Quantifier swapping in set theory

[WM]

For every FISON of the set of FISONs there exists a natural number not contained in that FISON. Obviously this easily provable theorem does not allow quantifier swapping resulting in: There exists a natural number that is not contained in any FISON.

Precisely this quantifier exchange however is required to "prove" the most impressive result of set theory: For every Finite Initial Segment of a given Cantor-List (FISCL) there exists a real number not contained in the FISCL. This is easy to show. But then the quantifier swapping yields: There exists a real number that is not contained in all FISCL of the given Cantor-list.

Between Cantor's list and Hilbert's hotel there is only the arbitrary difference that Hilbert's hotel is really infinite, unfinished, extendable whereas Cantor's list is not. Two different interpretations of one and the same infinity.

Only that allows to conclude that the antidiagonal, as a new guest differing from all resident guests or entries of the list, cannot be inserted, for instance into the first position when every other entry moves on by one room number. Without Cantor's arbitrary constraint even all infinitely many antidiagonals that ever could be constructed could be accommodated. Cantor's theorem would go up in smoke.

Regards, WM

[Jim Burns]

On 10/23/2020 4:17 AM, WM wrote:

> For every FISON of the set of FISONs there exists a
> natural number not contained in that FISON.
> Obviously this easily provable theorem

This strikes me as part of a good (indefinite) description
of a natural number. As such, it would be an easily provable
theorem, from that claim (and others) about natural numbers.

Of course, whether something is a theorem at all or not a
theorem depends upon what description (what axioms) we
reason from. I mention this because, although we agree that
this claim describes natural numbers, you and we don't
always agree. (See below.)

Often, the most important part of a mathematical argument
is over before any apparent math is begun. "The setting of
the terms of engagement", we might say.

> Obviously this easily provable theorem does not allow
> quantifier swapping resulting in: There exists a natural
> number that is not contained in any FISON.

The negation of this _not_ easily-provable-claim is
There does not exist a natural number that is not contained
in any FISON.

This also strikes me as part of a good (indefinite) description
of a natural number.

In fact, these two claims and their consequences seem to cover
the most important features of natural numbers, when we
reason from them as a description, using indefinite references
to _a_ natural number.

> Precisely this quantifier exchange however is required to
> "prove" the most impressive result of set theory:
> For every Finite Initial Segment of a given Cantor-List
> (FISCL) there exists a real number not contained in the FISCL.
> This is easy to show. But then the quantifier swapping yields:
> There exists a real number that is not contained in all FISCL
> of the given Cantor-list.

Sorry, no. That is not the argument.

The second claim here depends upon Dedekind completeness,
in some form or another, not upon quantifier swapping.

The quantifier swap contemplated is invalid.
| forall FISCL f(1),...,f(n), exists real x,
| x not in f(1),...,f(n)
does not imply
| exists real x, forall FISCL f(1),...,f(n),
| x not in f(1),...,f(n)

You note above that a swap of that form is invalid,
in the case where it is about natural numbers in FISONs.

Here, with infinite lists of numbers, if we changed what
we were talking about to rationals or algebraics or
something else not Dedekind complete, we would have
counter-examples to the claim that swap is valid.

As with "a FISON for each natural" describing what we
mean by natural numbers, "a point between every cut" describes
what we mean by real numbers -- or, better, what we mean by
a line, which is what the real numbers are intended to
describe.

The argument we actually make uses a list f() to determine a
cut of the line which cannot have any listed number
between the sides.

This part of the argument doesn't require Dedekind
completeness. If we had, for example, a list of all
rational numbers
(p/q indexed by k = (p+q-1)*(p+q-2)/2 + p or some other way)
then the cut thus determined would indeed not have any
_rational_ numbers between the sides of the cut. But the
existence of irrational numbers is well-known.
This tells us nothing about rationals.

However, the real numbers are Dedekind complete.
This allows us to step from
| There is a list of real numbers
to
| There is a cut without any real number between its sides
to (!)
| There is a real number not in the list.
to
| There is no list of all real numbers.

[Jim Burns]

Correction
s/real/listed/

On 10/23/2020 12:09 PM, Jim Burns wrote:

> However, the real numbers are Dedekind complete.
> This allows us to step from
> | There is a list of real numbers
> to

> | There is a cut without any *listed* number between its sides

[WM]

Am Freitag, 23. Oktober 2020 18:09:21 UTC+2 schrieb Jim Burns:
> On 10/23/2020 4:17 AM, WM wrote:

> > Precisely this quantifier exchange however is required to
> > "prove" the most impressive result of set theory:
> > For every Finite Initial Segment of a given Cantor-List
> > (FISCL) there exists a real number not contained in the FISCL.
> > This is easy to show. But then the quantifier swapping yields:
> > There exists a real number that is not contained in all FISCL
> > of the given Cantor-list.
>
> Sorry, no. That is not the argument.

It is.

>
> The second claim here depends upon Dedekind completeness,
> in some form or another,

The Cantor-list contains sequences of digits or symbols. That has nothing to do with Dedekind. When Cantor devised his argument he did not know about Dedekind-completeness. In fact the diagonal argument is applied everywhere in matheology. Don't blame it on Dedekind.

> You note above that a swap of that form is invalid,
> in the case where it is about natural numbers in FISONs.

And of course it is as invalid in the Cantor-list. It has been called the most important argument by matheologians - and in fact it is the most important argument in matheology. You will have to find a lot of completeness axioms.

>
> However, the real numbers are Dedekind complete.
> This allows us to step from
> | There is a list of real numbers
> to
> | There is a cut without any real number between its sides
> to (!)
> | There is a real number not in the list.
> to
> | There is no list of all real numbers.

So the argument fails in all cases other than "Dedekind-complete". Are you sure that you will get support for this mistake?

Regards, WM

[Dan Christensen]

On Friday, October 23, 2020 at 4:17:04 AM UTC-4, WM wrote:
> For every FISON of the set of FISONs there exists a natural number not contained in that FISON. Obviously this easily provable theorem does not allow quantifier swapping resulting in: There exists a natural number that is not contained in any FISON.
>
> Precisely this quantifier exchange however is required to "prove" the most impressive result of set theory: For every Finite Initial Segment of a given Cantor-List (FISCL) there exists a real number not contained in the FISCL. This is easy to show. But then the quantifier swapping yields: There exists a real number that is not contained in all FISCL of the given Cantor-list.
>

I'm not sure what you are getting at here, but, while it may not be true in general, in some cases, we do have:

[For all x in A: Exists y in B: P(x,y)] => [Exists y in B: For all x in A: P(x,y)]

If you are imagining that this is your long sought after contradiction in set theory, you have failed yet again, Mucke.


Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

[Ross A. Finlayson]

... the Dedekind/Cauchy real numbers as what as a continuous domain
have established the fundamental theorems of calculus as what have
countable additivity of all analytical character of real numbers.

The ran(EF) of course, as a model or set of individua of a continuous
domain, with rather restricted transfer principle and that for function
theory that it's a non-Cartesian function, is different.

[Jim Burns]

On 10/23/2020 2:12 PM, WM wrote:
> Am Freitag, 23. Oktober 2020 18:09:21 UTC+2
> schrieb Jim Burns:
>> On 10/23/2020 4:17 AM, WM wrote:

>>> Precisely this quantifier exchange however is required to
>>> "prove" the most impressive result of set theory:
>>> For every Finite Initial Segment of a given Cantor-List
>>> (FISCL) there exists a real number not contained in the FISCL.
>>> This is easy to show. But then the quantifier swapping yields:
>>> There exists a real number that is not contained in all FISCL
>>> of the given Cantor-list.
>>
>> Sorry, no. That is not the argument.
>
> It is.

No one can convict you of insufficient confidence.

Your Grand Theory is meant to repair flaws you see in
our arguments (I assume, having given you perhaps too much
benefit of the doubt).

The BIG problem you have is that you don't actually
know what the arguments you intend to repair are.
When you're shown
"THIS argument doesn't have the problem you see",
your response is to insist that we should use the other one,
the one that has the problem.

No thank you.

> In fact the diagonal argument is applied everywhere in
> matheology. Don't blame it on Dedekind.

The diagonal argument takes a _possible_ accounting (of reals,
of subsets, of whatever) and uses it describe something
(a real, a subset, a whatever) _not accounted for_

But the mere existence of a description does not grant
existence to the thing described. (At least, that's what
my friend, the flying rainbow sparkle pony Max, tells me.)

When we prove that the powerset of B is larger than B,
we assume some sort of matching f() of elements of B to
subsets of B and then _uses f() to describe a subset D_
It's easy to prove D is not matched to any element of B.

However, in order to prove that the subset described exists,
we need the axiom of separation -- described subsets exist.

When we prove that an unlisted real number exists,
we used the list to describe a real number -- a
single point gap such that, _of there were a point there_
that point would be not be any listed real.
It's not hard to prove that if that hypothetical point
existed, it would not be any listed real.

However, in order to prove that a point exists in
the described gap, we need Dedekind completeness.

The real numbers have Dedekind completeness.
The diagonal argument proves the reals are uncountable.

The rational numbers do not have Dedekind completeness.
The diagonal argument is invalid for rationals.
It doesn't prove the rationals are uncountable.
Also (separately), we can prove the rationals are countable.
For example, with k = (p+q-1)*(p+q-2)/2 + p.

Some form of the diagonal argument gets used in a wide variety
of placed in math. Everywhere it is used, some analogy
to Dedekind completeness or the axiom of separation is needed.

[Me]

On Friday, October 23, 2020 at 8:12:58 PM UTC+2, WM wrote:
> Am Freitag, 23. Oktober 2020 18:09:21 UTC+2 schrieb Jim Burns:
> > On 10/23/2020 4:17 AM, WM wrote:
> > >

> > > <BLA BLA BLA>

> > >
> > Sorry, no. That is not the argument.
> >
> It is.

No, it isn't.

[WM]

Am Freitag, 23. Oktober 2020 23:24:54 UTC+2 schrieb Jim Burns:
> On 10/23/2020 2:12 PM, WM wrote:

> The BIG problem you have is that you don't actually
> know what the arguments you intend to repair are.

I know that the original diagonal argument is not based on real numbers and therefore your Dedekind-completeness argument is completely nonsense.

> When you're shown
> "THIS argument doesn't have the problem you see",
> your response is to insist that we should use the other one,
> the one that has the problem.

