

You are just trying to impose your (false) philosophical notions onto people
in these newsgroups. You won't succeed because your ideas are incorrect.

+

[Curt Welch]

Graham Cooper <graham...@gmail.com> wrote:

> On May 26, 10:51=A0am, c...@kcwc.com (Curt Welch) wrote:
> > Graham Cooper <grahamcoop...@gmail.com> wrote:

> > > On May 25, 8:31=3DA0am, c...@kcwc.com (Curt Welch) wrote:
> > > > Shmuel (Seymour J.) Metz <spamt...@library.lspace.org.invalid>
> > > > wrote:
> > > You can KNOW 50% of position and 50% of momentum!
> >
> > > There are no AXIOMS to ascertain the universal fact that Russell's
> > > Set cannot exist.
> >
> > > ~E(R)xeR<->xex
> >
> > > this is an ABSOLUTE FACT!
> >

> > Then, tell me, what is a fact? =A0What does that word mean? =A0That is,
> > w=
> hat
> > does it mean in terms of physics. =A0When is are some atoms
> > representing =
> (or
> > creating, or whatever) a fact? =A0Where do we find "facts" in the the

> > behavior of atoms (without a human in the system to declare or test the
> > fact).
> >
> > And then, what is an absolute fact?
>
> a true statement without assumptions
>
> >
> > We can use language to make up nonsense, but for the language we

> > produce =

> to
> > have any purpose or use, we must create meaning, that is defined not in
> > terms of axioms, but in terms of of the physical universe.
> >
>
> which runs like clockwork, according to a small subset of mathematics
>
> > Axioms have no meaning, if their meaning is not first grounded to

> > physica=
> l
> > reality. =A0Any axiom we try to make up, is in fact, grounded in

> > concepts from physical reality.
> >
> > One concept however that people make a mistake about, is the concept
> > that "absolute truths can exist".
>
> of course they do!
>
> you cannot dispute you are reading words right now!

Oddly enough, I can. The odds that I'm not reading (and writing) words
right now is incredibly small, but still greater than zero.

When I read your question, I form an internal state in my brain which
represents what my brain has decided the meaning of the words are. This is
a perception process. My brain classifies the sensory signals as belonging
to some defined state. Though the odds of me having perceived the words
incorrectly is small, the odds are greater than zero. I might have failed
to correctly read the words. We all known examples where we failed to read
words correctly (or at least, you should be aware of it).

On a larger scale, the same problems exists. How do I know that I'm
currently sitting in my office, on a Saturday afternoon, reading and
writing a Usenet post? Again, this is all a brain perception problem. My
brain brain has translated a long stream of sensory data and come to the
conclusion that it indicates that I am in my office, reading and writing a
Usenet post.

When the brain is functioning correctly, it's perception system works very
well. It can correctly transform highly noise sensory data, into a very
rich and full and accurate representation of the environment. Only for
highly noisy and sparse data problems that occurs at the edge of the
ability to perceive does it make mistakes - but most the time, it is also
aware that the quality of perception is in question (as when we try to read
words that are too small to read accurately).

We become so accustomed to the amazingly accurately powers of perception
our brains possess that the perception action tends to fade away, leaving
us only with the sense of automatically "knowing" what the state of our
environment is. We in general, have no real understanding that the state
was actually calculated by a machine. It did not enter our mind by
osmosis. It was calculated by a process from noisy sensory data.

And that calculation can be in error - at any point in time.

When it really starts to mess up, people have hallucinations. They become
psychotic. The brain believes things are happening that simply are not
happening. Most people that suffer from this, have no understanding of
what is happening. They can't cope with the idea that the brain is
indicating something happening that is not happening. I had a friend of a
friend that suffered from mild issues like this. She would believe she had
seen someone, when noone was there. If you try to talk to her about the
fact there was no one there, she would reject your suggestion saying "I
know what I saw". The idea that her brain ad "made it up", was something
she could not accept - or understand.

So, when you suggest "you cannot dispute you are reading words right now!",
I most certainly can. I don't know if I'm really reading a post by you, or
if I just had a stroke 5 minutes ago, and I'm just laying on the floor
passed out, and in the middle of some psychotic episode where I believe I'm
reading your words, but in fact, I'm reading nothing - my brain is just
making all this up. There is always a non-zero chance that we are wrong
about anything we think we "know".

> >
> > You have not proven above that issue of whether they exist or not.

> > =A0You have just stated the opinion that they do.

