Math Folks: Set Theory Question

Great Debates
1

Have a question (or two-ish, as it turns out) about set theory:

I didn’t get a lot of post-high-school mathematical training (I know, my fault), so I’m going on memory here, but if I recollect correctly…

Every set has at least two subsets: the Empty Set (which contains NO elements of the original set), and the _____ Set (whose name escapes me…but which contains ALL elements of the original set).

My main question: how many subsets does the Empty Set contain? Is it considered to contain both the Empty and the _____ (both of which are empty, anyway), or not? In other words, is the Subset Population of the Empty Set 1 or 2?

(A subquestion: what is the name of the subset that contains the entire original set?)

(A second subquestion: are there other set theories than Cantor’s?)

Thanks, folks.

(If anyone wants to ask: this is, oddly enough, a theological question.)

2

A set which contains exactly the same elements as another set is an improper subset.

http://mathworld.wolfram.com/ImproperSubset.html

I don’t know about the part about the empty set.

3

To amplify, I believe that two sets that have the same elements can be considered distinct sets, though I don’t know if that extends to the empty set.

4

Two sets are equal exactly when they have the same elements. So no, they can’t be considered distinct. A single set may have multiple descriptions that aren’t necessarily obvious.

A set with n elements has 2[sup]n[/sup] subsets. The empty set has 0 elements, so it has one subset–the empty set itself.

No special name. I’d say that every set has two subsets: the empty set, and the set itself. In the case of the empty set, the two have the same elements and are therefore equal.

There are multiple competing set theories out there, but they all agree with Cantor’s notion that two sets are of the same size iff there’s a one-to-one correspondence between their elements.

5

Well, fatrats!

My thanks to all who answered. This is exactly what I was looking for…if it weren’t for the fact that it WASN’T what I was looking for.

Don’t get me wrong: I wanted the answer to this question; it just wasn’t the answer I was hoping for. I wanted TWO subsets in the Empty Set!

Now I have to go and think some more.

Thanks, everybody.

6

I am intrigued, Dijon Warlock. Why did you want the empty set to have two distinct subsets, and what link does it have with theology?

7

We’d generally say “the original set”. There’s no real term for it. The empty set has a name not because it’s a kind of a subset, but because it’s a particular set, which is a subset of every set.

Just itself: one set. Sets don’t count identical elements twice. Also, this preserves the pattern that the number of subsets in a given set is 2 to the number of elements it has.

You just asked this.

The best answer that can be given is this: there are various extensions to the basic axiomatic theory Cantor started to lay out, not all of which are identical. Further, each of these has (in general) many “models”, where a model is basically a way of assigning “something” to each term of the theory so that the axioms all hold.

BTW: in practice, we don’t speak of the standard set theory as being Cantor’s, but rather Zermelo and Fraenkel’s.

8

To riff on this a bit, it can be quite a bit deeper than this in practice. Two sets containing the exact same elements are exactly the same. Do you notice the inherent recursion in the definition here? We can get away with it when talking about the empty set because there are no elements to check, so all sets with no elements are identical.

What about two sets with a single element? There exists a one-to-one correspondance between them, obviously. If that’s all we know, though, we can’t call them identical. They’re merely isomorphic. Traditionally, you’d expect that term to apply to groups rather than sets, but we’re starting to realize that to identify sets and cardinal numbers (specially picked sets) is to “decategorify” and lose information. Mathematics has to this point been the progressive decatigorification of mathematical structures. Mere isomorphisms have been elevated to the status of identities.

So, what does it really mean to say that two sets have “the same number of elements”? The number of elements is the cardinal number (remember, a special kind of set) with which it can be put into a one-to-one correspondance. That is, a set S “has N elements” if there exists a set-isomorphism (left- and right-invertible function) between S and N. If S’ also has N elements, we know that there exists an isomorphism from S to S’ – the composite of the isomorphisms assumed to exist. But note that we have no idea what that isomorphism is. Saying two sets have the same number of elements says that there is an isomorphism, but forgets what it is.

The upshot is that {A} and {B} cannot be said to be identical unless A and B are identical, and now we have to check the definition recursively, with no guarantee of bottoming out eventually. Induction in set theory is now very much up in the air, and if we can’t even say when two sets are the same or not… well, it’s a hairier situation than most practicing mathematicians would like to admit.

9

Bit of a long story, but you’re willing to slog through it:

It was inspired by a debate on Larry King Live the other night over Evolution vs. Intelligent Design, and whether the latter should be taught in schools as an equally competing theory. :smack: :rolleyes: The ID people seemed to be saying that the complexity of everything necessitates a pre-existing intelligence for it to have been created. My thought was that this must hold even more truly for God, yet the ID people seem to believe that God just kind of happened with no precursor (He simply bootstrapped himself into existence), which (to me at least) seems to contradict their “theory” of necessitated precursorness.

(Yes, I know the word needs many more quotes than that)

So it got me thinking about God, and how we can explain the existence thereof…which feeds into what I believe to be a very profound question: “Why is there Something rather than Nothing?”

My thinking along this question has always been that it is because Something is a necessary consequence of Nothing. So the question then becomes How/why does Nothing become Something?

A few years ago, I was floating (mostly in my head, but I think I mentioned here once) a definition of God: The Set That Contains All Sets. It seemed to be the most inclusive definition I could come up with, and all-inclusiveness seemed necessary to correctness. Plus, it would make the opposite of God (Satan, or what have you) to be the Empty Set. The Empty Set would be a subset of God, but God would not be a subset of the Empty Set. That’s why God wins.