Infinite digit sequences without the attached powers of 10 (or another usable base) do not converge. With only few exceptions they accidentally jump to and fro and do not assume a limit because the trembling does never calm down. But just these sequences are produced by the diagonal argument. Cantor, in his original version [G. Cantor: "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der DMV I (1890-91) pp. 75-78], did not define limits at all. Therefore the invention of the rule to avoid antidiagonal numbers with periods of nines does merely show a big misunderstanding of facts. Of course the string 1,0,0,0,... differs from the string 0,9,9,9,...

Finally: The antidiagonal does not exist but is only constructed. If it had existed during the creation of the list, it would have been included, wouldn't it?

>
> > In fact the diagonal argument is applied everywhere in
> > matheology. Don't blame it on Dedekind.
>
> The diagonal argument takes a _possible_ accounting (of reals,
> of subsets, of whatever) and uses it describe something
> (a real, a subset, a whatever) _not accounted for_

by a simple quantifier swapping.

> When we prove that the powerset of B is larger than B,

That is Hessenberg's flawed argument. Here we are discussing Cantor's.

> However, in order to prove that a point exists in
> the described gap, we need Dedekind completeness.

Infinite digit sequences without the attached powers of 10 (or another usable base) do not converge. Therefore Dedekind-completeness cannot be applied.

>
> The real numbers have Dedekind completeness.
> The diagonal argument proves the reals are uncountable.
> The rational numbers do not have Dedekind completeness.

All digit sequences (with attached powers) are representing sequences of rational numbers. Irrational numbers are limits, not contained in any Cantor-list.

> Some form of the diagonal argument gets used in a wide variety
> of placed in math. Everywhere it is used, some analogy
> to Dedekind completeness or the axiom of separation is needed.

What about Cantor's original sequences mmm..., www..., and so on? Did Dedekind prove them complete?

Regards, WM

[WM]

Am Samstag, 24. Oktober 2020 01:04:23 UTC+2 schrieb Me:
> On Friday, October 23, 2020 at 8:12:58 PM UTC+2, WM wrote:
> > Am Freitag, 23. Oktober 2020 18:09:21 UTC+2 schrieb Jim Burns:
> > > On 10/23/2020 4:17 AM, WM wrote:

> > > > For every Finite Initial Segment of a given Cantor-List (FISCL) there exists a real number not contained in the FISCL. This is easy to show. But then the quantifier swapping yields: There exists a real number that is not contained in all FISCL of the given Cantor-list.

> > > Sorry, no. That is not the argument.
> > >
> > It is.
>
> No, it isn't.

Infinite digit sequences without the attached powers of 10 (or another usable base) do not converge. With only few exceptions they accidentally jump to and fro and do not assume a limit because the trembling does never calm down. But just these sequences are produced by the diagonal argument. Cantor, in his original version [G. Cantor: "Über eine elementare Frage der Mannigfaltigkeitslehre", Jahresbericht der DMV I (1890-91) pp. 75-78], did not define limits at all. Therefore the invention of the rule to avoid antidiagonal numbers with periods of nines does merely show a big misunderstanding of facts (cp. section "The nine-problem"). Of course the string 1,0,0,0,... differs from the string 0,9,9,9,...

Finally: The antidiagonal does not exist but is only constructed. If it had existed during the creation of the list, it would have been included, wouldn't it?

Regards, WM

[Me]

On Saturday, October 24, 2020 at 11:00:57 AM UTC+2, WM wrote:
> Am Samstag, 24. Oktober 2020 01:04:23 UTC+2 schrieb Me:
> > On Friday, October 23, 2020 at 8:12:58 PM UTC+2, WM wrote:
> > > Am Freitag, 23. Oktober 2020 18:09:21 UTC+2 schrieb Jim Burns:
> > > > On 10/23/2020 4:17 AM, WM wrote:
> > > > >

> > > > > <bla bla bla>

> > > > >
> > > > Sorry, no. That is not the argument.
> > > >
> > > It is.
> > >
> > No, it isn't.

Now:

> The antidiagonal does not exist but is only constructed. If it had existed
> during the creation of the list, it would have been included, wouldn't it?

Nothing is "constructed" or "created" here, dumbo.

The claim (in this case) is: ANY (EACH and EVERY) "list" (sequence) of infinite sequences (of binary digits) lacks at east one infinite sequence (of binary digits).

Proof: For any (each and every) "list" (sequence) L of infinite sequences (of binary digits), there's at least one infinite sequence (of binary digits) which is not a term/element in L.

From this we conclude that there is no "list" (sequence) of infinite sequences (of binary digits) which "contains" _all_ infinite sequences (of binary digits).

[Mostowski Collapse]

You can "construct" the diagonal by the
comprehension axiom. This axiom here:

Axiom schema of specification
https://en.wikipedia.org/wiki/Axiom_schema_of_specification

For example you can do for
f : X -> P(X), you can do:

d = { x e X | ~(x e f(x)) }

Then you will find ~(d e ran(f)). So although
the diagonal argument is only an existence

statement, it is proved by a "construction".
Without the above axiom, how would you

prove the diagonal argument Crrank "Me"?

[WM]

Am Samstag, 24. Oktober 2020 16:13:19 UTC+2 schrieb Me:


> The claim (in this case) is: ANY (EACH and EVERY) "list" (sequence) of infinite sequences (of binary digits) lacks at east one infinite sequence (of binary digits).

This is true for finite lists. For every FISCL, there is a sequence not contained therein.

>
> Proof: For any (each and every) "list" (sequence) L of infinite sequences (of binary digits), there's at least one infinite sequence (of binary digits) which is not a term/element in L.

Only finite lists and finite collections of symbols are availabe without formulas.

>
> From this we conclude that there is no "list" (sequence) of infinite sequences (of binary digits) which "contains" _all_ infinite sequences (of binary digits).

Wrong.

Regards, WM

[Ross A. Finlayson]

Least Upper Bound (LUB) which along with measure 1.0
is defined, not derived, for "standard real analysis",
finds a simpler and even derived establishment from some
"continuum infinitesimal analysis" as after Finlayson's slate.

[Ross A. Finlayson]

Because we know linear algebra and methods about diagonals in
matrix analysis and matrix algebras, besides generalized matrix
products, "diagonalization" in the usual set-theoretic setting is
instead disambiguated as "anti-diagonalization".

Nobody much remembers the orthogonant any-more.

[Jim Burns]

On 10/24/2020 5:00 AM, WM wrote:
> Am Freitag, 23. Oktober 2020 23:24:54 UTC+2
> schrieb Jim Burns:
>> On 10/23/2020 2:12 PM, WM wrote:

>>> In fact the diagonal argument is applied everywhere in
>>> matheology. Don't blame it on Dedekind.
>>
>> The diagonal argument takes a _possible_ accounting (of reals,
>> of subsets, of whatever) and uses it describe something
>> (a real, a subset, a whatever) _not accounted for_
>
> by a simple quantifier swapping.

No. That argument is not valid.

If all we know is...
exists x, forall y, P(x,y)

...then we also know
forall y, exists x, P(x,y)

*HOWEVER* if all we know is...
forall y, exists x, P(x,y)

...then *WE DO NOT KNOW*
exists x, forall y, P(x,y)

You want to use the second inference.
No, wait. You want *us* to use the second inference.
We don't use it -- not that you care about reality.
You will continue beating up your straw man.

This invalidity has nothing to do with infinity.
Take a domain {a,b} of two.
Define P(x,y) == (x=y)

This is false:
[?] Ay,Ex,P(x,y) -> Ex,Ay,P(x,y)

And we can extend this to any larger domain.

>> When we prove that the powerset of B is larger than B,
>
> That is Hessenberg's flawed argument.
> Here we are discussing Cantor's.

Make up your mind. I'm answering you here:

>>> In fact the diagonal argument is applied everywhere in
>>> matheology. Don't blame it on Dedekind.

There is nowhere that quantifier swap is enough to
show that the appropriate anti-diagonal _exists_
because that is not a valid quantifier swap.

That doesn't mean Ex,Ay,P(x,y) won't be _true_ in some
instances. But we need something more than Ay,Ex,P(x,y)
to _prove_ it.

For real numbers, something more is Dedekind completeness.

For sets, something more is the axiom of separation.

For endless sequences of 'm' and 'w', some principle is
needed to conclude that the anti-diagonal sequence exists.
I'll guess it sounds like the axiom of separation.
It might be that Cantor did not state it explicitly.
I don't know, I am not a scholar.
Perhaps it seemed too obvious to need stating.

Mathematics has at least a little experience with
principles too obvious to need stating that later were
found to need stating. For example, the Axiom of Choice.

Apart from the rest of this, though, that quantifier swap
is invalid. This is not debatable for { a,b,... }:
not( Ay,Ex,(x=y) -> Ex,Ay,(x=y) )

>> However, in order to prove that a point exists in
>> the described gap, we need Dedekind completeness.
>
> Infinite digit sequences without the attached powers of 10
> (or another usable base) do not converge.

We don't need to show that infinite digit sequences
converge in order to show that infinite digit sequences
are uncountable. We only need to show that an anti-diagonal
_digit sequence_ exists which is different from each
_digit sequence_ in a given sequence of sequences.

If we are showing that there are uncountably-many
_number-line points_ by representing points with
infinite digit sequences, then we will need to show that
our representatives represent, that, for an infinite digit
sequence, there is a number-line point. That's what we need
Dedekind completeness for.

> Therefore Dedekind-completeness cannot be applied.

Do your homework.
Find out what the argument is which you are "criticizing".

> All digit sequences (with attached powers) are representing
> sequences of rational numbers. Irrational numbers are limits,
> not contained in any Cantor-list.

Each finite initial segment of an infinite digit sequence
_represents_ certain _constituents_

For a k-digit initial segment, the number-line points
_below_ all its constituents and the number-line points
_above_ all its constituents come _as close as_
1/10^k + epsilon, for any epsilon > 0.

Define the constituents of an infinite digit sequence
to be those points which are constituents of all
finite initial segments.

Assume the Archimedean property for number-line points.
For two positive magnitudes x,y, there is some finite
natural k such that k*x > y

Let x and y be constituents of an infinite digit sequence.
forall _finite_ naturals k, -1/10^k < x-y < 1/10^k

Since Archimedean, x-y = 0

Therefore, an infinite digit sequence represents
_no more_ than one constituent.

----
Each finite initial segment of an infinite digit sequence
_represents_ certain _constituents_

Each number-line point x either has the property P
| there exists a finite initial segment of digits
| for which x is below all constituents
or x does not have that property.