>
> No I thoroughly explained
> 1 AXIOM OF SET SPECIFICATION
> 2 PROOF(THEOREM) PREDICATE
> 3 TABLE OF TAUTOLOGIES
> 4 TABLE OF THEOREMS
>
> in www.tinyurl.com/SETAXOIMS

That would be www.tinyurl.com/SETAXIOMS

> and detailed the difference between a statement proven inside and
> outside a Theory.

Your writing doesn't mean much to me.

However, it's not important. You need to understand that thinking is a
communication process - as much as trying to write a Usenet post is a
communication process. And just like you (and I) are unable to write a
posts without making mistakes, we are also unable to _THINK_ without making
mistakes. All communication in this universe includes noise - and that
noise means no communication is absolutely prefect.

For an absolute truth to EXIST - the universe would have to support
ABSOLUTELY prefect communication. Our universe does not do that.

What we can do however, is act so as to reduce the errors to an incredibly
small level - as we do when we design our computers. We build them so as
they are able to communicate a digital value from one part of the machine,
to another, billions of times in row without making a communication error.
We use large amounts of redundancy to achieve this effect. We do it not
only in our computers, but in how our language works. It is what allowed
me to understand what you really mean in the URL above, even though you got
it wrong - even though you had a communication error.

Our brain is makes massive use of redundancy to allow it to very accurately
decode the correct state of the environment. So when I think I'm reading
your words, the odds are very high, that I am n fact reading your words.

But it is NOT AN ABSOLUTE TRUTH. Absolute truths do not exist. We just
choose to pretend they do, when we do math.

> > Generally speaking, all of mathematics is based on the belief that

> > absolu=
> te
> > truths can exist. =A0It's an implied axiom of the entire field. =A0But
> > it=
> 's
> > wrong. =A0And as such, one must be careful in understanding the
> > separatio=

> n
> > between, reality, and mathematics.
>
> You've proven yourself to write unreliable statements if that is the
> case.

That's true. Just like you have shown the same ability, by for example,
getting the URL wrong above. This is just normal human behavior.

We train ourselves to minimize such errors - some of us far more so than
others. some of us seem to fear being sloppy. Others, like myself,
embrace the sloppiness at times.

> >
> > The most common belief of absolutes in mathematics is the belief that

> > humans have absolutely perfect perception. =A0That it's 100%
> > impossible, =

> for
> > language to misunderstood.
> >
> > If we say the symbol 1 is absolutely and always different from the
> > symbol 2, then we are also saying that perception of the symbol by all
> > machines that attempt to read and understand them, will be 100% perfect

> > 100% of th=
> e
> > time for the rest of eternity. =A0But that is never an absolute.

> >
>
> See: IFF every Natural Number was a SET in sci.logic
>

> ALL(x) ALL(y) ( ALL(a) TM-x(a)=3DTM-y(a) ) -> x =3D=3D y

>
> AXIOM OF COMPUTABLE EXTENSIONALITY
> (equivalent programs have the same output for every input)

Yes, we pretend computers are perfect and never make an error. They are
not. We just pretend they are in how we think about them (which is ok,
because when they are working correctly, they are so close to prefect it
doesn't cause us much of a problem for our normal use of computers). Some
applications however have to carefully track how likely a computer error
is - like when you use computers to control machines that can kill people
when they malfunction.

> > To make the statement that "1 !=3D 2" (absolutely), we are also
> > simplicit=

> y
> > making the statement that the language sequence will never be

> > misundersto=

> od
> > - because without someone's perception when they read the sentence, the

> > meaning of the symbols don't exist - the math doesn't exist. =A0Only
> > ligh=
> t
> > and dark spots on a screen exist. =A0It doesn't become math, until a
> > perception system reads it and interprets it. =A0And if there is any
> > chan=

> ce a
> > perception system will fail to read it correctly, then the light and
> > dark marks on the screen do not form in any sense an "absolute" truth.
> >

> > Mathematicians choose to ignore all this. =A0They choose to ignore the
> > fa=

> ct
> > that math doesn't just happen by the magical will of God (or the will
> > of the mathematician). "math" itself, exists only one way - as a

> > physical process. =A0Math is a physical process even though due to a

> > lack of understanding of the mind, mathematicians have chosen to
> > believe math exists somehow separate from the mind and separate from
> > any physical limitations of the body.
> >
> > But that is exactly where math goes off the rails and wanders into pure
> > useless nonsense.
>
> See my above post.
>
> These are the underlying assumptions of construction that allow the
> self-inconsistency of a specific syntactic form to be extended to a
> metaphysical assertion of reality.
>
> i.e. using FORALL(SETS) but only EXIST(MISSING_SET)
> or MISSING_REAL
>
> leads to people like Virgil stating SIZE(R)>INFINITY

I don't understand what you are saying there, but it sounds like you are on
to something valid.

