fom
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Re: set builder notation
Posted:
Aug 18, 2013 11:18 PM


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.. >