Assume Dedekind completeness.
If the number-line points are cut into D and U
such that forall x in D and forall y in U, x < y,
then there exists a number-line point z in the cut
such that x < z implies x in D and y > z implies y in U.

Every point x which _has_ property P comes before
every point y which _does not have_ property P.
The finite initial segments of the infinite digit sequence
determine a cut of the number-line.

By Dedekind completeness, there exists a number-line point
in that cut. That point in the cut is a constituent of
all finite initial segments. It is a constituent of
the infinite digit sequence (as we have defined it).

Therefore, an infinite digit sequence represents
_no less_ than one constituent.

[Me]

On Saturday, October 24, 2020 at 5:50:27 PM UTC+2, Mostowski Collapse wrote:

> You can "construct" the diagonal by the comprehension axiom.

We do not "construct it", you silly asshole.

The axiom just ensures its existence.

Learn some mathematics, learm some logic, you psychotic crank.

[Me]

On Saturday, October 24, 2020 at 5:50:27 PM UTC+2, Mostowski Collapse wrote nonsense.

Anyway, MC = microcephalus.

[Mostowski Collapse]

Well its a construction axioms. Set theory
has different set constructions:
- Pair sets
- Power sets
- And sets from other sets by comprehension
- Etc..

Thats how the constructible universe for
example emerges:

https://en.wikipedia.org/wiki/Constructible_universe

Its quite common to use the word construct
here, as WM uses it, and

not deny the involved construction as Crrank
"Me" wants it.

[Mostowski Collapse]

The only non-constructive thing is f in
the premisse of the diagonal argument.
It might be any function f : X -> P(X),
but the rest is constructive in a set

theoretic sense. Its not the same constructive
connotation that is for example understood
in constructive analysis, but still the
word construction for the diagonal is

adequate. Set theory has also non-constructive
axioms, like the axiom of choice. There
existence of a thing is postulated, but
is not a well defined definition that states

a unique object, not like the other construction
axioms of set theory.

[Mostowski Collapse]

Together with the extensionality axiom,
you can show that these axioms all lead
to well defined definition, the postulated

objects not only exist, they are also unique:

- Pair sets
- Power sets
- And sets from other sets by comprehension
- Etc..

So they have the same status like a
compass and ruler construction in geometry,
from a few objects you get a few new
objects in finite steps.

The comprehension axiom only uses
a finite comprehension formula.

[Mostowski Collapse]

So you created the word orthogonant? What
does it mean, that a chicken crossed the road?
Or was it Crrank "Me". No I dont guess so,
he is possibly colorblind and still waiting

in front of the traffic light. Same with
Crrank "Me", we are still waiting for his
awakening. A few months ago FOL functions
were set like. Whats on the plate now?

Set theory does never construct sets?

LoL

[WM]

Am Samstag, 24. Oktober 2020 20:53:28 UTC+2 schrieb Jim Burns:


> That doesn't mean Ex,Ay,P(x,y) won't be _true_ in some
> instances. But we need something more than Ay,Ex,P(x,y)
> to _prove_ it.

But you have not. Read any description of the digonal argument, read Fraenkel or Zermelo or any modern author. There is no reference to your idea.

>
> For real numbers, something more is Dedekind completeness.

There is no such principle. Dedekind said: "Every time when there is a cut (A1, A2) which is not produced by a rational number, we create a new, an irrational number  which we consider to be completely defined by this cut (A1, A2);" There is no word about the existence of all curs.

>
> For sets, something more is the axiom of separation.

It does not prove completeness.

>
> For endless sequences of 'm' and 'w', some principle is
> needed to conclude that the anti-diagonal sequence exists.

But there is no such principle. Cantor's claim is based on simple quantifier swapping.

Regards, WM

[Me]

On Sunday, October 25, 2020 at 10:28:51 AM UTC+1, WM wrote:

> Cantor's claim is based on simple quantifier swapping.

You are dumb like shit, crank!

[Jim Burns]

On 10/25/2020 5:28 AM, WM wrote:
> Am Samstag, 24. Oktober 2020 20:53:28 UTC+2
> schrieb Jim Burns:

>> For real numbers, something more is Dedekind completeness.
>
> There is no such principle.

https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Dedekind_completeness

| Dedekind completeness is the property that
| every Dedekind cut of the real numbers is
| generated by a real number.

This is pretty much what I've said. I've explained it by
re-writing with terms that you haven't found a way to
misinterpret _so far_

For "real number", I write "number-line point".
The number-line points are what the real numbers are
intended to describe. I cut out the middle-man.

A number-line point z _generates_ a cut L,R of number-line
points by being _in the cut_

The line has an intrinsic order '<'.
"Dedekind complete" is part of the description of '<'.

The pair of sets L,R is a _cut_ iff
every point x is in (L U R)
neither L nor R is empty
forall x in L and y in R, x < y

A point z is _in the cut_ L,R iff
forall x < z, x is in L, and
forall y > z, y is in R.

Dedekind completeness ==
for every Dedekind cut L,R, there exists a point z
such that z generates L,R.

It's _logically possible_ for what we mean by
"real numbers" to lack Dedekind completeness.
For example, someone might be _saying_ "real numbers" and
_meaning_ "definable numbers". But our intention is to
describe the number line. A lack of Dedekind completeness
would imply there's some way to separate the number line
into left and right parts without any point being between
the parts. This is not what we see the number line to be.

> Dedekind said: "Every time when there is a cut (A1, A2)
> which is not produced by a rational number, we create
> a new, an irrational number  which we consider to be
> completely defined by this cut (A1, A2);"

> There is no word about the existence of all [cuts].

I see two possibilities.
(i)
Dedekind and Cantor and all the Wikipedia citizen-editors
mistakenly believe that finite beings can perform infinite
tasks, and you have corrected them all.
(ii)
In order to reason about infinitely many objects and/or
infinitely many tasks, we (that is: Dedekind and Cantor
and Wikipedia and others not you), we describe an object
and/or a task indefinitely, one of infinitely many, and we
reason finitely from the description to new insights true of
infinitely many objects and/or tasks.

For example, we describe an infinite task of stepping
along "all" of an infinite sequence (something which I
suppose qualifies as "potentially infinite") by describing
one (indefinite) item k in the sequence as
k is finitely many steps from the beginning and
there are more steps after k.

If we suppose (i) holds, maybe it is at long last time for
you (WM) to just assume that you have the undying gratitude
from the universe at large for your pointing out that we
are finite, and now you should move on to other, more
challenging topics.

If we suppose (ii) holds, it seems completely reasonable
to me that Cantor et al. wrote so as to appeal to the
intuition of their reader and trusted their reader to
translate the prose without help into a form better
suited to finitely reasoning about the infinite.
(For example, by reading "endless steps" as "each step
finite, no step last".)

If we faulted them for that, it would have to be for
trusting their reader too much.

>> For sets, something more is the axiom of separation.
>
> It does not prove completeness.

Since I'm referring to the diagonal argument that proves
that a powerset P(B) is larger than B, I find myself
totally at peace with that judgment.

Do you think I should care what all the things are that
the axiom of separation does not prove? Why?

>> For endless sequences of 'm' and 'w', some principle is
>> needed to conclude that the anti-diagonal sequence exists.
>
> But there is no such principle. Cantor's claim is based on
> simple quantifier swapping.

My skepticism is based on the belief that Cantor was not
stupid. That he knew quite well that that quantifier swap
is not valid.

You (WM) apparently don't realize what it says _about you_
that, in order to appear smarter than Cantor, you need
Cantor to be stupid.

No surprise there.
https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect

[Ross A. Finlayson]

Deduction simply arrives at what the "divided" is after
"division" that it's the transfer principle in effect.

Don't confuse "WM's grab-bag from Hodges' hopeless" with
"extra-standard integers and bridge results in the transfer principle".

Many of the classic "paradoxes" of continuous motion about
Zeno and the heap are simply frameworks to establish limit in
effect and then for that a ray of time naturally falls out from
a more-than-less simple framework of deductive inference.

Keep in mind that adding Archimedes' spiral to the other two
tools of classical construction solves most of the circle-squaring
angle-trisecting problems of classical geometry.

[WM]

Am Sonntag, 25. Oktober 2020 20:00:16 UTC+1 schrieb Jim Burns:
> On 10/25/2020 5:28 AM, WM wrote:
> > Am Samstag, 24. Oktober 2020 20:53:28 UTC+2
> > schrieb Jim Burns:
>
> >> For real numbers, something more is Dedekind completeness.
> >
> > There is no such principle.
>
> https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Dedekind_completeness
>

"There is a construction of the real numbers based on the idea of using Dedekind cuts of rational numbers to name real numbers"

Note that every construction is restricted to countably many elements.

"If one were to repeat the construction of real numbers"

then one would turn out as a fool.

Anyhow it is irrelevant for the diagonal argument.

>
> It's _logically possible_ for what we mean by
> "real numbers" to lack Dedekind completeness.

Then the diagonal argument would fail in your opinion?

>
> > Dedekind said: "Every time when there is a cut (A1, A2)
> > which is not produced by a rational number, we create
> > a new, an irrational number  which we consider to be
> > completely defined by this cut (A1, A2);"
> > There is no word about the existence of all [cuts].
>
> I see two possibilities.
> (i)
> Dedekind and Cantor and all the Wikipedia citizen-editors
> mistakenly believe that finite beings can perform infinite
> tasks, and you have corrected them all.

Everybody knows that we cannot perform infinite tasks step by step, i.e., without a defining finite formula.

> > But there is no such principle. Cantor's claim is based on
> > simple quantifier swapping.
>
> My skepticism is based on the belief that Cantor was not
> stupid. That he knew quite well that that quantifier swap
> is not valid.

He did not recognize that it was a quantifier swap - as little as you did hitherto. But you explanation with Dedekind completeness is completely silly.

> You (WM) apparently don't realize what it says _about you_
> that, in order to appear smarter than Cantor, you need
> Cantor to be stupid.

Your trust in the existing theory would prevent to discover any faults therein. Cantor has made four attempts to teach others, three of them have been recognized as nonsense by everybody:

Cantor knew about the true Father of Christ, the true Author of Shakespeare's Writings, the true Jakob Böhme, and the true Infinity. Nowadays his disciples endorse only one of his four findings.

The fourth is nonsense too as your ridiculous defence has shown.

