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Topic: On the diagonal argument again
Replies: 199   Last Post: Jun 10, 2012 10:12 AM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,044 Registered: 5/20/10
Re: On the diagonal argument again
Posted: May 19, 2012 6:25 PM

On May 20, 7:58 am, "K_h" <KHol...@SX729.com> wrote:
> "Graham Cooper"  wrote in message
>
>
> On May 19, 10:20 am, "K_h" <KHol...@SX729.com> wrote:
>

> > > > Your point seems to be 'you made typos so you are wrong and confused'.
>
> > > Those are hardly just typos.  You actually claimed that well-ordering is
> > > an
> > > axiom of ZFC!  This indicates a big unfamiliarity with the subject.

>
> > I cut and pasted a text version from Transfer Principle of ZFC axioms
> > since they are images in Wikipedia.

>
> Odd.  First you claimed you made typos then you claimed it is somebody else
> who made typos.  No matter.
>

Not exactly.

>
> > > > It looks similar to ZFC AOI
>
> > > > FUNCTION P(x, INF)
> > > > BEGIN
> > > >   IF (x = 0) RETURN TRUE
> > > >   ELSE RETURN P(x-1, INF)
> > > > END

>
> > > It looks like you have replaced the bi-conditional with a conditional
> > > (as I
> > > recommended) so your P() function only counts down to zero and can never
> > > count up  (assuming it starts with a natural).  If your function starts
> > > with
> > > a natural number n then it will generate sets of the form
> > > {0,1,2,...,n-1,n}.
> > > It will also generate infinite sets: if it starts with X=pi then you
> > > will
> > > generate the infinite set {...,pi-5,pi-4,pi-3,pi-2,pi-1,pi}.  It's not a
> > > well ordered set but it is a genuine set in ZF as well as satisfying
> > > your
> > > `set membership predicate'.

>
> > It's a DEFINITION OF Predicate P to be SUBSTITUTED into the
> > SPECIFICATION axiom.

>
> > There is no "->" suitable here.
>
> Previously you had a thing like P(X,INF)<-->(stuff) and I suggested you
> write P(X,INF)-->(stuff) so you only "count down" instead of "counting up".
> Now you write P(X,INF) as a function that only counts down instead of being
> part of a bi-conditional assertion.  Now you write that "-->" is not
> suitable here yet the bi-conditional <--> is the conjunction of two
> conditionals -->.  So is <--> suitable here and, if so, why and, if not, why
> not?

It's a SET MEMBERSHIP PREDICATE straight out of NAIVE SET THEORY.

The predicate P() is defined as "x ~e x"
when defining Russell's Set.

Because Predicates and their definitions both have truth values, a
biconditional is shorthand (and works) for predicate definitions.

e.g.

DEFINE PLUS(a,b) = "a + b" // a numeric value function

DEFINE IS-IN-RUSSELL-SET(a, RS) = "a ~e a" // uses truth values

or more concisely in LOGIC

IS-IN-RUSSELL-SET(a, RS) <-> a ~e a

Predicates can be defined using bi-conditionals in logic

Then we can safely SUBSTITUTE the formula for P for any P in other
formula.

Strictly when talking about many different Predicates we should name
them differntly.

P-RS(x, RS) <-> x ~e x
P-INF(n, INF) <-> ...

or P1() P2() ...

P just stands for any Predicate.

----

In Naive Set Theory any definable collection is a set.

N.S.T. Y = { x | P(x,Y) }

OR

x e Y <-> P(x,Y)

NOTICE NST is similar format to AXIOM OF SPECIFICATION

----

If you **define** P to be

P(x,Y) <-> x ~e x

then you get

Y e Y <-> Y ~e Y

A weakness in NST making it INCONSISTENT.

that's why ZFC was invented to circumvent Predicate Set Instantiation
with subset instantiation only.

>
>
>

> > RIGHT!
>
> > The following must hold true for INF to stratify as a set.
>
> > INF = { x | (x = 0) v (E(z) x = S(z) & P(z, INF)) }
> > IFF
> > NOT(PROOF(NOT(EXIST(INF) INF = { x | (x = 0) v (E(z) x = S(z) & P(z,
> > INF)) } )))

>
> > So all you do is ASSUME the 3rd line just above!
>
> Yes.  There is no proof that infinity does not exist because infinity does

the only Self Evident Truths I accept are anti-paradoxes!

'There is no Russell's Set!'

> exist.  So the third line is more than a speculative assumption, it is a
> fact for reasonable domains like N.  You've defined INF to be the set of all
> things satisfying :
>
>   { x | (x = 0) v (E(z) x = S(z) & P(z, INF)) }
>
> and every natural number satisfies this.  So, assuming your domain is N,
> then INF=N and obviously INF is not a member of N since it is not a natural
> number.
>

> > The Predicate version of INF is
>
> > P(x,INF) = (x = 0) v (E(z) x = S(z) & P(z, INF))
>
> > which only holds if x is a Natural Number, otherwise no z would exist
> > to satisfy E(z)

>
> ALEPH_0 is not a natural number so naturally it won't have a predecessor.
> But your relation above does hold true for hyper-natural numbers and surreal
> numbers.  Consider the infinite hyper-natural number w-1:
>
> P(w-1, INF) = (x = 0) v (E(w-2) w-1 = S(w-2) & P(w-2, INF))
>
> So the domain is only the natural numbers if you specify it.  So to specify
> it you need to add the statement AxAz((xeN)^(zeN)).  So your predicate needs
> to be:
>
> P(x,INF) = (x = 0) v (E(z) x = S(z) & P(z, INF)) ^ (AxAz  (xeN)^(zeN) )
>
>
>

> > P(Pi, INF) = (x = 0) v (E(z) x = S(z) & P(z, INF))
> > P(Pi, INF) = E(z) x = S(z) & P(z, INF)
> > P(Pi, INF) = FALSE

>
> > Pi is not the successor of a [successor of] 0
>
>

The central tenet of formalism is context-free deduction.

i.e. being able to do maths WITHOUT understanding why.

