# Math Folks: Set Theory Question
**URL:** https://boards.straightdope.com/t/math-folks-set-theory-question/318608
**Category:** Great Debates
**Created:** [August 25, 2005, 7:03pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608 "2005-08-25T19:03:40Z")
**Posts on this page:** 20
**Page:** 1
### Author:  [@Dijon\_Warlock](https://boards.straightdope.com/u/Dijon_Warlock)
#### Post date: [August 25, 2005, 7:03pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/1 "2005-08-25T19:03:40Z")
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.)
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### Author:  [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)
#### Post date: [August 25, 2005, 7:08pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/2 "2005-08-25T19:08:23Z")
A set which contains exactly the same elements as another set is an improper subset.
[http://mathworld.wolfram.com/ImproperSubset.html](http://mathworld.wolfram.com/ImproperSubset.html)
I don’t know about the part about the empty set.
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### Author:  [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)
#### Post date: [August 25, 2005, 7:09pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/3 "2005-08-25T19:09:34Z")
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.
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### Author:  [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)
#### Post date: [August 25, 2005, 7:17pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/4 "2005-08-25T19:17:19Z")
> [@CookingWithGas](#):
>
> 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.
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.
> [@Dijon Warlock](#):
>
> 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 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.
> [@](#):
>
> (A subquestion: what is the name of the subset that contains the entire original set?)
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.
> [@](#):
>
> (A second subquestion: are there other set theories than Cantor’s?)
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.
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### Author:  [@Dijon\_Warlock](https://boards.straightdope.com/u/Dijon_Warlock)
#### Post date: [August 25, 2005, 7:34pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/5 "2005-08-25T19:34:13Z")
> [@ultrafilter](#):
>
> 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.
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.
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### Author:  [@Hypnagogic\_Jerk](https://boards.straightdope.com/u/Hypnagogic_Jerk)
#### Post date: [August 25, 2005, 8:15pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/6 "2005-08-25T20:15:42Z")
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?
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [August 26, 2005, 2:28am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/7 "2005-08-26T02:28:50Z")
> [@Dijon Warlock](#):
>
> 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).
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.
> [@](#):
>
> 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?
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.
> [@](#):
>
> (A subquestion: what is the name of the subset that contains the entire original set?)
You just asked this.
> [@](#):
>
> (A second subquestion: are there other set theories than Cantor’s?)
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.
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [August 26, 2005, 2:44am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/8 "2005-08-26T02:44:17Z")
> [@ultrafilter](#):
>
> 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.
> [@](#):
>
> 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.
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.
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### Author:  [@Dijon\_Warlock](https://boards.straightdope.com/u/Dijon_Warlock)
#### Post date: [August 27, 2005, 12:26am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/9 "2005-08-27T00:26:14Z")
> [@severus](#):
>
> 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?
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.
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [August 27, 2005, 12:53am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/10 "2005-08-27T00:53:28Z")
> [@Dijon Warlock](#):
>
> oating (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.
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.
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### Author:  [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)
#### Post date: [August 27, 2005, 1:20am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/11 "2005-08-27T01:20:49Z")
> [@Mathochist](#):
>
> So, if your assumption is correct as to the nature of God, you’ve managed to rather conclusively prove that God doesn’t exist.
_“Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic._
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### Author:  [@TJdude825](https://boards.straightdope.com/u/TJdude825)
#### Post date: [August 27, 2005, 6:11am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/12 "2005-08-27T06:11:50Z")
> [@Dijon Warlock](#):
>
> Why is there Something rather than Nothing?
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…
😃
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### Author:  [@thirdname](https://boards.straightdope.com/u/thirdname)
#### Post date: [August 27, 2005, 10:48am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/13 "2005-08-27T10:48:44Z")
> [@Mathochist](#):
>
> So, if your assumption is correct as to the nature of God, you’ve managed to rather conclusively prove that God doesn’t exist.
It’s airtight! Mods, we can’t let this little doozy get out.
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### Author:  [@Monty](https://boards.straightdope.com/u/Monty)
#### Post date: [August 27, 2005, 11:12am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/14 "2005-08-27T11:12:36Z")
> [@Bytegeist](#):
>
> _“Oh dear,” says God, “I hadn’t thought of that,” and promptly vanishes in a puff of logic._
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.
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### Author:  [@Giles](https://boards.straightdope.com/u/Giles)
#### Post date: [August 27, 2005, 2:29pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/15 "2005-08-27T14:29:26Z")
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.
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [August 27, 2005, 7:56pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/16 "2005-08-27T19:56:59Z")
> [@Giles](#):
>
> Another aspect of the analogy is that the empty set is a subset of every set (regardless of what universe of sets you take)
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.
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### Author:  [@Dijon\_Warlock](https://boards.straightdope.com/u/Dijon_Warlock)
#### Post date: [August 28, 2005, 6:39am UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/17 "2005-08-28T06:39:47Z")
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:
> [@Mathochist](#):
>
> Well, the rather unbased assumption that a dichotomy is necessary aside, here’s the problem: there’s no set of all sets.
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.
> [@Mathochist](#):
>
> So, if your assumption is correct as to the nature of God, you’ve managed to rather conclusively prove that God doesn’t exist.
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]
> [@Bytegeist](#):
>
> “Oh dear,” says God, “I hadn?t thought of that,” and promptly vanishes in a puff of logic.
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]
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [August 28, 2005, 7:05pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/18 "2005-08-28T19:05:17Z")
> [@Dijon Warlock](#):
>
> 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:
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?
> [@](#):
>
> 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 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.
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### Author:  [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)
#### Post date: [August 28, 2005, 8:50pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/19 "2005-08-28T20:50:20Z")
> [@Mathochist](#):
>
> 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…
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._
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### Author:  [@Mathochist](https://boards.straightdope.com/u/Mathochist)
#### Post date: [August 28, 2005, 9:16pm UTC](https://boards.straightdope.com/t/math-folks-set-theory-question/318608/20 "2005-08-28T21:16:54Z")
> [@Bytegeist](#):
>
> 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_.
Fine. Now prove that Z-F obtains in the real world, and further that reductios are valid (no, it’s not a settled point).
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