Regards, WM

[Mostowski Collapse]

WM: ... Between Cantor's list and Hilbert's hotel ...

Hilberts Hotel is not a proof. Its an example
to illustrate the notion of Dedekind Infinite.
Cantors Theorem on the other hand has proofs.

How on earth would a sane person mixup
examples and proofs in an argument? There
is no "the Cantor's List" like there is

"the Hilber Hotel". The list in Cantors Theorem
is arbitrary, its a multiplicity, leading to a
contradiction of existence.

Whats wrong with you Cranks, how on earth
can one even talk about Cantor's List as if
there were such a list per se.

On Friday, October 23, 2020 at 10:17:04 AM UTC+2, WM wrote:
> For every FISON of the set of FISONs there exists a natural number not contained in that FISON. Obviously this easily provable theorem does not allow quantifier swapping resulting in: There exists a natural number that is not contained in any FISON.

>
> Precisely this quantifier exchange however is required to "prove" the most impressive result of set theory: For every Finite Initial Segment of a given Cantor-List (FISCL) there exists a real number not contained in the FISCL. This is easy to show. But then the quantifier swapping yields: There exists a real number that is not contained in all FISCL of the given Cantor-list.
>

> Between Cantor's list and Hilbert's hotel there is only the arbitrary difference that Hilbert's hotel is really infinite, unfinished, extendable whereas Cantor's list is not. Two different interpretations of one and the same infinity.
>
> Only that allows to conclude that the antidiagonal, as a new guest differing from all resident guests or entries of the list, cannot be inserted, for instance into the first position when every other entry moves on by one room number. Without Cantor's arbitrary constraint even all infinitely many antidiagonals that ever could be constructed could be accommodated. Cantor's theorem would go up in smoke.
>
> Regards, WM

[WM]

Am Montag, 26. Oktober 2020 13:48:49 UTC+1 schrieb Mostowski Collapse:
> WM: ... Between Cantor's list and Hilbert's hotel ...
>
> Hilberts Hotel is not a proof.

Hilbert's hotel shows the basic feature of infinity.

> Its an example
> to illustrate the notion of Dedekind Infinite.
> Cantors Theorem on the other hand has proofs.

Proofs by people who have only superficially thought about it because it seems to be so easy.

Fact is this: For every Finite Initial Segment of a given Cantor-List (FISCL) there exists a real number not contained in the FISCL. This is easy to show. But then the quantifier swapping yields: There exists a real number that is not contained in all FISCL of the given Cantor-list.

Regards, WM

[Mostowski Collapse]

There is no such thing as "proofs by people".
Once a proof has been written down, it can
be even fed into a computer, and verified.

Whats wrong with you?

There are a couple of different proofs to
show Cantors theorem. Usually one type of
proof, the diagonal argument,

goes along showing that an arbitrary
f : X -> P(X) cannot be surjective. To
show this, it is sufficient that the

constructed diagonal d not in ran(f).

[Ross A. Finlayson]

Ah, but EF or sweep is defined as a limit of functions,
it's not Cartesian and its elements can not be re-ordered,
there's only one anti-diagonal, and it's always at the end.

[Mostowski Collapse]

This shows Cantors theorem for arbitrary
X, i.e. it shows |X| < |P(X)|. The construction
uses the comprehension axiom:

Axiom schema of specification
https://en.wikipedia.org/wiki/Axiom_schema_of_specification

The above axiom is used to construct the
diagonal. In reverse mathematics weaker
forms of comprehension are used.

This is done by restricting the formula that
is used in the set builder. Crrank "Me", note
the wording set "builder". Its a construction!

So the unrestricted set "builder" is:

{ x e X | phi(x) }

Where phi(_) is an arbitrary first order
Formula. I guess if we even restrict phi(_)
in some way, look at fragments of set

theory, still sometimes diagonals can be
constructed. The diagonal argument is also
used in recursion theory for

recursive jumps. Would need some time to
look it up. But even under MathRealism, the
diagonal argument might be put to use,

to distinguish different kinds of sets. Not
only countable and uncountable, but also
r.e. and other stuff.

[Mostowski Collapse]

The main ingredient is the comprehension
axiom, which allows the construction of a
diagonal. Maybe Crrank "Me" knows a proof of

the Cantor Theorem without comprehension axiom?

[Me]

On Monday, October 26, 2020 at 3:42:01 PM UTC+1, WM wrote:

> FISCL

See: https://en.wikipedia.org/wiki/Fistula

[WM]

Am Montag, 26. Oktober 2020 15:46:59 UTC+1 schrieb Mostowski Collapse:
> There is no such thing as "proofs by people".
> Once a proof has been written down, it can
> be even fed into a computer, and verified.

And it can be contradicted nevertheless in cases, like the present one, where the underlying theory is inconsistent.

One simple counter proof among man others: All that is used in a Cantor-list is the digit sequence from d_1 to d_n, hence of a rational number. Cantor proves that the digit sequence of the antidiagonal is not one of the rationals of his list. Ridicoulos that this has never been pointed out.

Regards, WM

Regards, WM

[Jim Burns]

On 10/26/2020 7:39 AM, WM wrote:
> Am Sonntag, 25. Oktober 2020 20:00:16 UTC+1
> schrieb Jim Burns:

>> It's _logically possible_ for what we mean by
>> "real numbers" to lack Dedekind completeness.
>
> Then the diagonal argument would fail in your opinion?

_Logic_ does not dictate _what we mean_ by "real numbers".

>> For example, someone might be _saying_ "real numbers" and
>> _meaning_ "definable numbers". But our intention is to
>> describe the number line. A lack of Dedekind completeness
>> would imply there's some way to separate the number line
>> into left and right parts without any point being between
>> the parts. This is not what we see the number line to be.

_Logic_ does not tell us what we see the number line to be.

Suppose that we saw "only the rationals" in the number line.

Consider the endless list of "real" (rational) numbers
wherein p/q is at index k,
k = (p+q-1)*(p+q-2)/2 + p

Each entry p/q has a k_th decimal digit (p/q)#k.

_If some number-line point d exists_ such that,
_for all indexes k_, the k_th digit of d is sufficiently
different from the k_th digit of p/q at index k,
then, _for all indexes k_ d is not p/q at k.
An example of sufficiently different would be if we defined
d#k = ((p/q)#k+ 5) mod 10

Does such a number-line point d exist? No.

Suppose d exists.

Given what we mean by "number-line point", d is rational.
There are naturals r,s such that d = r/s.
Therefore d is at index j = (r+s-1)*(r+2-2)/2 + r

But d and the number at index j differ at their j_th digits
by 5. d cannot be at index j. Contradiction.

Therefore, d (the rationals-only anti-diagonal)
does not exist _as one of the rationals_

The diagonal argument fails for the rationals.

tl,dr it would fail. Yes.

[Jim Burns]

On 10/26/2020 10:41 AM, WM wrote:

> Proofs by people who have only superficially thought
> about it because it seems to be so easy.
>
>
> Fact is this: For every Finite Initial Segment of
> a given Cantor-List (FISCL)

For every human being,

> there exists a real number not contained in the FISCL.

there exists that human being's mother,
the unique woman who gave birth to them.

> This is easy to show.
> But then the quantifier swapping yields:
> There exists a real number that is not contained in
> all FISCL of the given Cantor-list.

But then quantifier swapping yields:

There exists a unique woman who gave birth to every human.

This kind of quantifier swapping is not valid. *That means*
It is _not impossible_ for both
| forall x, exists y, P(x,y)
and
| not exists y, forall x, P(x,y)
to be true.

[Mostowski Collapse]

To contradict it, you need to show us
an f : X -> P(X), which would be surjective.

So far you did not exhibit such an f. For
sets X it is likely impossible. For classes

it is known that for example:

id : V -> P(V)

id(x) = x

where V is the universal class.

[Mostowski Collapse]

id is surjective since V = P(V).
Hint: every set is a set of sets.

[Mostowski Collapse]

So basically the diagonal argument
would fail for V. Problem is the diagonal
is not anymore a set, so that ~(g e ran(f))
is not anymore false, and therefore the

reductio ad absurdum proof doesn't go
true. The proof is also not applicable,
since there is no set-like f : V -> P(V),
id is not a set-like function. So id

is not really a counter example to Cantors
theorem. To exhibit a counter example to
Cantors theorem, you would need to exhibit
a set-like function f : X -> P(X), which

is surjective. We are waiting, take your time!

[Ross A. Finlayson]

Building a countable continuous domain is a bit different.

[Mostowski Collapse]

You mean orthogonant?

[Ross A. Finlayson]

You can read Muir for that, if your Googlay hasn't forg-otten it.

The orthogonant is part of an approach to the coordinate-free
in linear algebra and matrix mathematics.

[Ross A. Finlayson]

About the powerset result proper, and
_numbering_ besides _counting_, is
for a notion of "ubiquitous ordinals",
that for a usual successor function with
_succession_ as _containment_, that
the successor function results in the
missing set being the empty set, which in
this projection of ubiquitous ordinals,
isn't missing.

(I.e. the subsets are not empty these numbers.)

Of course, this reads a bit more into
the set-theoretic universe as having _numbers_,
than the usual ord-inary and its reg-ular (increment),
about the extra-ord-inary and its reg-ular (distribution).

With interest in these sorts of things, it's
more-than-less for _numbering_ before _counting_,
where all the objects are already _numbers_.

Dolt.

[Mostowski Collapse]

And why did the chicken choose the moebius strip?

[Ross A. Finlayson]

Couldn't get off of the Klein bottle?

[Ralf Bader]

Cantor proves that the digit sequence of the antidiagonal does not start
with any of the digit sequences "d_1 to d_n". That is, the antidiagonal
differs from every number starting with those digits. The rational
number 0.d_1...d_n0... is just one of these, but there are many more.

The idiocy of your crap is really amazing.

[WM]

Am Montag, 26. Oktober 2020 18:07:20 UTC+1 schrieb Jim Burns:
> On 10/26/2020 7:39 AM, WM wrote:
> > Am Sonntag, 25. Oktober 2020 20:00:16 UTC+1
> > schrieb Jim Burns:
>
> >> It's _logically possible_ for what we mean by
> >> "real numbers" to lack Dedekind completeness.
> >
> > Then the diagonal argument would fail in your opinion?
>
> _Logic_ does not dictate _what we mean_ by "real numbers".

Therefore the diagonal argument should hold also without Dedekind completeness. But according to you it would not.

