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IFF every Natural Number was a SET of Natural Numbers
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Graham Cooper
May 21, 2012, 3:15:52 AM5/21/12
to
On May 21, 8:31 am, Jim Burns <burns...@osu.edu> wrote:
> I am curious if you have any objections to the general diagonal argument.
> Let A be a set, P(A) be its powerset, and f: A -> P(A).
> Then there is at least one element D of P(A) (D subset A)
> such that, for all x in A, f(x) =/= D.
>
> Proof: Consider D = { y in A | y not in f(y) }.
In a Countable Sets UoD this is d = { n | n ~e n }
Using a PURE SET THEORY analogue of PURE MATHEMATICS
where each nat corresponds to some set of nats
e.g.
1 = {1,2,3}
2 = {2,4,6,8...}
...
the MISSING SET or ANTIDIAGONAL of Cantor's Powerset Proof
is { x | x ~e x }
Herc
> I am curious if you have any objections to the general diagonal argument.
> Let A be a set, P(A) be its powerset, and f: A -> P(A).
> Then there is at least one element D of P(A) (D subset A)
> such that, for all x in A, f(x) =/= D.
>
> Proof: Consider D = { y in A | y not in f(y) }.
In a Countable Sets UoD this is d = { n | n ~e n }
Using a PURE SET THEORY analogue of PURE MATHEMATICS
where each nat corresponds to some set of nats
e.g.
1 = {1,2,3}
2 = {2,4,6,8...}
...
the MISSING SET or ANTIDIAGONAL of Cantor's Powerset Proof
is { x | x ~e x }
Herc
Virgil
May 21, 2012, 4:47:45 AM5/21/12
to
In article
<a4fff1f4-c359-405d...@x6g2000pbh.googlegroups.com>,
I know of a set theory in which 1 = {{}} and 2 = { {}, {{}} }, but none
in which 1 = {1,2,3} or 2 = {2,4,6,8...}.
Do you have any references to such a set theory in the literature?
Particularly any in which { x | x ~e x } is a set at all?
--
<a4fff1f4-c359-405d...@x6g2000pbh.googlegroups.com>,
in which 1 = {1,2,3} or 2 = {2,4,6,8...}.
Do you have any references to such a set theory in the literature?
Particularly any in which { x | x ~e x } is a set at all?
--
Graham Cooper
May 21, 2012, 5:18:36 AM5/21/12
to
On May 21, 6:47 pm, Virgil <vir...@ligriv.com> wrote:
>
> > On May 21, 8:31 am, Jim Burns <burns...@osu.edu> wrote:
> > > I am curious if you have any objections to the general diagonal argument.
> > > Let A be a set, P(A) be its powerset, and f: A -> P(A).
> > > Then there is at least one element D of P(A) (D subset A)
> > > such that, for all x in A, f(x) =/= D.
>
> > > Proof: Consider D = { y in A | y not in f(y) }.
>
> > In a Countable Sets UoD this is d = { n | n ~e n }
>
> > Using a PURE SET THEORY analogue of PURE MATHEMATICS
> > where each nat corresponds to some set of nats
>
> > e.g.
> > 1 = {1,2,3}
> > 2 = {2,4,6,8...}
> > ...
>
> > the MISSING SET or ANTIDIAGONAL of Cantor's Powerset Proof
> > is { x | x ~e x }
>
> > Herc
>
> I know of a set theory in which 1 = {{}} and 2 = { {}, {{}} }, but none
> in which 1 = {1,2,3} or 2 = {2,4,6,8...}.
>
> Do you have any references to such a set theory in the literature?
>
Let the set { {{}}, { {}, {{}} } } have the name {{}}
>
> > On May 21, 8:31 am, Jim Burns <burns...@osu.edu> wrote:
> > > I am curious if you have any objections to the general diagonal argument.
> > > Let A be a set, P(A) be its powerset, and f: A -> P(A).
> > > Then there is at least one element D of P(A) (D subset A)
> > > such that, for all x in A, f(x) =/= D.
>
> > > Proof: Consider D = { y in A | y not in f(y) }.
>
> > In a Countable Sets UoD this is d = { n | n ~e n }
>
> > Using a PURE SET THEORY analogue of PURE MATHEMATICS
> > where each nat corresponds to some set of nats
>
> > e.g.
> > 1 = {1,2,3}
> > 2 = {2,4,6,8...}
> > ...