> > > > All we have, are events which are highly likely, and highly
> > > > unlikely.
> >
> > > > It's highly unlikely that gravity will work differently starting

> > > > tomorrow for example. =3DA0But it's not an absolute fact that it

> > > > will work the same tomorrow.
> >
> > > > What we do however, is _pretend_ such high probability events are
> > > > absolutes, because they are so close to being absolutes, it would

> > > > be =

> a
> > > > waste of time and computational energy in most applications, to do

> > > > anythi=3D

> > > ng
> > > > other than plan our future based on the assumption gravity will
> > > > keep working the same.
> >

> > > > In the language of logic, we pretend absolutes can exist. =3DA0We
> > > > pre=
> tend
> > > > "true" and "false" are valid absolute axioms. =3DA0We pretend they
> > > > ex=

> ist
> > > > as
> >
> > > They DO EXIST as solidly as HOT and COLD exist.
> >

> > The concept of an absolute exists. =A0Just as the concept of pink
> > flying elephants exist. =A0The elephants however don't exist, and
> > neither do the absolutes. =A0It is easy to use language to define
> > things that are nonsen=

> se.
> >
> > > Here is the INEVITABLE SEQUENCE.
> >
> > > What is Truth?
> >
> > > IN THE BEGINNING..
> > > there was the physical world
> > > from that came evolution
> > > from that came event associated positive and negative awareness
> >
> > Ok, but it seems that you have assumed a form of absolutes here that is

> > t=
> he
> > start of all the problems. =A0What came from evolution, was
> > probabilistic event responses, not "positive and negative". =A0Positive
> > and negative implies absolute differences. =A0Human behavior is only
> > understood (and produced) probabilistically. =A0We can not do anything

> > absolutely.
> >
>
> No, Schrodingers Cat is not a Theory In Physics.
>
> It's a hypothesis because the exact mathematics of quantum state
> reduction is unknown.

ok.

> I am strongly going to suggest that you look at orthomodular logics
> and quantum logics for this in addition to Bayesian decision models
> and whatever else you find. I had been looking a Paul Halmos paper on
> Hilbert spaces and discovered one of the inadequacies of quantum logic
> for quantum mechanics--the Heisenberg uncertainty principle cannot be
> represented.

What do you mean by "can not be represented"? I'm not trying to question
you, or catch you in some error, I just don't follow what you are trying to
say.

> > > then also came language
> > > from that came proposition associated true and false properties
> > > from that came predicate() associated true and false properties
> > > from that came logic
> > > from that and the physical world came computers
> >

> > Yes, but even computers are not absolutes. =A0They are not guaranteed
> > to produce the answer 2, every time they add 1 + 1. =A0The sometimes
> > make mistakes. =A0We build them so as to reduce the odds of that type
> > of mista=

> ke
> > to such low levels that the machines remain practically useful to us,
> > but good enough to be practically useful is not the same as having
> > absolute behaviors.
> >
> > Math is based on the abstraction of absolutes, which is fine, but it
> > must be kept in mind, when doing math, that what the abstraction
> > represents, does't actually exist in this universe.
> >
> > For most of basic math, this difference between reality and the

> > abstractions used by math, doesn't really cause any issues. =A0We can
> > sti=

> ll
> > do math correctly, most the time, just like a computer can add numbers

> > correctly, most the time. =A0But when we get down to debating the
> > foundat=

> ion
> > of mathematics, ignoring the difference between the reality of the

> > machin=

> es
> > in this universe that "do math", and what math defines itself to be,
> > must the same, or else, the split with reality creates endless

> > pointless nonsense. =A0It's just becomes made up fiction, and not fact.

> >
>
> ABSOLUTE is just "in all domains"
>
> ALL (x) ..... x ....
>
> is just ..... x ....
>
> Removing ABSOLUTE from the lexicon is fruitless.

There is no such thing as absolute language. It does not exist. All
language is riddled with errors at some level of probability. It makes no
difference what you write - nothing you write is absolute. Nothing you can
write about is absolute.