>
> Suppose that we saw "only the rationals" in the number line.
>
> Consider the endless list of "real" (rational) numbers
> wherein p/q is at index k,
> k = (p+q-1)*(p+q-2)/2 + p
>
> Each entry p/q has a k_th decimal digit (p/q)#k.
>
> _If some number-line point d exists_ such that,
> _for all indexes k_, the k_th digit of d is sufficiently
> different from the k_th digit of p/q at index k,
> then, _for all indexes k_ d is not p/q at k.
> An example of sufficiently different would be if we defined
> d#k = ((p/q)#k+ 5) mod 10
>
> Does such a number-line point d exist? No.

Neither does it exist in a Cantor list. Note that every digit of the antidiagonal sits on a place belonging to the left-hand part of the list, between left side and diagonal (inclusive). That means it is taken from a rational number and defines a rational number. The antidiagonal is confined to finite digit sequences defining rationals.

>
> The diagonal argument fails for the rationals.

And other numbers are not available for constructing the antidiagonal.

Regards, WM

[WM]

Anyhow it is not possible to use it as an argument as Cantor did.

Regards, WM

[Me]

On Tuesday, October 27, 2020 at 6:21:37 PM UTC+1, WM wrote:

> it is not possible to use it as an argument as Cantor did.

CANTOR didn't use "it" as an argument, you silly crank.

[WM]

Am Montag, 26. Oktober 2020 19:07:54 UTC+1 schrieb Mostowski Collapse:
> To exhibit a counter example to
> Cantors theorem, you would need to exhibit
> a set-like function f : X -> P(X), which
> is surjective.

No, it is sufficient to show that countability is nonsense. That has been proved in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf.

Your argument, usually stated by matheologians, is the summit of nonsense. Here is a statememt nearly as silly: If you want to show that no devil exists, you have to prove that his grandmother was infertile.

Regards, WM

[WM]

Am Dienstag, 27. Oktober 2020 07:50:21 UTC+1 schrieb Ralf Bader:
> On 10/26/2020 05:40 PM, WM wrote:

> > One simple counter proof among man others: All that is used in a
> > Cantor-list is the digit sequence from d_1 to d_n, hence of a
> > rational number. Cantor proves that the digit sequence of the
> > antidiagonal is not one of the rationals of his list. Ridicoulos that
> > this has never been pointed out.
>
> Cantor proves that the digit sequence of the antidiagonal does not start
> with any of the digit sequences "d_1 to d_n". That is, the antidiagonal
> differs from every number starting with those digits.

But he can exclude the presence of such a number only for Finite Initial Segments of the Cantor-List (FISCL), because every definable entry belongs to a (FISCL). For every FISCL there is a larger FISCL. Potentiaö infinity.

An actually infinite list must have more lines. The mistaken argument is to conclude from

For all FISCL, there exists a real number not contained in it
to
There exists a real number not contained in all FISCL.

Regards, WM

[WM]

For every Finite Initial Segment of a given Cantor-List (FISCL) there exists a real number not contained in the FISCL. This is easy to show. But then the quantifier swapping yields: There exists a real number that is not contained in all FISCL of the given Cantor-list.

Regards, WM

[Mostowski Collapse]

There is no nonsense "Proofs by people".

If you were to find an inconsistency of ZFC,
this could be also fed into a computer. Its
just two proofs:

Proof 1: Deriving A

ZFC |- A

Proof 2: Deriving ~A

ZFC |- ~A

Both proofs would be finite objects that you
could feed into a computer step by step,
and the computer would say:

Proof 1: Verified, Yes

Proof 2: Verified, Yes

This way you could verify an inconsistency of
ZFC. But so far you didn't present any
inconsistency.

On Monday, October 26, 2020 at 3:42:01 PM UTC+1, WM wrote:

[Mostowski Collapse]

You only need counter examples to show
that something is not provable. But
it is impossible that something

is provable and not provable at the
same time. So Cantors Theorem is set
in stone. You cannot make it

unhappen. It happened.

P.S.: Here is how a counter example
model works. If you find a model M,
that satisifies a theory T and the

negation of a formula A, then you
can conclude:

T |/- A

i.e. that A is not provable from the
theory T.

[Mostowski Collapse]

You don't need counter examples
for inconsistency. Inconsistency
is two proofs T |- A and T |- ~A.

[WM]

Am Dienstag, 27. Oktober 2020 20:56:57 UTC+1 schrieb Mostowski Collapse:
> You only need counter examples to show
> that something is not provable. But
> it is impossible that something
> is provable and not provable at the
> same time.

It is possible that a proof is wrong.

> So Cantors Theorem is set
> in stone.

In sandstone.

> You cannot make it
> unhappen. It happened.

It happened the mistake that some people believed that all natnumbers were definable. This is wrong but it is the basis of countability.

[1/2, 1/1] U [1/3, 1/2] U [1/4, 1/3] U ... U [1/(n+1), 1/n] contains only all definable intervals and misses aleph_0 unit fractions.

[1/2, 1/1] U [1/3, 1/2] U [1/4, 1/3] U ... U [1/(n+1), 1/n] U ... contains all intervals including dark intervals and misses nothing.

Regards, WM

[WM]

Am Dienstag, 27. Oktober 2020 20:58:48 UTC+1 schrieb Mostowski Collapse:
> You don't need counter examples
> for inconsistency. Inconsistency
> is two proofs T |- A and T |- ~A.

A theory need not be inconsistent if the interpretation of its results is wrong. To show the latter a counter example is sufficient.

Regards, WM

[Mostowski Collapse]

You do not show an inconsistency by denying
a proof. You constantly tell us Cantor Theorem
doesn't exist.

But inconsistency is two proofs, two existencies,
Proof 1: Deriving A and Proof 2: Deriving ~A.
But you want to tell us

that Cantor Theorem is flawed. Youre in the
wrong track showing inconsistency of ZFC. Also
Cantor Theorem was long ago fed

into computers, and verified. Its not flawed.

[Mostowski Collapse]

63. Cantor's Theorem
HOL Light, John Harrison: statement
Isabelle, Larry Paulson: statement
Metamath, Norman Megill: statement
Coq, Jorik Mandemaker: statement
Mizar, Grzegorz Bancerek: statement
Lean, mathlib: statement
ProofPower, Rob Arthan: statement
NuPRL, Jim Caldwell

https://www.cs.ru.nl/~freek/100/

[Jim Burns]

On 10/27/2020 1:19 PM, WM wrote:
> Am Montag, 26. Oktober 2020 18:07:20 UTC+1
> schrieb Jim Burns:
>> On 10/26/2020 7:39 AM, WM wrote:
>>> Am Sonntag, 25. Oktober 2020 20:00:16 UTC+1
>>> schrieb Jim Burns:

>>>> It's _logically possible_ for what we mean by
>>>> "real numbers" to lack Dedekind completeness.
>>>
>>> Then the diagonal argument would fail in your opinion?
>>
>> _Logic_ does not dictate _what we mean_ by "real numbers".

I'm going to try again.

The reasons that we describe number-line points as
one of the points existing _between_ the sides of each
_cut_ are not reasons of _logic_

We have reasons; they are not reasons of logic.

Basically, we describe number-line points that way because
we don't want points to be missing from our description
of "all" the real numbers. We want to use a _complete_
number line.

( How one _expresses_ this completeness (in order to describe
( the number line as complete) might not be immediately obvious.
( Whatever a variable's domain may be, it is what it is.
( A variable ranges _completely_ over that (incomplete?) domain.
( If we describe a domain which misses intended number-line
( points, any variable over that domain does not refer to any
( missing point. A domain is what it *is*, it isn't what
( it *isn't* So, how do we say "There are no missing points"
( without saying "missing point"?
(
( This is the problem that Dedekind cuts solve. A Dedekind
( cut describes where a point _should_ exist relative to
( _all the other points_ -- to the right of all these points
( and to the left of all those points. We can let a variable
( range over the _other_ points in order to describe a
( conceivably-missing point, and then, for the number line,
( go on to say the conceivably-missing point is _not_ missing.

> Therefore the diagonal argument should hold also
> without Dedekind completeness.
> But according to you it would not.

*If* that quantifier swap
('forall exists' implies 'exists forall')
was valid, then Dedekind completeness would not be
needed to prove that the real numbers are uncountable.

That's a big 'if'. Too big an 'if', actually.

That quantifier swap is not a valid step.
If the diagonal argument depended upon that step,
then the diagonal argument would be invalid.
Fortunately(?), the diagonal argument does not depend
upon that step.

If Dedekind completeness was unnecessary to prove
uncountability, we would be able to prove that the
not-Dedekind-complete rational numbers were uncountable
_for logical reasons. It's not valid, and we can't do that.

We can assign an index k to each rational number p/q

k = (p+q-1)*(p+q-2)/2 + p

That proves the rationals are countable.
If we could do both, the theory without Dedekind completeness
would be inconsistent.

Fortunately(?), we can't do both of those, and absolutely
not by using this quantifier swap, which is invalid.

The diagonal argument, which depends upon Dedekind
completeness to be valid, is valid when applied to
the Dedekind complete real numbers and invalid when applied
to the not-Dedekind-complete rational numbers.

[WM]

Am Mittwoch, 28. Oktober 2020 20:01:18 UTC+1 schrieb Jim Burns:
> On 10/27/2020 1:19 PM, WM wrote:

> > Therefore the diagonal argument should hold also
> > without Dedekind completeness.
> > But according to you it would not.
>
> *If* that quantifier swap
> ('forall exists' implies 'exists forall')
> was valid, then Dedekind completeness would not be
> needed to prove that the real numbers are uncountable.

But it is not valid. And Dedekind completeness is not necessary.

> That quantifier swap is not a valid step.
> If the diagonal argument depended upon that step,
> then the diagonal argument would be invalid.

It does depend and is invalid.

> If we could do both, the theory without Dedekind completeness
> would be inconsistent.

It is inconsistent. The diagonal argument does use only the left side of the list up to the diagonal. There all digit sequences are finite and therefore rational.

>
> Fortunately(?), we can't do both of those, and absolutely
> not by using this quantifier swap, which is invalid.

Cantor does it. He did not know about completeness. And no logician applies your argument.

>
> The diagonal argument, which depends upon Dedekind
> completeness to be valid, is valid when applied to
> the Dedekind complete real numbers and invalid when applied
> to the not-Dedekind-complete rational numbers.