>
> > the MISSING SET or ANTIDIAGONAL of Cantor's Powerset Proof
> > is { x | x ~e x }
>
> > Herc
>
> I know of a set theory in which 1 = {{}} and 2 = { {}, {{}} }, but none
> in which 1 = {1,2,3} or 2 = {2,4,6,8...}.
>
> Do you have any references to such a set theory in the literature?
>
i.e. 1 = {1,2}
But since you all insist N is a SELF EVIDENT FACT
I don't see what you're problem is using Natural Numbers as a
primitive.
> Particularly any in which { x | x ~e x } is a set at all?
>
Herc
Tonico
May 21, 2012, 6:34:39 AM5/21/12
to
Naive idiocy
George Greene
May 21, 2012, 6:45:22 AM5/21/12
to
> > Let A be a set, P(A) be its powerset, and f: A -> P(A).
> > Then there is at least one element D of P(A) (D subset A)
> > such that, for all x in A, f(x) =/= D.
>
> > Proof: Consider D = { y in A | y not in f(y) }.
> > Then there is at least one element D of P(A) (D subset A)
> > such that, for all x in A, f(x) =/= D.
>
> > Proof: Consider D = { y in A | y not in f(y) }.
On May 21, 3:15 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> In a Countable Sets UoD this is d = { n | n ~e n }
No, it isn't. In the usual set theory, sets are NEVER elements of
> In a Countable Sets UoD this is d = { n | n ~e n }
themselves.
Therefore, d would be the universal set. But in the usual set theory,
the universal set
DOES NOT EXIST.
The question is NOT about n such that n~en.
The question IS about n such that n~ef(n).
Stop skipping the f! The question is REALLY about the f, not the d !
> Using a PURE SET THEORY analogue of PURE MATHEMATICS
> where each nat corresponds to some set of nats
>
> e.g.
> 1 = {1,2,3}
> 2 = {2,4,6,8...}
encoding, every natural number IS a set of natural numbers, namely,
the set of ALL SMALLER natural numbers:
0 = { }
1 = {0}
2 = {0,1}
3 = {0,1,2}
etc. One of the best things about this encoding is that for all n,
the set encoding n has cardinality n.
You CAN'T HAVE a "countable sets universe of discourse" for a set
theory AND a powerset axiom, unless all your countable sets happen to
be finite.
Graham Cooper
May 21, 2012, 8:06:34 AM5/21/12
to
On May 21, 8:45 pm, George Greene <gree...@email.unc.edu> wrote:
> > > Let A be a set, P(A) be its powerset, and f: A -> P(A).
> > > Then there is at least one element D of P(A) (D subset A)
> > > such that, for all x in A, f(x) =/= D.
>
> > > Proof: Consider D = { y in A | y not in f(y) }.
>
> On May 21, 3:15 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > In a Countable Sets UoD this is d = { n | n ~e n }
>
> No, it isn't. In the usual set theory, sets are NEVER elements of
> themselves.
I meant in my 'PST'.
> > > Let A be a set, P(A) be its powerset, and f: A -> P(A).
> > > Then there is at least one element D of P(A) (D subset A)
> > > such that, for all x in A, f(x) =/= D.
>
> > > Proof: Consider D = { y in A | y not in f(y) }.
>
> On May 21, 3:15 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > In a Countable Sets UoD this is d = { n | n ~e n }
>
> No, it isn't. In the usual set theory, sets are NEVER elements of
> themselves.
There is a difference between a CYCLIC GRAPH
and a ACYCLIC GRAPH that is REFLEXIVE
So REGULARITY, if it was required, would just allow XeX but not XeYeX.
There is no recurring loop in IDEAS = {IDEAS, THOUGHTS, FEELINGS}
> Therefore, d would be the universal set. But in the usual set theory,
> the universal set
> DOES NOT EXIST.
> The question is NOT about n such that n~en.
> The question IS about n such that n~ef(n).
> Stop skipping the f! The question is REALLY about the f, not the d !
>
> > Using a PURE SET THEORY analogue of PURE MATHEMATICS
> > where each nat corresponds to some set of nats
>
> > e.g.