You can not create something which is absolute, if you don't have some type
of "absolute" material to build it from. You do not have that material.
Everything you work with in this universe is non-absolute.

We certainly can talk about something being absolute, and pretend our talk
is a absolutely true statement. We do this all the time in math. But
pretending it's a true statement, does not make it true. Trying to create
by saying it's an axiom of absolute truth, does not mean that absolute
truth exists or can exist in this universe.

The _idea_ of absolute truth most definitely does exist. It's the idea we
are talking about here. But an idea is only a reference to something else.
The idea of a pink flying elephants does not mean pink flying elephants can
exist.

Generally speaking, all of math and logic grows from the assumption that
absolute truth exists. It's an exploration of a big "what if" of whether
absolute truth exited. We can explore the idea of absolute truth, even
though we can't actually have it. That is what we do in mathematics and
logic.

To say 1 + 1 = 2 is an absolute truth, assumes that our ability to read the
symbols "1" and "+" and "=" and "2" is an absolute. It assumes the meaning
of the symbols are n themselves an absolute. They aren't. We don't have
any way to communicate an absolute truth. We don't have any fundamental
system that is able to create an absolute truth, on which we could build
the rest of mathematics.

But we can put forth (with high probability), the IDEA of an absolute truth
existing, and then explore (again with high probability), the consequences
that would result if this WERE a truth. This is what we do in mathematics.

But, when we are done, we must realize that all our work, has to be
"adjusted" for the fact that we can't actually create this process of
manipulating absolutes, in this universe. This does not stop us from
making good use of math in day to day lives, it's just a fact we have to
deal with. The fictional world of absolutes we play with in math (and
computer science) is only that - a fiction which happens to be close enough
to realty, to make it very useful.

[Curt Welch]

That is correct, but not relevant to whether he is right or wrong.

[K_h]

"Curt Welch" wrote in message news:20120526001338.545$w...@newsreader.com...

>
> concept of events happening in a sequence). They hide the contradictions

What contradictions?

> They form a set of axioms, which can't be implemented, but yet they act as
> if it's valid to pretend they can be implemented and that they are valid
> axioms when they are not.

More unsubstantiated assertions. What axiom do you think is invalid and
why?

> they have diverged from reality, into pure fiction - just making stuff up
> that can't exist, and pretending that it can to have something fun to talk

Baloney.

> Clearly, no one has ever done that, but yet, we still have all the

No, it has been done. People have added an infinite number of terms in
calculus without enumerating every term. So add up 1/2 plus 1/4 plus 1/8
plus 1/16 plus 1/32 and so on ad-infinitum and the result is 1. There, I
just did it.

> It's this tricky little side step from "there's no end to the number of
> points we could calculate", to saying "they all exist!", which is where it
> all goes off the rails when it never needed to. Mathematicians have been
> trained to believe that "can be calculated" is the same thing as "exist".

There are all sorts of numbers that exist that cannot be calculated. Every
digit of pi exists and is totally fixed but nobody can ever enumerate them
all.

> you have taken a real word (exist), and turned it into total fiction.

Untrue. The axiom of infinity is not fiction. All naturals exist. Anybody
with common sense can see that simple self-evident truth.

> Using the word "exist" so carelessly is in the same way, harmless, up
> until
> the point it stops being harmless. It stops being harmless, the minute
> you
> start messing around with the axiom of infinity.

Who is being harmed by the axiom of infinity (LOL )??

+

[K_h]

"Curt Welch" wrote in message news:20120525231626.639$D...@newsreader.com...

>
> You just need to understand that where you have used "infinite set" in the
> past, you must substitute "process that never terminates". All of
> calculus

Mathematically, infinite `processes' can sometimes terminate. Consider an
algorithm that produces the nth digit of pi at time t=1-(1/n) for the nth
digit of pi. For example, the first digit of pi, 3, is produced at t(n=1)=0
and the second digit, 1, at t(n=2)=1/2. The process is terminated at time
t=1 with all aleph_0 digits produced. Infinite sets are obviously needed in
mathematics. Just consider the set of locations in the unit interval --
obviously there are an infinite number of such locations.

+

[Graham Cooper]

> c...@kcwc.com                                        http://NewsReader



Curt your philosophy is old hat. The brain in a VAT experiment was a
movie called THE MATRIX. It led to nowhere.

Your argument is cyclic.

CURT |- Deny(Universal Language) <-> Deny(100% Communication)
<-> Deny(Concrete Observations) <-> Deny(Abstractions)
<-> Deny(Concepts) <-> Deny(Truth) <-> Deny(Certainty)

You're no better than the Mathematicians in denial that LOGIC formula
are all countable.