And invalid when applied to not Dedekind-complete real numbers.

Here are examples for arbitrariness in set theory:

(1) The sum over all finite terms of the sequence 0.9, 0.09, 0.009, ... is less than 1. The infinite sum is 1.
Here it does not matter that all finite terms fail.

(2) ∀n ∈ ℕ: The digit sequence 0.d1d2d3...dn of the antidiagonal in Cantor's list differs from the first n entries of the list. And in the limit the antidiagonal is said to differ from all entries of the Cantor-list too.

Here we can conclude, from "each dn fails" to the "failure of all". Unlike in case of the digits 9, it does matter in the limit that all finite terms fail.

(3) ∀n ∈ ℕ: The nth level of the Binary Tree has N(n) = 2^n nodes. The limit is said to be actually infinite. Here again, like in (1) it does not matter in the limit that all finite terms fail to be infinite.

Rather arbitrary.

Regards, WM

[Jim Burns]

On 10/29/2020 2:17 PM, WM wrote:
> Am Mittwoch, 28. Oktober 2020 20:01:18 UTC+1
> schrieb Jim Burns:

>> If we could do both, the theory without Dedekind
>> completeness would be inconsistent.
>
> It is inconsistent.

It often seems as though you don't read the posts to which
you respond.

"It" refers to a theory of an ordered field without
Dedekind completeness. For example, of the rationals.

Do you mean to say that the rationals are inconsistent?

----
To review:

On the one hand, we can prove
| for any list f(k) of rationals and any natural k,
| there is a rational p/q not equal to any of the first k
| listed rationals.

If we swapped the quantifiers (which would be invalid),
we would get
| for any list f(k), there is a rational p/q which,
| for any natural k, is not equal to any of the first k
| listed rationals

This is contradicted by f(k) = p/q defined as the inverse of

k = (p+q-1)*(p+q-2)/2 + p

That inverse is:

p+q = floor( sqrt(2*k-7/4) + 3/2 )

p = k - (p+q-1)*(p+q-2)/2

q = p+q - p

*If that quantifier swap was valid* then this would prove that
the rationals are inconsistent.

But maybe Cantor didn't think it was worth mentioning
that he could prove the rationals are inconsistent.
Sure, why not?

> The diagonal argument does use only the left side of
> the list up to the diagonal. There all digit sequences
> are finite and therefore rational.
>
>> Fortunately(?), we can't do both of those, and absolutely
>> not by using this quantifier swap, which is invalid.
>
> Cantor does it. He did not know about completeness.
> And no logician applies your argument.

Google "complete ordered field" sometime.

[Ross A. Finlayson]

Whose members of course in set theory are
"the equivalance classes of sequences what are Cauchy".

[WM]

Am Freitag, 30. Oktober 2020 18:28:05 UTC+1 schrieb Jim Burns:
> On 10/29/2020 2:17 PM, WM wrote:
> > Am Mittwoch, 28. Oktober 2020 20:01:18 UTC+1
> > schrieb Jim Burns:
>
> >> If we could do both, the theory without Dedekind
> >> completeness would be inconsistent.
> >
> > It is inconsistent.
>
> It often seems as though you don't read the posts to which
> you respond.

If there are wrong derivations or derivations from wrong premises, I usually don't continue reading.

>
> "It" refers to a theory of an ordered field without
> Dedekind completeness. For example, of the rationals.
>
> Do you mean to say that the rationals are inconsistent?

No. You said that the reals could exist without Dedekins-completeness.

>
> ----
> To review:
>
> On the one hand, we can prove
> | for any list f(k) of rationals and any natural k,
> | there is a rational p/q not equal to any of the first k
> | listed rationals.
>
> If we swapped the quantifiers (which would be invalid),
> we would get
> | for any list f(k), there is a rational p/q which,
> | for any natural k, is not equal to any of the first k
> | listed rationals

But the same argument shows that the irrationals are uncountable.

>
> *If that quantifier swap was valid* then this would prove that
> the rationals are inconsistent.
>
> But maybe Cantor didn't think it was worth mentioning
> that he could prove the rationals are inconsistent.

He did not recognize that his theory is inconsistent because he did not wish to recognize it - like most of his followers.

> > The diagonal argument does use only the left side of
> > the list up to the diagonal. There all digit sequences
> > are finite and therefore rational.
> >
> >> Fortunately(?), we can't do both of those, and absolutely
> >> not by using this quantifier swap, which is invalid.
> >
> > Cantor does it. He did not know about completeness.
> > And no logician applies your argument.
>
> Google "complete ordered field" sometime.

That exists but has nothing to do with the diagonal argument.
Google: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Any hint on complete orderd field?

Regards, WM

[Me]

On Friday, October 30, 2020 at 8:28:44 PM UTC+1, WM wrote:

> If there are wrong derivations or derivations from wrong premises,
> I usually don't continue reading.

Dumb disgusting asshole.

[Jim Burns]

On 10/30/2020 3:28 PM, WM wrote:
> Am Freitag, 30. Oktober 2020 18:28:05 UTC+1
> schrieb Jim Burns:
>> On 10/29/2020 2:17 PM, WM wrote:
>>> Am Mittwoch, 28. Oktober 2020 20:01:18 UTC+1
>>> schrieb Jim Burns:

>>>> If we could do both, the theory without Dedekind
>>>> completeness would be inconsistent.
>>>
>>> It is inconsistent.
>>
>> It often seems as though you don't read the posts to which
>> you respond.
>
> If there are wrong derivations or derivations from
> wrong premises, I usually don't continue reading.

Have considered just not having an opinion about
things you don't know about -- such as posts you
haven't read?

>> "It" refers to a theory of an ordered field without
>> Dedekind completeness. For example, of the rationals.
>>
>> Do you mean to say that the rationals are inconsistent?
>
> No. You said that the reals could exist without
> Dedekins-completeness.

No, I didn't say that.

An ordered field has the four operations '+','-,'*','/'
and an order '<' which obey the axioms you would expect.

Any ordered field _other than_ the complete ordered field,
that is, any ordered field _other than_ the real numbers
is "a theory of an ordered field without Dedekind
completeness". The rational numbers is one example.

"A theory of an ordered field without Dedekind completeness"
isn't necessarily _inconsistent_ It's _not the real numbers_

>> To review:
>>
>> On the one hand, we can prove
>> | for any list f(k) of rationals and any natural k,
>> | there is a rational p/q not equal to any of the first k
>> | listed rationals.
>>
>> If we swapped the quantifiers (which would be invalid),
>> we would get
>> | for any list f(k), there is a rational p/q which,
>> | for any natural k, is not equal to any of the first k
>> | listed rationals
>
> But the same argument shows that the irrationals are
> uncountable.

It's an invalid inference for both rationals and
real numbers. It's invalid, period.

For a _valid_ inference,
_whenever the *premises* are true, the *conclusion* is true_

For example, _modus ponens_

If P and P->Q are true, then Q is true.
Always, always, always:

There are four cases to consider.
The truth values of P and Q could be TT, TF, FT, FF.
The truth values of P, Q, P->Q are TTT, TFF, FTT, FFT.
Of the four, only TTT has P and P->Q true, and, in that
one case, Q is true.
Always, always, always being true is
what makes modus ponens valid.

That quantifier swap you use is not valid.
The conclusion is NOT true always, always, always.
I gave you a counter-example some posts ago.

Given...
for each human h, there is a woman w,
who is their mother w = mother(h).

...it is incorrect to infer
there is a woman w who, for each human h,
is their mother w = mother(h).

----
The reason we care about valid (truth-preserving)
inference:

Suppose we have an indefinite reference to
a natural number k which has a finite linear set
{ 0,1,...,k } of predecessors.

"k has a finite linear set { 0,1,...,k } of predecessors"
is something true we can say about k.
We know it's true even though we DON'T KNOW WHICH
natural number with finite linear predecessors k is.
We know that it's true because we set it up that way.

Suppose that we use "{ 0,1,...,k } is finite linear"
as a premise for some _valid_ inference.
It's valid: whenever the premises (about k) are true,
the conclusion is true (about k).
And all this goes on, start to finish, while we
still DON'T KNOW WHICH natural number with finite
linear predecessors k is.

[WM]

Am Samstag, 31. Oktober 2020 00:39:20 UTC+1 schrieb Jim Burns:
> On 10/30/2020 3:28 PM, WM wrote:


> > No. You said that the reals could exist without
> > Dedekins-completeness.
>
> No, I didn't say that.

JB: It's _logically possible_ for what we mean by "real numbers" to lack Dedekind completeness. (25 Oct)

> >> If we swapped the quantifiers (which would be invalid),
> >> we would get
> >> | for any list f(k), there is a rational p/q which,
> >> | for any natural k, is not equal to any of the first k
> >> | listed rationals
> >
> > But the same argument shows that the irrationals are
> > uncountable.
>
> It's an invalid inference for both rationals and
> real numbers. It's invalid, period.

Yes, but it's Cantor's argument.

Google: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
Any hint on complete orderd field?

> That quantifier swap you use is not valid.

I do not use it but I point out that Cantor uses it, as well as every mathematician except you.

> Given...
> for each human h, there is a woman w,
> who is their mother w = mother(h).
>
> ...it is incorrect to infer
> there is a woman w who, for each human h,
> is their mother w = mother(h).

That is wrong because mother means *direct* predecessor. If you drop direct, then all humans may well have a predessor or prime mother.
>
Regards, WM

[Mostowski Collapse]

Zuviel RTL Frauentausch im TV gesehen?

Frauentausch: Best of - RTL II
https://www.youtube.com/watch?v=SCUGPI-1BiA

LoL

[Me]

On Saturday, October 31, 2020 at 12:47:23 PM UTC+1, WM wrote:

>> Given...
>>
>> for each human h, there is a woman w,
>> who is their mother w = mother(h).
>>
>> ...it is incorrect to infer
>>
>> there is a woman w who, for each human h,
>> is their mother w = mother(h).
>>
> That is wrong because mother means *direct* predecessor. If you drop direct,
> then all humans may well have a predessor or prime mother.

And that prime mother is her own mother?

Holy shit!

Hint: From

For each integer h, there is an integer w,
which is the predecessor of h: w = h-1.

we can't infer

There is an integer w which, for each integer h,
is the predecessor of h: w = h-1.

And no, there's no "common predecessor" of "all integers" (direct or not).

