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

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Shmuel (Seymour J.) Metz

Posts: 3,336
Registered: 12/4/04
Re: set builder notation
Posted: Aug 18, 2013 9:21 PM
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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.

>(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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