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Tue, Jan. 25th, 2005, 04:56 pm

do any of you have any good suggestions for books on set theory? i'm interested in the subject but lack the information.

in addition: how can one best describe the null set?

Tue, Jan. 25th, 2005 11:19 pm (UTC)

Patrick Suppes: Axiomatic Set Theory

Comprehensive. But quite hardcore.

The null set is, simply, the unique set containing 0 elements. Notationally, either {}, or, as I prefer, the symbol in my icon.
(Deleted comment)

Wed, Jan. 26th, 2005 03:21 am (UTC)

But... why would anyone think that you're talknig about the null set when you say "X is nonempty"? Doesn't the fact that it is nonempty mean it is _not_ the null set?

Wed, Jan. 26th, 2005 03:22 am (UTC)

Oops. talking*

Wed, Jan. 26th, 2005 12:26 am (UTC)

It really depends on what you mean by "best." There are tons of interesting and rememberable examples that illustrate it... if that's what you're wanting to do.

BTW, probably the most common illustration, "the set containing the current king of france" or any variation thereof, is usually attributed falsely to Russell. It was actually stated before (up to 11 years prior I do believe) by Husserl. But that's a mute point when most scholars of logic in English speaking countries don't even recognize the name...

The null set becomes an intuitive concept that is used is lots of mathematics. Take the real line and remove a point x. Now take x and intersect it with the real line.. you get the null set.

Wed, Jan. 26th, 2005 06:23 am (UTC)

i like this definition:

{x: x doesn't equal x}

with this and the axiom of extentionality, it's pretty formal, i think.

Mon, Mar. 7th, 2005 05:32 am (UTC)

null-set can be represented as the zero-information system that always returns "no" when asked the predicate "does X belong to you?". I like this definition because it can be formally recognized in turing machines and in lambda-calculus.

Cute side-note: the /other/ zero-information system is the set of everything, which always returns "yes" when asked the preducate "does X belong to you?" even if X is the set of everything. Since the answer for null and set-of-everything is constant, they are the two trivial ZIS. This provides us very handily with the solution to causation, a question olden philosophers ask: we are caused by something, which is caused by something, and so forth until a seminal causation. In all their babblings, they pretty much conclude that the null-set, a zero-dimensional information-less void, is the seminal causation but are then baffled by how it begets our universe. The modern man's retort would be if it stands to reason a ZIS like null-set is a seminal causation, so too is a ZIS like the set-of-everything. Under the latter, we very easily exist under the predicate "does our unverse belong to you?". Thus, our information-rich asymmetrical universe can reside within a zero-information symmetrical system.

Wed, Aug. 3rd, 2005 07:54 pm (UTC)
madvolf: the null set

The null set is subset of any set. :)

(it's not a definition, but it's unique property, i guess, since it has a proof)

Wed, Aug. 3rd, 2005 10:27 pm (UTC)
wetpanda: Re: the null set

do you happen to know the proof?

Thu, Aug. 4th, 2005 07:52 am (UTC)
madvolf: Re: the null set

Yes, it's not difficult.

Lemma: Null set is subset of any set.

Proof: Suppose that the lemma is wrong. Then there has to exist such a set B, that null set is not subset of B.

If null set is not a subset of B, that means that there exists such element x that belongs to null set and doesn't belong to B.
But no element can belong to null set. That's a condtradition, which proves the lemma.

Thu, Aug. 4th, 2005 08:02 am (UTC)
madvolf: Re: the null set

^contradiction, sorry

Fri, Aug. 5th, 2005 02:31 pm (UTC)
madvolf: book

As for books on set theory, i would recommend Grimaldi "Discrete and combinatorical mathematics". Not exactly on the sets theory, but first chapters are good introduction to sets, i believe.