[Jim Burns]

On 10/31/2020 7:47 AM, WM wrote:
> Am Samstag, 31. Oktober 2020 00:39:20 UTC+1
> schrieb Jim Burns:
>> On 10/30/2020 3:28 PM, WM wrote:

>>> No. You said that the reals could exist without
>>> Dedekins-completeness.
>>
>> No, I didn't say that.
>
> JB:
> It's _logically possible_ for what we mean by
> "real numbers" to lack Dedekind completeness.
> (25 Oct)

It's not _logically true_ that what we mean by
"real numbers" is Dedekind complete.
Nonetheless, it's _true_ that what we mean by
"real numbers" is Dedekind complete.

If you're sincerely confused by the distinction I'm
drawing, that would explain pretty much all your posts.

[WM]

Am Samstag, 31. Oktober 2020 17:06:48 UTC+1 schrieb Jim Burns:
> On 10/31/2020 7:47 AM, WM wrote:

> > JB:
> > It's _logically possible_ for what we mean by
> > "real numbers" to lack Dedekind completeness.
> > (25 Oct)
>
> It's not _logically true_ that what we mean by
> "real numbers" is Dedekind complete.
> Nonetheless, it's _true_ that what we mean by
> "real numbers" is Dedekind complete.
>

So you said that the reals could logically exist without Dedekind-completeness.

However this detail is irrelevant. Try to find support of your view that the diagonal argument needs Dedkind completeness or somethin equivalent.

Regards, WM

[WM]

Am Samstag, 31. Oktober 2020 13:16:45 UTC+1 schrieb Me:
> On Saturday, October 31, 2020 at 12:47:23 PM UTC+1, WM wrote:
>
> >> Given...
> >>
> >> for each human h, there is a woman w,
> >> who is their mother w = mother(h).
> >>
> >> ...it is incorrect to infer
> >>
> >> there is a woman w who, for each human h,
> >> is their mother w = mother(h).
> >>
> > That is wrong because mother means *direct* predecessor. If you drop direct,
> > then all humans may well have a predessor or prime mother.
>
> And that prime mother is her own mother?

Her mother could have had too little similarity with a human.

Regards, WM

[Jim Burns]

On 10/31/2020 7:47 AM, WM wrote:

> Am Samstag, 31. Oktober 2020 00:39:20 UTC+1
> schrieb Jim Burns:
>> On 10/30/2020 3:28 PM, WM wrote:

>>>> If we swapped the quantifiers (which would be invalid),
>>>> we would get
>>>> | for any list f(k), there is a rational p/q which,
>>>> | for any natural k, is not equal to any of the first k
>>>> | listed rationals
>>>
>>> But the same argument shows that the irrationals are
>>> uncountable.
>>
>> It's an invalid inference for both rationals and
>> real numbers. It's invalid, period.
>
> Yes, but it's Cantor's argument.
> Google: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

I find that literally unbelievable, but it doesn't matter.
If Cantor was wrong, he was wrong.[1]
The result I state does not require his blessing.

An Archimedean-finite infinite list of Dedekind-complete
real numbers misses at least one real number. Because...

...such a list determines two sets L,R of real numbers such that
(i) Dedekind completeness requires a point to exist
between L and R, and
(ii) no point in the list is between L and R.

[1]
In an excess of charity, I consider the possibility that
you saw _the equivalent_ of the assumption of Dedekind
completeness, and you didn't recognize it.
You, a quasi-teacher of quasi-mathematics, ought to know
that Dedekind completeness is equivalent to
| If two continuous curves cross,
| then they meet at some point

https://en.wikipedia.org/wiki/Intermediate_value_theorem

> Any hint on complete orderd field?

Seriously?
Ask someone how to use Google.

https://en.wikipedia.org/wiki/Real_number#%22The_complete_ordered_field%22

>> That quantifier swap you use is not valid.
>
> I do not use it but I point out that Cantor uses it,
> as well as every mathematician except you.

Oh! I feel so special!

https://www.whitman.edu/mathematics/higher_math_online/section01.04.html
1.4 Mixed Quantifiers

| In general, if you compare ∃y∀x P(x,y) with ∀x∃y P(x,y)
| it is clear that the first statement implies the second.
| If there is a fixed value y0 which makes P(x,y) true
| for all x, then no matter what x we are given, we can find
| a y (the fixed value y0) which makes P(x,y) true. So the
| first is a stronger statement. As in example 1.4.3, it is
| usually the case that this implication cannot be reversed.

Well, maybe not all that special.

Does that surprise you?
https://en.wikipedia.org/wiki/Dunning%E2%80%93Kruger_effect

>> Given...
>> for each human h, there is a woman w,
>> who is their mother w = mother(h).
>>
>> ...it is incorrect to infer
>> there is a woman w who, for each human h,
>> is their mother w = mother(h).
>
> That is wrong

That is wrong.
It doesn't matter why it's wrong or what else might be
right. It's wrong. _That's the point_

"Each human has a mother" does not force
"There exists an All-Mother" to be true.

"Each listed real misses a real" does not force
"There exists an All-Missed" to be true.

It's a bad argument. It _should not_ force the conclusion.
(That is, it's _invalid_ )

That doesn't mean that no All-Missing real exists.
It does exist. _But not for the reason you want to use_

A different argument which uses Dedekind completeness
_or its equivalent_ is valid. This is how we know
know an All-Missing real exists.

> That is wrong because mother means *direct* predecessor.
> If you drop direct, then all humans may well have
> a predessor or prime mother.

A favorite rhetorical tactic of yours is to say
"You're wrong if you say this other thing you're not saying".

When I say "natural numbers", I'm saying
"Archimedean-finite natural numbers".
When I say "real numbers", I'm saying
"Archimedean-finite and Dedekind-complete real numbers".

[WM]

Am Samstag, 31. Oktober 2020 19:26:10 UTC+1 schrieb Jim Burns:
> On 10/31/2020 7:47 AM, WM wrote:


> >>
> >> It's an invalid inference for both rationals and
> >> real numbers. It's invalid, period.
> >
> > Yes, but it's Cantor's argument.
> > Google: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
>
> I find that literally unbelievable, but it doesn't matter.

Ask someone whether he shares your opinion. For instance look into the other answers in this thread.

> If Cantor was wrong, he was wrong.[1]
> The result I state does not require his blessing.

But your statement is nonsense. It has no-ones blessing.
>
>
> [1]

Es läßt sich aber von jenem Satze ein viel einfacherer Beweis liefern, der unabhängig von der Betrachtung der Irrationalzahlen ist.
Sind nämlich m und w irgend zwei einander ausschließende Charaktere, so betrachten wir den Inbegriff M von Elementen

E = (x1, x2, ..., x, ...),

welche von unendlich vielen Koordinaten x1, x2, ... x, ... abhängen, wo jede dieser Koordinaten entweder m oder w ist. M sei die Gesamtheit aller Elemente E.
Zu den Elementen von M gehören beispielsweise die folgenden drei:

EI = (m, m, m, m, ...),
EII = (w, w, w, w, ...),
EIII = (m, w, m, w, ...).

Ich behaupte nun, daß eine solche Mannigfaltigkeit M nicht die Mächtigkeit der Reihe 1, 2, 3, ..., , ... hat.
Dies geht aus folgendem Satze hervor:
"Ist E1, E2, ..., E, ... irgendeine einfach unendliche Reihe von Elementen der Mannigfaltigkeit M, so gibt es stets ein Element E0 von M, welches mit keinem E übereinstimmt."

No hint to completeness. Even the domain of irrational numbers has been left.

> > Google: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

Any hint on the complete ordered field in this article?

Regards, WM

[Mostowski Collapse]

Everytime some posts some nonsense
"CANTOR IS DISPROVEN" a Bat dies.

LoL

[Jim Burns]

On 10/31/2020 1:59 PM, WM wrote:

> Try to find support of your view that
> the diagonal argument needs Dedkind completeness or
> somethin equivalent.

1.
The axioms of an ordered field together with
Dedekind completeness (that is, the real numbers)
prove the diagonal argument.

Because

An Archimedean-finite infinite list of Dedekind-complete
real numbers misses at least one real number. Because...

...such a list determines two sets L,R of real numbers such that

(i) Dedekind completeness would require a point to exist

between L and R, and
(ii) no point in the list is between L and R.

2.
The axioms of an ordered field WITHOUT Dedekind completeness
cannot prove the diagonal argument -- not without those
axioms being inconsistent.

Because
The rational numbers are an ordered field, and
they do not have Dedekind completeness, and
There is a complete list of rationals, and
this would contradict the diagonal argument.

----
Therefore, for an ordered field,
valid diagonal argument <-> Dedekind complete

[Jim Burns]

On 10/31/2020 2:55 PM, WM wrote:
> Am Samstag, 31. Oktober 2020 19:26:10 UTC+1
> schrieb Jim Burns:
>> On 10/31/2020 7:47 AM, WM wrote:

>>>> It's an invalid inference for both rationals and
>>>> real numbers. It's invalid, period.
>>>
>>> Yes, but it's Cantor's argument.
>>> Google: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
>>
>> I find that literally unbelievable, but it doesn't matter.
>
> Ask someone whether he shares your opinion.
> For instance look into the other answers in this thread.

"... but it doesn't matter."

>> If Cantor was wrong, he was wrong.[1]
>> The result I state does not require his blessing.
>
> But your statement is nonsense. It has no-ones blessing.

"...does not require his blessing."

----

An Archimedean-finite infinite list of Dedekind-complete
real numbers misses at least one real number. Because...

...such a list determines two sets L,R of real numbers
such that

(i) Dedekind completeness would require a point to exist

[Ross A. Finlayson]

On Saturday, October 31, 2020 at 12:19:24 PM UTC-7, Jim Burns wrote:
> ----
> Therefore, for an ordered field,
> valid diagonal argument <-> Dedekind complete

... for an ordered field, though ran(EF) has LUB property
with regards to that being same as "Dedekind complete".

[WM]

Am Samstag, 31. Oktober 2020 20:19:24 UTC+1 schrieb Jim Burns:
> On 10/31/2020 1:59 PM, WM wrote:
>
> > Try to find support of your view that
> > the diagonal argument needs Dedkind completeness or
> > somethin equivalent.
>
> 1.
> The axioms of an ordered field together with
> Dedekind completeness (that is, the real numbers)
> prove the diagonal argument.

Try to find support of your view that Dedkind completeness or something equivalent is required for the diagonal argument. Ask some set theorists.

