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Old 2011-01-27, 22:59   #331
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Quote:
Originally Posted by Mr. P-1 View Post
You're almost there.

If A is a 1-element set, then the only subsets of A are TEX]\empty[/TEX] and A itself.

A cannot be an element of A. What could be an element of A?
anything but A or \empty ? as long as A can be a collection or a set of sets

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Old 2011-01-27, 23:07   #332
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The set S itself isn't a member of S -- ZF doesn't allow that, so that won't work.
I would argue that Z, rather than ZF is a closer formalisation of the informal set theory we've been doing here, because 1. nothing has been said which relies upon the axioms of replacement or regularity, and 2, we have admitted individual elements, which most treatments of ZF do not admit.

S\inS still doesn't work, even in Z, because while not forbidden, it is impossible to prove such a set exists.
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Old 2011-01-27, 23:10   #333
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anything but A or \empty ? as long as A can be a collection or a set of sets
Stick with your idea of a 1-element set. A has two subsets: \empty and A. A's sole element is a subset. The element is not A. What is the element?

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Old 2011-01-27, 23:13   #334
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Stick with your idea of a 1-element set. A has two subsets: \empty and A. A's sole element is a subset. The element is not A. What is the element?
A[1] in PARI terms, last time I checked.
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Old 2011-01-27, 23:38   #335
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A[1] in PARI terms, last time I checked.
A is a one-element set with two subsets: \empty and A. One of the two subsets I have just named is an element of A. The element is not A. What is the element?

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Old 2011-01-27, 23:45   #336
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Quote:
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A is a one-element set with two subsets: \empty and A. One of the two subsets I have just named is an element of A. The element is not A. What is the element?
\empty is what you seem to imply.
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Old 2011-01-27, 23:52   #337
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\empty is what you seem to imply.
If \empty is an element of a 1-element set. What is the set?
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Old 2011-01-27, 23:55   #338
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If \empty is an element of a 1-element set. What is the set?
\empty by the sounds of it.
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Old 2011-01-28, 01:49   #339
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Originally Posted by Mr. P-1 View Post
I would argue that Z, rather than ZF is a closer formalisation of the informal set theory we've been doing here
Yep, agreed. But I didn't want to bring up issues of the foundation axiom and I wanted to help sm along the path so I wrote what was needed.
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Old 2011-01-28, 01:58   #340
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\empty by the sounds of it.
\empty's not a member of that set, though; that has no elements.
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Old 2011-01-28, 12:48   #341
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Quote:
Originally Posted by science_man_88 View Post
A[1] in PARI terms, last time I checked.
this is the final thing I come up with if it's not that I have no idea because I obviously don't know the question.

Last fiddled with by science_man_88 on 2011-01-28 at 12:51
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