SCI.MATH |- !E(U)<->!E(RS)<->E(GS)<->E(t)t~eT<->E(p)p~eHALT

<->E(r)r=/=fn<-E(s)s~ePS(N)<->~E(U)

This is the cyclic argument you are disputing,

all the same antidiagonal 'proof'!

SCI.MATH are disputing a consistent Universal System exists.
Now you are disputing a coherent Universal System exists too!

'With this formulation, the language can support reference to a

greatest class. The assertion of "almost universality" ensures that
"proper classes" different from the class universe cannot be referred

to within the theory.'

i.e. people on drugs watching THE MATRIX aren't considered legally
responsible to have communication.

Herc

[Curt Welch]

"K_h" <KHo...@SX729.com> wrote:
> "Curt Welch" wrote in message
> >news:20120526001338.545$w...@newsreader.com...
> > concept of events happening in a sequence). They hide the
> > contradictions

> No, it has been done. People have added an infinite number of terms in
> calculus without enumerating every term. So add up 1/2 plus 1/4 plus 1/8
> plus 1/16 plus 1/32 and so on ad-infinitum and the result is 1. There, I
> just did it.

I really don't have a clue how to respond to that.

> Who is being harmed by the axiom of infinity (LOL )??

:)

The truth is being harmed. :)

> +

[Uergil]

In article <20120527005538.532$i...@newsreader.com>,

cu...@kcwc.com (Curt Welch) wrote:

> "K_h" <KHo...@SX729.com> wrote:
> > "Curt Welch" wrote in message
> > >news:20120526001338.545$w...@newsreader.com...
> > > concept of events happening in a sequence). They hide the
> > > contradictions
>
> > No, it has been done. People have added an infinite number of terms in
> > calculus without enumerating every term. So add up 1/2 plus 1/4 plus 1/8
> > plus 1/16 plus 1/32 and so on ad-infinitum and the result is 1. There, I
> > just did it.
>
> I really don't have a clue how to respond to that.
>
> > Who is being harmed by the axiom of infinity (LOL )??
>
> :)
>
> The truth is being harmed. :)

Not at all, because what is derived from an axiom system is only held to
be true IF the axioms are all true, and does not impose any of its
conclusions on any "reality" in which they are not all true.

But those, like WM, who cannot, or will not, come up with any complete
set of axioms for anything cannot claim to have any certainty of truth.

That is the great virtue of starting from an axiom system, you never
have to claim anything you cannot prove in the form:
"If my axioms are all true then...."
--
"Ignorance is preferable to error, and he is less
remote from the- truth who believes nothing than
he who believes what is wrong.
Thomas Jefferson

[Curt Welch]

"K_h" <KHo...@SX729.com> wrote:
> "Curt Welch" wrote in message
> >news:20120525231626.639$D...@newsreader.com...
> > You just need to understand that where you have used "infinite set" in
> > the past, you must substitute "process that never terminates". All of
> > calculus
>
> Mathematically, infinite `processes' can sometimes terminate. Consider
> an algorithm that produces the nth digit of pi at time t=1-(1/n) for the
> nth digit of pi. For example, the first digit of pi, 3, is produced at
> t(n=1)=0 and the second digit, 1, at t(n=2)=1/2. The process is
> terminated at time t=1 with all aleph_0 digits produced.

That process can't exist because it assumes the next digit of pi can always
be calculated in an arbitrary small unit of time. That's just not possible.
It assumes there is no time limit to how fast a digit of PI can be computed
- that is equally as bad as assuming it can be done in zero time.

> Infinite sets
> are obviously needed in mathematics. Just consider the set of locations
> in the unit interval -- obviously there are an infinite number of such
> locations.

Your logic is begging the question. You are assuming your conclusion, and
using that assumption to try and support your conclusion as "obvious".

This is because you have been trained by your culture to think as a
mathematician thinks. You believe the space you exist in is made up of
points defined by infinite precision real numbers.

Points in space don't exist however. The idea that they do, is only an
abstraction created by mathematicians.

How do you know if something exists at a point in space? What is the
location of a grain of sand? what is the location in space of an electron?
Does it exist at some exact location? How would you know if it didn't?
How do you know if the location of elections in space are discrete instead
of continuous? How do you know if the space we exist in is discrete and
not infinity continuous?