> > 1 = {1,2,3}
> > 2 = {2,4,6,8...}
>
> Now, you're just being an idiot -- in the usual (von Neumann)
> encoding, every natural number IS a set of natural numbers, namely,
> the set of ALL SMALLER natural numbers:
> 0 = { }
> 1 = {0}
> 2 = {0,1}
> 3 = {0,1,2}
> etc. One of the best things about this encoding is that for all n,
> the set encoding n has cardinality n.
0 <=> {}
1 <=> {0}
I mean VARIABLE NAMES - SPECIFICATION
1 <=> {0}
EXIST(5) 5 = {1,2,3}
BTW how on Earth do you have a set {{0,1,2}} distinct from {3} ???
>
> You CAN'T HAVE a "countable sets universe of discourse" for a set
> theory AND a powerset axiom, unless all your countable sets happen to
> be finite.
~e n }
If UTM^2 can enumerate 1X2X3X4X5... permutations of N, should be easy!
Herc
George Greene
May 21, 2012, 11:12:42 PM5/21/12
to
On May 21, 8:06 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> I meant in my 'PST'.
> There is a difference between a CYCLIC GRAPH
> and a ACYCLIC GRAPH that is REFLEXIVE
>
> So REGULARITY, if it was required, would just allow XeX but not XeYeX.
>
> There is no recurring loop in IDEAS = {IDEAS, THOUGHTS, FEELINGS}
OK, it's a distinction.
> I meant in my 'PST'.
> There is a difference between a CYCLIC GRAPH
> and a ACYCLIC GRAPH that is REFLEXIVE
>
> So REGULARITY, if it was required, would just allow XeX but not XeYeX.
>
> There is no recurring loop in IDEAS = {IDEAS, THOUGHTS, FEELINGS}
But that doesn't change the fact that
> > The question is NOT about n such that n~en.
> > The question IS about n such that n~ef(n).
> > Stop skipping the f! The question is REALLY about the f, not the d !
> > Now, you're just being an idiot -- in the usual (von Neumann)
> > encoding, every natural number IS a set of natural numbers, namely,
> > the set of ALL SMALLER natural numbers:
> > 0 = { }
> > 1 = {0}
> > 2 = {0,1}
> > 3 = {0,1,2}
> > etc. One of the best things about this encoding is that for all n,
> > the set encoding n has cardinality n.
> No this is
> 0 <=> {}
> 1 <=> {0}
You have stopped too soon! So far, your encoding IS THE SAME as von
> > encoding, every natural number IS a set of natural numbers, namely,
> > the set of ALL SMALLER natural numbers:
> > 0 = { }
> > 1 = {0}
> > 2 = {0,1}
> > 3 = {0,1,2}
> > etc. One of the best things about this encoding is that for all n,
> > the set encoding n has cardinality n.
> No this is
> 0 <=> {}
> 1 <=> {0}
Neumann's!
I assume you meant it to be different in SOME way but YOU HAVEN'T SAID
HOW!
The only obvious difference is that you are using <=> instead of = ,
which simply makes no sense.
>
> I mean VARIABLE NAMES - SPECIFICATION
either, or any kind of variable, either.
5 IS A NUMBER which makes it A CONSTANT, NOT a variable, and, again, A
NUMBER, THEREFORE, NOT a name.
>
> EXIST(5) 5 = {1,2,3}
No, there doesn't; EXIST(5) 5 is incoherent because 5 is not a
variable.
And EXIST(5) 5 = whatever is even MORE incoherent; what goes after the
right parenthesis has to be some sort of delimiter separating the
quantifier from the open sentence being quantified OVER. If you mean
Ex[x={1,2,3}] THEN SAY that. And it would be much better to say
Ex[x={0,1,2}] because THEN everybody WOULD KNOW that the x in question
WAS 3. And you canNOT say EXIST(3) 3= {0,1,2} to mean that. You have
to KNOW SOME GRAMMAR for YOUR OWN language.
> BTW how on Earth do you have a set {{0,1,2}} distinct from {3} ???
{0,1,2} distinct from 3 ! Von Neumann et al think that 3 = {0,1,2}.
YOU are the one proposing 5 = {1,2,3}. WE said that 5={0,1,2,3,4}.
> > You CAN'T HAVE a "countable sets universe of discourse" for a set
> > theory AND a powerset axiom, unless all your countable sets happen to
> > be finite.
>
> Why not? the missing set in the PowerProof is Russell's Set { n | n ~e n }
IS AN ALLEGED BIJECTION BETWEEN
A and p(A) for some set A, and all the n's are in A.