So…the question became: How do we get from the Empty-Set to the Set-That-Contains-All-Sets? How does God manage to bootstrap him/her/it/self into existence without being there to do it?

One thing that distinguishes the latter state from the former is plurality: The Empty-Set is uniform, whilst the STCAS is diverse. My thinking was that if the Empty Set contained within itself an inherent diversification (two subsets), then I might have pinpointed how Nothing necessarily begats Something.

As I said, I’ll have to think some more.

10

Well, the rather unbased assumption that a dichotomy is necessary aside, here’s the problem: there’s no set of all sets.

One of the axioms is that given any set S and a predicate P, there is a set of all elements of S which satisfy P. We can easily construct the predicate (using other axioms) P(x) = “x does not contain itself (as an element)”. So, if there is a set of all sets then S’ = {x in S | P(x)} must exist. However, if S’ is an element of itself, it can’t be, and if it isn’t, it must be. Russel’s antinomy is why we use this axiom rather than the original (naïve) one that any set described by a predicate (rather than any subset of a given set) must exist.

So, if your assumption is correct as to the nature of God, you’ve managed to rather conclusively prove that God doesn’t exist.

11

“Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic.

12

Because if there were nothing, there wouldn’t be any intelligent beings capable of asking the question, “Why is there Something rather than Nothing?” Incidentally, the answer to this question is that if there were nothing, there wouldn’t be any…

:smiley:

13

It’s airtight! Mods, we can’t let this little doozy get out.

14

Remember the saw about René Descartes drinking beers in a bar? The bartender asked, “Hey, René! Want another beer?” When Descartes answered, “I think not,” Descartes disappeared.

15

There’s an analogy to the issue of every set having at least two subsets, itself and the empty set, but these twobeing the same in the case of the empty set.

Every positive integer has two factors, itself and 1. (In the case of prime numbers, these are the only factors of the integer). Here, the number 1 is a special case, because those two factors are the same, so it only has one factor (and hence is not prime, since primes have two factors).

Another aspect of the analogy is that the empty set is a subset of every set (regardless of what universe of sets you take), and 1 is a factor of every positive integer.

And it’s called the empty set because there is only one set with no elements: there is only one empty set.

16

Be very, very careful here. Now you’re bringing up a choice of universe, which raises choices of models, and you have no guarantee that the representor of the empty set in two distinct models is the same.

17

Truly, you folks make my head asplode! (GOOD thing, not bad!)

(Only from the joy of learning, I assure you…and the overwhelming desire to use the word “asplode”.)

To address the posts since my last contribution:

My basis for that assumption is that Existence demonstrates Multiplicity rather than Uniformity: that is, there are lots of different things rather than one undifferentiated thing. For that to happen (which it clearly has–witness US), then Diversity must be a ubiquitous component of observed Existence. The most fundamental manifestation of Diversity (or Plurality, if you prefer) is Dichotomy, since the minimum parts that a Single thing can be divided into is Two. Therefore, to explain Existence as we observe it, Dichotomy must be included.

I find this rather intriguing for a couple of reasons:

First of all, my assumption (based on my misremembering of set theory) doesn’t appear to be correct at all (which means I must go back and revise my thinking).

Second, even if it was (of which I am no longer convinced), it would match the conclusion reached by Bart Kosko in his book Fuzzy Logic (not necessarily agreeing with it, but it was intriguing to read), wherein he mathematically disproved the existence of God, as well. I find that rather interesting, as well as my revised theory…which I won’t bother people with unless they ask.

I still DO think that Nothingness MUST contain within itself the necessary components to bootstrap itself into Somethingness, or else none of us would be here.

I’m just not finalized on how that happened…but I’m getting close.

[sub]Now watch me get “disappeared” by the makers of Reynold’s Wrap…by coincidence.[/sub]

Now THAT made me laugh.

Sorry about getting rid of God, folks. D’ya think Pat Robertson will shut up now?

I’m still convinced that the Empty Set must be non-Empty in order to explain why there is Something rather than Nothing…but the plurality of “two subsets” might not be the Answer anymore. Perhaps it’s the Set-and-its-only-Subset that I’m digging about.

Don’t know, but I’m betting I’m close to something…
[sub]…if nothing else, more nothing…[/sub]

18

Okay, please stop with all the Capitalized Words. The more capitals you gratuitously throw in, the more psychoceramic you look. Unless you mean a specific concept differing from the normal definition of “existence” – a concept you would need to explicitly define – don’t capitalize the word.

Now, what I meant by assuming a dichotomy is that you haven’t given any reason that just because there is a supreme being (“God”) there must be an antithesis (“Satan”). Why should this be?

I wouldn’t trust any mathematical “proof” of God’s non-existence any more than I would of His existence. The divine is simply beyond the scope of mathematical structures, which clearly delineate their own scope in their definitions. If you really want to attack theology with mathematics, for God’s sake go back to college and learn some real mathematics and how it works rather than just half-remembering a collection of formulas and factoids.

19

Oh I don’t know. Can’t we try assuming that God doesn’t exist, and show that that leads to a contradiction within Zermelo-Fraenkel set theory? Seems promising enough. Just your basic, vanilla reductio ad absurdum.

Me, I once proved God’s existence using Euler’s formula and planar graph theory. Of course, my definition of God is, “a convex regular polyhedron with regular pentagonal faces.” Unfortunately most faiths seem to regard this belief as against canon, if not outright heretical, despite its obvious appeal.

Fine. Roast in hell, unbelievers. But I know my God exists.

20

Fine. Now prove that Z-F obtains in the real world, and further that reductios are valid (no, it’s not a settled point).