The answer is, we can't know. We can only measure location down to some
finite limit of precision. We can't measure location of anything down to
an infinite precision - a real number of infinite length. If we can't
measure it, then we can't know if it exists. We don't get the make stuff
up just because we want it to be that way. We only get to know what is
knowable. And what is knowable about the space we exist in, is that we
can't measure location to an infinite precision, and as such, we don't know
how many "points" in space exist between two locations.

But yet, you think that if we consider the number of points between two
locations, that it is "obvious" the number is infinite. What you call
"obvious" is in fact unknowable. It's only obvious, because you have been
trained by your culture to BELIEVE that space is infinity continuous.
Locations in space could very well be quantized, and not continuous. The
idea of euclidean space being infinitely continuous is just that - an idea
created by the same people who created the idea of the axiom of infinity.
It's an unprovable assumption, not the obvious fact you try to make it out
to be.

Space doesn't need to be infinitely continuous, for everything in physics
to work exactly how it seem to work for us. It could easily be quantized
at a level below what we can measure. The fact that we use abstract models
of infinitely continuous space, does not mean the space that we exist in,
actually works (exists) that way.

Your logic is begging the question, because you start by assuming space is
infinitely continuous, and then you show that it's "obvious" that space of
the type you are assuming we exist in, is as you assumed it is. No where
do you provide evidence to show your assumption is valid.

[Uergil]

In article <20120527014358.924$t...@newsreader.com>,

cu...@kcwc.com (Curt Welch) wrote:

> Your logic is begging the question

Without begging some questions, i.,e., making some assumptions about
something, there cannot be any mathematics, since there is nothing of
objective physical evidence that anything mathematical can or does exist.

[WM]

On 27 Mai, 06:55, c...@kcwc.com (Curt Welch) wrote:

> "K_h" <KHol...@SX729.com> wrote:
> > "Curt Welch"  wrote in message
> > >news:20120526001338.545$w...@newsreader.com...
> > > concept of events happening in a sequence).  They hide the
> > > contradictions
> > No, it has been done.  People have added an infinite number of terms in
> > calculus without enumerating every term.  So add up 1/2 plus 1/4 plus 1/8
> > plus 1/16 plus 1/32 and so on ad-infinitum and the result is 1.  There, I
> > just did it.
>

You reached the state after that which is never reached?
Congratulations!

I, for one, can only calculate the limit of the series, never sum up
all its terms.

Why don't you prove your skills with non-converging sequences like
1 - 1 + 1 - 1 + 1 - 1 +-...?
If being able of what you boasted, then you should know the result.

> I really don't have a clue how to respond to that.
>
> > Who is being harmed by the axiom of infinity (LOL )??
>
> :)
>
> The truth is being harmed. :)

So it is.

Regards, WM

[Curt Welch]

Uergil <Uer...@uer.net> wrote:
> In article <20120527014358.924$t...@newsreader.com>,
> cu...@kcwc.com (Curt Welch) wrote:
>
> > Your logic is begging the question
>
> Without begging some questions, i.,e., making some assumptions about
> something, there cannot be any mathematics, since there is nothing of
> objective physical evidence that anything mathematical can or does exist.

Your wording there is a little confusing to me, but it sounds like you are
saying there is no objective physical evidence that math exists.

If that is what you were saying, you are wrong. Pick up a math book.
That's physical evidence that math exists. Read the posts in sci.math.
That's physical evidence that math exists. Pick up a calculator. That's
physical objective evidence that math exists.

The reason anyone would think math existed in some non-physical domain, is
just again, the mind body problem. People have long believed and accepted
the idea that the thoughts are not physical, but yet, they "exist". This
gives them free license to separate the meaning of the word "exist" from
"physical existence". It gives them free licence to believe that "math
exists" in a domain separate from the physical.

That's all wrong. There is only one type of existence. Physical and
mental existence are the same thing. Or, more accurately, the mind is just
just the brain and the brain is just a subset of all physical existence.
This belief that there are two separate types of existence is an error that
grows from a simple illusion in the brain. But it's ALL WRONG. It's all
an error.

This mind body error is not something that just happens to touch on the
field of math, the error is the FOUNDATION of the field of math. Math, as
it's often practiced, is foundational flawed, because those that practice
it have talked themselves into believing math "exists" separate from
physical reality, and due to that separation, nothing in math (so they
believe) need be limited to the constraints of the physical world. They
believe that anything they can imagine, "exists" in math.

