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Topic: set builder notation
Replies: 12   Last Post: Aug 24, 2013 1:38 AM

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fom

Posts: 1,968
Registered: 12/4/12
Re: set builder notation
Posted: Aug 18, 2013 11:18 PM
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On 8/18/2013 8:21 PM, Shmuel (Seymour J.) Metz wrote:
> In <g39v09h4tlqmni3hif6fn1n7g8ti39dp1a@4ax.com>, on 08/17/2013
> at 11:30 AM, dullrich@sprynet.com said:
>

>> (2) {x | x in A and P(x)}.
>
>> No:
>
>> No, because (2) is actually not a "legal"
>> construction of a set!

>
> It may not be legal in ZF, but it's perfectly legal in, e.g., NF. Of
> course, in NF P(x) can't be arbitray, e.g., {x \in A| x \in x} is not
> legal.
>


But Quine uses certain syntactic tricks that bind
the membership relation to class abstracts. So,
there is more here than simply depending on stratified
formulas.

Specifically,

( y = { x | Fx } ) <-> Ax( ( x in y ) <-> Fx )

Taking 'Fx' as 'x in y' one obtains

y = { x | x in y }

as well as all sentences of the form,

{ z | Fz } = { x | x in { z | Fz } }

These expand as

{ z | Fz } = { x | Ez( ( z = x ) /\ Fz ) }

which contracts to

{ z | Fz } = { x | Fx }

by virtue of Quine's analysis of identity
statements.

The relevant formula is

Fz <-> Ex( ( x = z ) /\ Fx)

The reverse conditional direction is an application
of the indiscernibility of identicals. The conditional
direction follows from

Fz -> Fz

Fz -> ( ( z = z ) /\ Fz )

Fz -> Ex( ( x = z ) /\ Fx )


These formulas are from his explanations in
"Set Theory and Its Logic".


>> (3) {x | Q(x)},
>> and things of the form (3) are officially not allowed.
>> Not allowed because they lead to contradictions:

>
> Again, that depends on the set theory that you're using. You can avoid
> Russel's Paradox by imposing restrictions on Q(x).
>

>> S = {x | x is not an element of x}.
>
> In, e.g., NF, that's not a valid construction, although
>
> S = {x | x = x} is..
>







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