>
> If UTM^2 can enumerate 1X2X3X4X5... permutations of N,
Graham Cooper
May 22, 2012, 4:24:39 AM5/22/12
to
On May 22, 1:12 pm, George Greene <gree...@email.unc.edu> wrote:
> > No this is
> > 0 <=> {}
> > 1 <=> {0}
>
> You have stopped too soon! So far, your encoding IS THE SAME as von
> Neumann's!
> I assume you meant it to be different in SOME way but YOU HAVEN'T SAID
> HOW!
SEE BELOW!
> > No this is
> > 0 <=> {}
> > 1 <=> {0}
>
> You have stopped too soon! So far, your encoding IS THE SAME as von
> Neumann's!
> I assume you meant it to be different in SOME way but YOU HAVEN'T SAID
> HOW!
> > > You CAN'T HAVE a "countable sets universe of discourse" for a set
> > > theory AND a powerset axiom, unless all your countable sets happen to
> > > be finite.
>
> > Why not? the missing set in the PowerProof is Russell's Set { n | n ~e n }
>
> NO, dumbass, the missing set in Cantor's proof is {n| n~eF(n)} WHERE F
> IS AN ALLEGED BIJECTION BETWEEN
> A and p(A) for some set A, and all the n's are in A.
P(N) = P(V)
-> MISSING SET = RUSSELL SET
What's the missing set here?
(1, {1,2,3})
(2, {2,3,4,5})
(3, {1,2})
(4, {})
>
>
>
> > If UTM^2 can enumerate 1X2X3X4X5... permutations of N,
>
> well, it CAN'T, so give THAT up.
< 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 TMs can NO UTM emulate?
TM4, TM7, TM1, ... ?
There exists a UTM that will produce that sequence!
*********NUMBER SETS***************
Originally called Pure Set Theory.
ALPHABET = {0,1,2,3,4,5,6,7,8,9, e, {, }, ,, =, A, E, ^, v, !, >, <,
n, m, o, p ,q, f, g, h, i, j}
DICTIONARY = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ...}
Note 1+1 = 2 is irrelevant at this level.
OPTIONAL# AXIOM OF SPECIFICATION
E(n) m e n <> f(m)
IMPLIES IS A SINGLE CHAR >
#by being an optional axiom you can upgrade it later from Naive Set
Theory!
#this is the thrux of the argument anyway
Herc
George Greene
May 25, 2012, 3:14:25 PM5/25/12
to
On May 22, 4:24 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
> 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?
OF COURSE we are saying that!
> 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?
There are WAY TOO MANY such permutations for you to list them!
There are ZILLIONS AND ZILLIONS of sequences missing!
You don't even HAVE A WAY to choose JUST ONE of the millions missing!
Graham Cooper
May 25, 2012, 6:31:40 PM5/25/12
to
<BB(1) , 1 , BB(2) , 2 , BB(3) , 3 , BB(4) , 4 ... >
is a missing permutation of N from UTM^2 since the sequence BB() in
uncomputable.
note in the original reference sequence TM1, TM2, TM3, TM4,...
where
EXIST(n) TMn = 1
EXIST(o) TMo = 2
EXIST(p) TMp = 3
EXIST(q) TMq = 4
...
there are duplicates, which suggests an AXIOM OF EXTENSIONALTIY for
computable objects/naturals.
ALL(x) ALL(y) ( ALL(z) TM-x=z ^ TM-y=z ) -> x = y
x=y could be x==y, i.e. two computer programs compute exactly the same
thing.
a paramaterized version would be better.
ALL(x) ALL(y) ( ALL(z) ALL(a) TM-x(a)=z ^ TM-y(a)=z ) -> x == y
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
Graham Cooper
May 25, 2012, 6:36:03 PM5/25/12
to
On May 26, 8:31 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> a paramaterized version would be better.
>
> ALL(x) ALL(y) ( ALL(z) ALL(a) TM-x(a)=z ^ TM-y(a)=z ) -> x == y
>
Better still..
>
> a paramaterized version would be better.
>
> ALL(x) ALL(y) ( ALL(z) ALL(a) TM-x(a)=z ^ TM-y(a)=z ) -> x == y
>
ALL(x) ALL(y) ( ALL(a) EXIST(z) TM-x(a)=z ^ TM-y(a)=z ) -> x == y
or
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)
Herc
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