If one can imagine the concept of a finished infinity (we can), then they
believe the finished infinity is a valid part of "math". It is not. It's
just fictional nonsense that is made to look valid, by obscuring the
objective facts that show it invalid.

In physics, people know better than to believe something exists just
because they can imagine it. Existence is only defined by objective
evidence - by measurements. In math, this is not known. People think they
are free to make up anything they want, and call it the "foundation of
math".

Yes, we are free to make up anything we want, but we are not free to call
it the foundation of math. If you make stuff up like this, and call it the
foundation of math, when it's not the actual physical foundation of math,
then your concept of what math "is" will be totally wrong. For someone
that calls themselves a mathematician, I'm surprised more of them don't
care whether they have the foundation of their work right or wrong.

[Ross A. Finlayson]

On May 27, 8:47 am, c...@kcwc.com (Curt Welch) wrote:
> Uergil <Uer...@uer.net> wrote:

> > In article <20120527014358.924...@newsreader.com>,

> c...@kcwc.com                                        http://NewsReader.Com/

Curt:

"We can talk about what the slope of the line is at x=1 because we can
take
a number dx, and calculate the slope of the line between the points
f(x-dx)
and f(x+dx) and see what the limit of that is as dx approaches zero.
We
can calculate limits without ever having to compute an infinite number
of
points.

We can do all of calculus without anyone every having to compute an
infinite number of values."

But - that's a special case - integrating the line with calculus uses
a theorem, here the fundamental theorem of calculus. That's just as
above where certain consequences OF an axiom of infinity here ZF's as
not always so are still true, because they are true of infinity,
whatever the axiom says.

'The reason anyone would think math existed in some non-physical

domain, is
just again, the mind body problem. People have long believed and
accepted
the idea that the thoughts are not physical, but yet, they "exist".
This
gives them free license to separate the meaning of the word "exist"
from
"physical existence". It gives them free licence to believe that "math

exists" in a domain separate from the physical.'

For Platonists mathematics is real in that it's as real as anything
else.

Whether I believe it or not, mathematics is real. That doesn't
interfere with the free domain of thought at all.

I believe that mathematics is quite sufficient to represent the large
part of anything I could think. Then simply having an objectionist
(for objectivist), rationalist, reason, helps a lot in that
mathematics is no different than the concrete.

Than as to whether you do or not believe of the average other that
people conflate their mathematical thoughts with mathematical reality,
they do and don't. Surely though you'll find that visual fact is
large evidence for most.

About the qualia of the individual or being or the mind body problem,
what you'll find is that for example the modern technical Being from
Hegel and Kant is still quite satisfactory for most in Platonist
foundations. Then it's generally Being and Nothing, all and the void,
body and mind.

It's satisfactory largely whether they care.

"On a larger scale, the same problems exists. How do I know that I'm
currently sitting in my office, on a Saturday afternoon, reading and
writing a Usenet post? Again, this is all a brain perception problem.
My
brain brain has translated a long stream of sensory data and come to
the
conclusion that it indicates that I am in my office, reading and
writing a
Usenet post."

Hmm, well, if you wrote a program that reads and writes Usenet posts,
might that not be its mode?

Now later in reading this, the fair reader would observe that it was
likely so upon reasonable expectations that you were so in your office

reading and writing a Usenet post.

Then though if you are expressing or noting a simple fact that you can
rationalize externally to your own situation then, it seems reasonable
that you can rationalize internally.

'Yes, we are free to make up anything we want, but we are not free to

call
it the foundation of math. If you make stuff up like this, and call it
the
foundation of math, when it's not the actual physical foundation of
math,

then your concept of what math "is" will be totally wrong.'

The foundation is at the bottom, by the time there is much work the
workers don't need the foundations, they are using reasonable
expectations that result from the foundations.

Well I'll assure you there are most assuredly foundations in
mathematics. In as to whether anyone has much say into what those
are, the rules and laws of mathematics can quite well be seen as
immutable. That's where there's already space in the conversation for
mathematical truth in its consequences and all its consequences. Then
abstract things defined mathematically are as so. But, abstract
things are defined, here into feelings and temperament. Basically I
don't see pink elephants, but, imagine they have a pleasant color.
Abstract things are defined. By the time they are well-defined, it's
mathematically.

'For someone that calls themselves a mathematician, I'm surprised more

of them don't care whether they have the foundation of their work

right or wrong.'

There's plenty more to it, in terms of fundamental results useful for
applications, compared to each having a firm grasp of the foudnations
as simply reasonable and in accord with fact and following being a
general tenet of rational discourse. There's plenty more to research
in foundations.