Regards, WM

[Mostowski Collapse]

Dedekind completness is not needed.
Only the comprehension axiom.

[Jim Burns]

On 11/1/2020 8:04 AM, WM wrote:
> Am Samstag, 31. Oktober 2020 20:19:24 UTC+1
> schrieb Jim Burns:
>> On 10/31/2020 1:59 PM, WM wrote:

>>> Try to find support of your view that
>>> the diagonal argument needs Dedkind completeness or
>>> somethin equivalent.
>>
>> 1.
>> The axioms of an ordered field together with
>> Dedekind completeness (that is, the real numbers)
>> prove the diagonal argument.
>
> Try to find support of your view that

... the quantifier swap that you say we use is invalid?

There is no All-Mother, even though we all have mothers.
Done.

... any list of reals (reals which are Dedekind-complete)
misses at least one real?

Such a list determines two sets L,R of real numbers

such that
(i) Dedekind completeness would require a point to exist
between L and R, and
(ii) no point in the list is between L and R.

Done,
except for filling in details you will refuse to read anyway.

> Try to find support of your view that Dedkind completeness
> or something equivalent is required for the diagonal argument.

Done -- in the post to which you respond. It's still there.

The rational numbers, which form an ordered field which is
not Dedekind-complete, would be a counter-example to
"The diagonal argument can be validly applied to any
ordered field, Dedekind-complete or not".
Done again.

> Ask some set theorists.

They will tell me that sets do not form an ordered field.
The question of the Dedekind-completeness of sets is
cousin to "What flavor is up?"

The "something equivalent" for sets that I specifically
mentioned earlier, which makes "card(P(B)) > card(B)" valid,
is the axiom of separation. If you think there's a
problem with the axiom of separation, you go and ask
some set theorists about it.

[Jim Burns]

On 11/1/2020 8:51 AM, Mostowski Collapse wrote:
> Dedekind completness is not needed.
> Only the comprehension axiom.

For an ordered field, Dedekind completeness is needed to
prove any list of elements misses at least one element.
Proof: the rationals can be listed.

For sets, restricted comprehension/separation is sufficient
to prove card(P(B)) > card(B). I haven't thought about
whether it's needed.

Separation is one example of "or something equivalent"
I gave upthread when WM was skeptical that Dedekind
completeness was used in the diagonal argument.
His thought was that the diagonal argument shows up
throughout mathematics. How could it be valid elsewhere,
where Dedekind completeness would make no sense?

The over-all structure of a diagonal argument, somewhere,
about something, is to suppose there is a list of things
('list' read broadly) and then use the list to describe
a thing not on the list.

My claim is that "or equivalent" is needed in order
to step from _a description_ to _the existence_ of what
is described. Dedekind completeness for reals, separation
for sets, etc. As appropriate.

We do not always have what is needed. For example,
we can use a list of _rationals_ the same way we use
a list of _reals_ and we will get a description of
a rational not on the list. But does that rational exist?
Typically, no.

Without assuming "or equivalent", the diagonal argument fails.
Sometimes "or equivalent" is so obviously true that we do not
notice that we've assumed it. Nevertheless, it's there in
our assumptions, or we would not be able to step from
description to existence.

[Mostowski Collapse]

Cantors Theorem deals with |X| < |P(X)|
and not with |omega| < |F| where F is some
field. Cantors Theorem is much simpler.

[Mostowski Collapse]

https://en.wikipedia.org/wiki/Cantor's_theorem

[Mostowski Collapse]

Dedekind complete possibly allows you
to show |R| = |P(omega)|. And then you
can use Cantors theorem.

But |F| might be greater than |P(omega)|.
But not if it is Dedekind complete?

"3.6 Corollary. All Dedekind complete fields
are mutually order-isomorphic. Consequently
they have the same cardinality, which is
usually denoted by c."

Completeness of Ordered Fields
James Forsythe Hall - 29.01.2011
https://arxiv.org/pdf/1101.5652.pdf

[Ross A. Finlayson]

But, the rationals if they were "Dedekind complete"
would suffer under nested intervals ("Cantor's first
after the mw proof"). So, LUB's negated for rationals.

[WM]

Am Sonntag, 1. November 2020 15:47:08 UTC+1 schrieb Jim Burns:


> The "something equivalent" for sets that I specifically
> mentioned earlier, which makes "card(P(B)) > card(B)" valid,
> is the axiom of separation.

That holds for rationals as well as for irrationals.
Cantor's diagonal argument does not presuppose Dedekind-completeness. It is based on a simple quantifier swapping - like his list:

For every list there exists a larger list is correct.
There exists a list that is larger than all other lists is wrong.

Regards, WM

[Mostowski Collapse]

Nope the axiom of comprehension is
not some quantifier swap.

[Jim Burns]

On 11/2/2020 7:17 AM, WM wrote:
> Am Sonntag, 1. November 2020 15:47:08 UTC+1
> schrieb Jim Burns:

>> The "something equivalent" for sets that I specifically
>> mentioned earlier, which makes "card(P(B)) > card(B)" valid,
>> is the axiom of separation.
>
> That holds for rationals as well as for irrationals.

The axiom of separation holds for sets, including
for sets of rationals.

For F: Q -> P(Q), by separation,
a subset with the *description* { p e Q | p ~e F(p) } *exists*

We can easily prove F does not map to { p e Q | p ~e F(p) }.
Therefore, card(P(Q)) > card(Q)

That's what this version of the diagonal argument concludes.
Given also that the rationals are countable,
we can prove _sets_ of rationals are not countable.

> Cantor's diagonal argument ...

... would be invalid if the argument were
what you think it is.

_From Dedekind completeness_ there is a proof that
there are uncountably many real numbers. That *is* valid.

There is no proof from Dedekind completeness for
the rationals, since they are not Dedekind complete.
This non-existence of a proof non-conflicts with
the existence of lists of all the rationals.
For example, with p/q at k = (p+q-1)*(p+q-2)/2 + p

----
We can also look at some of the sets of rationals,
the Dedekind cuts of rationals, and show that the cuts
are uncountable. And, of course, that would show the
whole set was uncountable.

Of course, we already showed that, above, and more simply.
But the Dedekind cuts can be proven to be Dedekind complete.

With respect to order-by-set-inclusion, the least upper bound
of a bounded non-empty set C of cuts of the rationals
provably is the set union UC of C.

The proof that UC = LUB(C) depends upon the axiom of union.
If you're skeptical of the axiom of union, I suggest that
you talk to set theorists.

Since the Dedekind cuts of the rationals form the complete
ordered field, the proof that the complete ordered field
of real numbers is uncountable applies here.

> For every list there exists a larger list is correct.
> There exists a list that is larger than all other lists
> is wrong.

There is no list that contains all real numbers.

If there were such an All-Reals list, it would contain its
anti-diagonal. We know its anti-diagonal exists because
the reals are Dedekind complete. From the description of
its anti-diagonal, there is no item of the list that is
its antidiagonal. Contradiction.

Therefore, there is no All-Reals list.

----
On the other hand, if each rational p/q is at
(Archimedean natural) index k = (p+q-1)*(p+q-2)/2 + p
then each rational is at one and only one index.

By the pigeonhole principle, there are not more
rationals than Archimedean naturals.

[Mostowski Collapse]

Yes, it says there is such a set. But whether you
can use this set theoretic assumption in your current
context is another question.

For Dedekind cuts it seems like a present from
heaven. Or does it? Does Dedekind completness
need it at all?

We could also use a sigma Algebra, and wouldn't
need full set theory. We only need countable
union of cuts.

[Mostowski Collapse]

So WM has found sigma Algebras inconsistent?

LoL

[Ross A. Finlayson]

The ran(EF) is interesting with at least _two_ sigma algebras.

[Mostowski Collapse]

What do you mean by _two_ sigma algebras ?

[Mostowski Collapse]

Ebola and Corona?

[Jim Burns]

On 11/2/2020 2:19 PM, Mostowski Collapse wrote:

> Yes, it says there is such a set. But whether you
> can use this set theoretic assumption in your current
> context is another question.

I think the current context is unavoidably vague.
I'm answering WM
| The Cantor-list contains sequences of digits or symbols.
| That has nothing to do with Dedekind. When Cantor devised
| his argument he did not know about Dedekind-completeness.
| In fact the diagonal argument is applied everywhere in
| matheology. Don't blame it on Dedekind.
(Date: Fri, 23 Oct 2020 11:12:55 -0700 (PDT))

What is a diagonal argument, when it's not about reals?
What is "or equivalent" to Dedekind completeness?

My take on these questions is that, broadly,
a diagonal argument proves that the elements of X and Y
cannot be matched by using a presumed match to describe
an unmatched element -- and showing that unmatched element
exists.

It seems to me that the existence part will often be much
easier than the description part, but the argument as a
whole won't be valid without it.

Mueckenheim thinks "set theorists" use an invalid
quantifier swap to justify the existence part, instead
of Dedekind completeness. (Although, he has conceded
that *I* don't do that. I only had to repeat myself twenty
or so times.)

For UC = LUB(C), the context is the Dedekind cut
construction of the real numbers.

'UC = LUB(C)' is an elegant theorem.
It's one of the reasons I like to go to the Dedekind cut
construction of the reals, given the smallest excuse.

It's a theorem about the construction. I suppose that
the axiom of union is the existence part in this case.

> For Dedekind cuts it seems like a present from
> heaven. Or does it? Does Dedekind completness
> need it at all?
>
> We could also use a sigma Algebra, and wouldn't
> need full set theory. We only need countable
> union of cuts.

I don't see what you want to do here.

If we only had countable unions, would only
countable sets of cuts have least upper bounds?
This would strike me as defeating the purpose of
constructing the reals.

[Mostowski Collapse]

You only need countable (infinite) unions.
A digit sequence for example is only
countably long. And a infinite sum with
more than countably (infinite) many non-zero

summands diverges. A dedekind cut over rational
numbers has also only countably (infinite)
elements, since the rational numbers are
countably (infinite).

[Mostowski Collapse]

But I dont remember how LUB M, where
M is subset of R, is reduced to a countable
(infinite) union, if at all? Since M

can be uncountable. You could do the following:

M' = { q e Q | exists r e M /\ q =< r }

Then, as a real number:

LUB M = LUB M' ??

And M' is obviously countable (infinite). The
construction of M' needs the comprehension
axiom I guess.

[Mostowski Collapse]

Didn't Weyl observe these things?
Feferman wrote about Weyl in this respect.