A computer won't know that N EXISTS because it's evident!

This is just SYNTAX not a SET.

Y = { x | P(x) }

This is what the computer sees:

x e y <-> P(x) <-> x e x

where e is a RELATION

TABLE e
------------
MEMBER SET
tom people
1 N
2 N
3 N
7 PRIMES
..

Herc

Date Subject Author
5/15/12 LudovicoVan
5/15/12 LudovicoVan
5/15/12 William Hughes
5/15/12 LudovicoVan
5/15/12 Virgil
5/16/12 Graham Cooper
5/15/12 Shmuel (Seymour J.) Metz
5/15/12 LudovicoVan
5/15/12 Virgil
5/16/12 mueckenh@rz.fh-augsburg.de
5/16/12 Daryl McCullough
5/16/12 LudovicoVan
5/16/12 LudovicoVan
5/16/12 Virgil
5/16/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 Graham Cooper
5/18/12 Michael Stemper
5/18/12 Daryl McCullough
5/18/12 Shmuel (Seymour J.) Metz
5/18/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Shmuel (Seymour J.) Metz
5/16/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/18/12 Virgil
5/18/12 Graham Cooper
5/18/12 David Bernier
5/18/12 Graham Cooper
5/18/12 K_h
5/18/12 |-| E R C
5/18/12 K_h
5/18/12 Graham Cooper
5/18/12 Graham Cooper
5/19/12 K_h
5/19/12 Graham Cooper
5/19/12 Graham Cooper
5/19/12 David Bernier
5/19/12 Graham Cooper
5/19/12 Graham Cooper
5/20/12 K_h
5/20/12 Graham Cooper
5/21/12 K_h
5/21/12 Graham Cooper
5/29/12 mueckenh@rz.fh-augsburg.de
5/29/12 Virgil
5/19/12 K_h
5/19/12 Graham Cooper
5/18/12 LudovicoVan
5/18/12 LudovicoVan
5/18/12 Shmuel (Seymour J.) Metz
5/17/12 Graham Cooper
5/17/12 Shmuel (Seymour J.) Metz
5/18/12 LudovicoVan
5/19/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/20/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/19/12 LudovicoVan
5/20/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/19/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Virgil
5/18/12 K_h
5/15/12 dilettante
5/16/12 LudovicoVan
5/16/12 dilettante
5/16/12 LudovicoVan
5/16/12 dilettante
5/16/12 Graham Cooper
5/16/12 dilettante
5/16/12 Graham Cooper
5/16/12 dilettante
5/16/12 Graham Cooper
5/16/12 LudovicoVan
5/16/12 dilettante
5/16/12 LudovicoVan
5/16/12 Virgil
5/17/12 William Hughes
5/17/12 LudovicoVan
5/17/12 William Hughes
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Virgil
5/16/12 |-| E R C
5/16/12 LudovicoVan
5/16/12 Virgil
5/16/12 LudovicoVan
5/16/12 Jim Burns
5/16/12 LudovicoVan
5/16/12 Graham Cooper
5/16/12 Jim Burns
5/16/12 LudovicoVan
5/16/12 Jim Burns
5/16/12 |-| E R C
5/17/12 LudovicoVan
5/17/12 Jim Burns
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/18/12 Virgil
5/17/12 |-| E R C
5/18/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/16/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 Graham Cooper
5/18/12 Daryl McCullough
5/18/12 Graham Cooper
5/18/12 Virgil
5/17/12 LudovicoVan
5/17/12 Virgil
5/17/12 LudovicoVan
5/17/12 YBM
5/17/12 LudovicoVan
5/18/12 Virgil
5/18/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/17/12 Shmuel (Seymour J.) Metz
5/19/12 LudovicoVan
5/20/12 Shmuel (Seymour J.) Metz
5/16/12 Virgil
5/29/12 Aatu Koskensilta
5/29/12 Graham Cooper
5/30/12 K_h
5/30/12 Virgil
5/30/12 Curt Welch
5/30/12 Graham Cooper
5/30/12 ross.finlayson@gmail.com
5/30/12 Graham Cooper
5/30/12 K_h
6/1/12 ross.finlayson@gmail.com
6/8/12 Albert van der Horst
5/29/12 Aielyn
5/29/12 Aielyn
5/29/12 Graham Cooper
5/29/12 Aielyn
5/29/12 Mike Terry
5/31/12 Aielyn
5/31/12 Virgil
5/31/12 Aielyn
5/31/12 Virgil
5/31/12 Aielyn
5/31/12 Shmuel (Seymour J.) Metz
5/31/12 Shmuel (Seymour J.) Metz
5/31/12 Mike Terry
6/1/12 Graham Cooper
6/1/12 Guest
6/1/12 LudovicoVan
6/1/12 LudovicoVan
6/1/12 LudovicoVan
6/1/12 LudovicoVan
6/1/12 Graham Cooper
6/1/12 Virgil
6/1/12 Graham Cooper
6/1/12 LudovicoVan
6/1/12 |-| E R C
6/10/12 Aielyn
6/10/12 Shmuel (Seymour J.) Metz
5/29/12 Tim Little
5/31/12 LudovicoVan
6/5/12 Shmuel (Seymour J.) Metz
6/4/12 Aatu Koskensilta
6/4/12 Graham Cooper
6/4/12 ross.finlayson@gmail.com
6/4/12 ross.finlayson@gmail.com
6/4/12 Graham Cooper
6/4/12 ross.finlayson@gmail.com
6/4/12 |-| E R C