Also, besides deep foundations, there are simply tremendous
opportunities in most of the fields of mathematics in as to where, for
example, entire books are full of tables of these things. This is
where, an average person could find, given ten or twenty years of
mathematical research or less, structural features of mathematics
worth them filling a book.

Well, Curt, yeah I don't care much either whether their foundations
are right or wrong. The foundations aren't right or wrong, they just
are.

Regards,

Ross Finlayson

[Uergil]

In article
<a9dccb2f-bbc4-4802...@d6g2000vbe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 27 Mai, 06:55, c...@kcwc.com (Curt Welch) wrote:
> > "K_h" <KHol...@SX729.com> wrote:
> > > "Curt Welch"  wrote in message
> > > >news:20120526001338.545$w...@newsreader.com...
> > > > concept of events happening in a sequence).  They hide the
> > > > contradictions
> > > No, it has been done.  People have added an infinite number of terms in
> > > calculus without enumerating every term.  So add up 1/2 plus 1/4 plus 1/8
> > > plus 1/16 plus 1/32 and so on ad-infinitum and the result is 1.  There, I
> > > just did it.
> >
> You reached the state after that which is never reached?
> Congratulations!
>
> I, for one, can only calculate the limit of the series, never sum up
> all its terms.

Which makes you, as a mathematician, unable to do what is often done by
students in introductory calculus.

>
> Why don't you prove your skills with non-converging sequences like
> 1 - 1 + 1 - 1 + 1 - 1 +-...?
> If being able of what you boasted, then you should know the result.

But WM has just confessed to being unable to add up
1 + 1/2 + 1/4 + 1/8 + ...

>
> > I really don't have a clue how to respond to that.
> >
> > > Who is being harmed by the axiom of infinity (LOL )??
> >
> > :)
> >
> > The truth is being harmed. :)
>
> So it is.

By WM!
>
> Regards, WM

[Shmuel Metz]

In <20120526145816.354$4...@newsreader.com>, on 05/26/2012

at 06:58 PM, cu...@kcwc.com (Curt Welch) said:

>Well, that comment is an exmaple of abstracting out what you want
>from reality, while at the same time, ignoring other aspects of
>reality

Water is wet. Mathematics is all about abstracting patterns common to
several different things. That turns out to be immensely useful to
physicists and even to engineers.

>There are no sequences in reality that aren't also part of a
>process.

I no of no sequences in reality that *are* part of a process. Or do
you have private definitions of those words grossly at variance with
the normal definitions?

>You can't just pretend the process is not there,

There's nothing to pretend. Your statement is like claiming that an
engineer pretends that string instruments don't exist simply because
he doesn't find them relevant when building a bridge.

>See physics for process.

In other words, you admit that processes are outside the scope of
Mathematics.

>We normally abstract out some subset of the universe and talk
>about the properties of that subset and refer to that as a process.

Only when we're modelling the evolution of a physical system. When we
do that, we already need the machinery of Real Analysis.

>I don't know how they defined convergence in math (I'm sure they
>have multiple formal ways to do it), but I would attack it along the
>lines of saying that a process that is outputting a sequence of
>measurements (numbers) is said to converge towards a target if the
>distance between the output value and the target grows smaller as
>the number of steps increases, AND if for any arbitrary small
>difference DY, there is always some number of steps N, where all the
>remaining distance to target values drops below the value DY.

That's close to the definition of convergence of a sequnce, except
that it would have {1 + (1+(-1)^i)(1/2)^i} not converging to 1. But
you still haven't defined "process" *in Mathematics*, nor given a set
of axioms if you want to take it as an undefined term.

>Note that there is no talk of the process reaching the target in
>this sort of definition.

Nor is there in Mathematics.

>no.
>A:
> while true
> print 1
>B:
> print 0
> while true
> print 1
>Why would it bother me that these two programs exist?

Because they produce different sequences, but those sequences have the
same limit, while you wrote 'So now we have an infinite number of

processes producing different sequences, which must be claimed to be

"the same real".'

>Well, the BELIEF that the proof can exist free of processes is
>exactly what bothers me.

Marks on paper don't exist?

>The brain is a process.

No. The brain is a physical structure.

>If you removed all the brains from the earth,
>where exactly on the earth, would the proof be located?

On paper. Or do you take the solipsistic position that the paper would
disappear if the brains vanished?

--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>

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