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Topic: UNCOUNTABILITY
Replies: 59   Last Post: Dec 24, 2012 2:06 PM

 Messages: [ Previous | Next ]
 Graham Cooper Posts: 4,495 Registered: 5/20/10
Re: UNCOUNTABILITY
Posted: Dec 20, 2012 4:05 AM

On Dec 20, 7:11 am, Zuhair <zaljo...@gmail.com> wrote:
> Some people are trying to prove from definability that only countably
> many sets can exist. The idea is that a set is an object extension of
> a predicate that is definable after a parameter free finite formula,
> and since we have countably many such formulas and since under
> Extensionality each parameter free formula can define only ONE set,
> then there is One-One correspondence between sets and their defining
> formulas and accordingly we can only have countably many sets.
>
> In symbols:
>
> X is a parameter free definable set <-> [Exist Phi. for all y. y in X
> <-> Phi(y)]
>
> where Phi(y) is a formula having y as the sole free variable.
>
> Now is the above rationale correct?
>
>
> If one insists that EVERY set must be parameter free definable, then
> we will end up having UNCOUNTABLY many parameter free definable sets!
>
> Proof:
> Lets assume we have countably may parameter free definable sets.
> Then the set of all parameter free definable reals must be countable.
> Accordingly there exist a bijection between the set N of all naturals
> and the set R* of all parameter free definable reals.
> Now since EVERY set is parameter free definable, then this bijection
> is definable!
> By The diagonal argument of Cantor, then the diagonal defined after
> that bijection would be PARAMETER FREE definable real that is NOT in
> the set of ALL parameter free definable reals. A CONTRADICTION!

Pardon my French this is UTTER FNG DRIVEL

The SETS are 1-TO-1 with some PREDICATE.

A *F_I_N_I_T_E* sentence in Predicate Calculus.

|N| = |GODEL NUMBERS|
|GODEL NUMBERS| = |FUNCTIONS|
|FUNCTIONS| = |CHOICE FUNCTIONS|
|CHOICE FUNCTIONS| = |SETS|
|SETS| > |N|

YOU ARE ALL FOXES IN HEADLIGHTS DUMB!

Herc

Date Subject Author
12/19/12 george
12/20/12 Zaljohar@gmail.com
12/20/12 Barb Knox
12/20/12 ross.finlayson@gmail.com
12/20/12 Zaljohar@gmail.com
12/20/12 ross.finlayson@gmail.com
12/20/12 Graham Cooper
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/20/12 Graham Cooper
12/20/12 mueckenh@rz.fh-augsburg.de
12/20/12 Zaljohar@gmail.com
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Zaljohar@gmail.com
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/21/12 William Hughes
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 William Hughes
12/22/12 Virgil
12/22/12 William Hughes
12/22/12 Graham Cooper
12/23/12 Virgil
12/23/12 William Hughes
12/24/12 Graham Cooper
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/23/12 mueckenh@rz.fh-augsburg.de
12/23/12 Virgil
12/23/12 mueckenh@rz.fh-augsburg.de
12/23/12 William Hughes
12/24/12 mueckenh@rz.fh-augsburg.de
12/24/12 Virgil
12/23/12 Virgil
12/22/12 Virgil
12/21/12 Zaljohar@gmail.com
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 William Hughes
12/22/12 Virgil
12/21/12 Zaljohar@gmail.com
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/22/12 Zaljohar@gmail.com
12/22/12 Virgil
12/22/12 Zaljohar@gmail.com
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 Virgil
12/22/12 mueckenh@rz.fh-augsburg.de
12/22/12 Virgil
12/23/12 mueckenh@rz.fh-augsburg.de
12/23/12 Virgil
12/21/12 Virgil
12/20/12 Virgil
12/20/12 Graham Cooper
12/20/12 Graham Cooper
12/21/12 mueckenh@rz.fh-augsburg.de
12/21/12 Virgil
12/21/12 Graham Cooper
12/21/12 Graham Cooper