Continuum Hypothesis

[mith]

I know this is kinda math-geekish, but does anyone have any opinions about this?

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

Do you tend to agree or disagree with it? Or do you just have no clue what any of that means?

[Chuck]

Doesn't it just say that the situation is undecidable unless new axioms are adopted? Anything can be decided if you can make up new rules.

Infinite sets are a lot of trouble anyway. The concept should be discarded.

[extropalopakettle]

Yes, but if something is not decideable based on the currently accepted axioms, then you want to decide which new axioms to adopt based on which you believe are actually true (if that makes sense). I feel that it's nice, if one accepts as an axiom that there is some set bigger than the integers and smaller than the reals, to have some idea what that set is like. I find it impossible to imagine such a set.

[mith]

Me too. I'm still trying to find any information about the results of negating it.

[Chuck]

There are no infinite sets of anything in real life. It's all just made up. None of its axioms are actually true. Maybe satisfying would be a better word.

Maybe someone, not me of course, could assume that such a set exists and explore what properties it must have. If the investigation leads to something unsatisfying then some axiom could be added that kills the idea.

[extropalopakettle]

Yeah, I know (about infinite sets not corresponding to anything real, I think).

It's just that I'm comfortable with the notion of infinite sets - and infinite sets of different sizes. The integers being the smallest size of infinite set. The reals (or the set of all sets of integers) being larger. And for any infinite set, the set of all it's subsets being a larger (next larger?) infinite set. But I can't imagine infinite sets "in between" these in size. And I don't think there's any way to describe how to construct one. That is, for instance, one can describe how, given any infinite set, to construct a larger infinite set: Just take all the subsets of the first infinite set - the set of all those subsets is larger. But there's no way to describe the construction of a set of size in between those two, despite the fact that it seems to be OK (no contradiction arises) to just assume one exists.

[Chuck]

How do we know that there are no smaller infinities than the integers?

Does the power set of the set of all sets contain sets that aren't in the set of all sets?

[extropalopakettle]

Here's an interesting problem: Take the smallest set that contains all it's own finite subsets. Smallest here means it contains nothing that isn't implied by the statement that it contains all it's own finite subsets. Now show a 1-1 mapping between the elements of that set and the non-negative integers.
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"no smaller infinities than the integers" - I think there could be an intuitive argument for this, with the premise that if I have an infinite set, I can remove an element, and still have an infinite set. Then I can remove an element, call it element 0, remove another, call it element 1, etc... Every integer is eventually assigned an element. So the infinite set is at least as big as the integers.


[This message has been edited by extropalopakettle (edited 02-07-2002 07:45 AM).]

[Bicho the Inhaler]

I too am comfortable with infinite sets. It's infinite numbers and the theory behind numbers that are infinitely large that I find contrived. The continuum hypothesis doesn't need to specifically mention infinity, though, so I guess I don't have a problem with it.

I see where Chuck's coming from, but I think we need infinite sets in the theory; set theory would seem very incomplete without them.

[Luna]

mith, I'm going to kill you for this thread.

just so you know.

[Elayne]

errr....any reason this thread warrents mith being killed Luna?

[Luna]

Oh, he'll know why.

[extropalopakettle]

Hmmm... sounds like there's an interesting inside story behind it.

Perhaps the Continuum Hypothesis is what's causing Luna to lose sleep.

[mith]

extro, do you mean properly contains? the empty set contains all it's own finite subsets, so it would be the smallest, and there is clearly a 1-1 mapping to the naturals (since there aren't any elements in the empty set).

*~hides from luna~*

[extropalopakettle]

no, contains as an element. So any set that contains all its own (finite) subsets, as elements, would have to contain the empty set, {}, as an element. Then the set containing that element would also be a subset, and thus an element, etc... So such a set would have to contain {}, {{}}, {{},{{}}} ...

[mith]

ah, yeah, i see what you mean. i read about that on some website, in a little different terms.

[Correllia1]

In some math course that I took I heard something about countably infinite and uncountably infinite sets. Countably infinite sets are any infinite sets of numbers which can be represented as a 1-1 mapping of the natural numbers (1,2,3,...). For example, the integers are countably infinite because each one can be assigned a "place value" within the sequence. 0 can be first, 1 can be second, -1 can be third, etc. Someone please correct me if I'm wrong. I didn't dig out my book.

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[Lilifreid]

Now prove that there are no other levels of infinity between countable and uncountable

[extropalopakettle]

Correllia1, yes, you're right.
The topic of different sized infinite sets comes up occasionally around here. Here's a recent thread about infinite sets that touches on a lot of the salient points.

[CrystyB]

i wonder how come i missed this...

extro, are you sure those two are equivalent in size? It's a little counterintuitive. I might prove otherwise if i had the time.

Chuck, accepting one choice or the other would not lead to anything unsatisfying. That's why it can't be either proven or disproven...

[extropalopakettle]

[CrystyB]

mith, i get an Access Denied when opening that - care to mirror somewhere? TY!

- - -

ok extro, it wan't counterintuitive, it was just pretty da*n HARD to grasp that set that contains all its finite subsets. (the empty set IS finite, right?)

When i read your reply 14 i thought "Hey, this resembles the new construction i just learnt - how the Set Theory creates N! Then it kind-of includes the natural numbers!". After that, i realised it would have to contain ANY finite set of natural numbers, and that's what was difficult to imagine.

However, i think a better definition of smallest would be handy. (the one used in maths would be great!) I can't imagine what you mean by "isn't implied by the statement ..." - are you thinking about a recursive definition with only one starting generator?

If so, there is a mapping, although not really straightforward. And it doesn't map n to n either!

- - -

What i meant by the other comment was about reply #4. "Smaller than the reals, bigger than the integers - that can't be proved either way." That's exactly what i was talking about!

- - -

Chuck: "Does the power set of the set of all sets contain sets that aren't in the set of all sets?" There are none, so yes it contains them! And btw, please stop making (?up?) stuff like these - i get a small headache whenever i hear the expression "set of all sets".

[This message has been edited by CrystyB (edited 03-11-2002 05:11 PM).]

[Chuck]

But all infinite sets are made up. If no one ever made any up we'd have none to talk about.

[extropalopakettle]

[Bicho the Inhaler]

CrystyB, the "set of all sets" problem is actually highly non-trivial because you can prove that no one-to-one mapping exists between any set and its power set (the set of all its subsets):

Suppose we have such a mapping between a set and its power set. Then each element of the set either is or isn't a member of the subset with which it is paired. Take all the elements that are not members of the subsets to which they are mapped. This comprises a subset of your set, but is the element paired with it a member of it? You can't avoid contradiction.

Just think about the "set of all sets" for a minute; just think of all the silly and incongruous objects it must contain! This is proof enough for some people that this "set" is ill-defined, but for the others we have the above.

Bicho

[Bicho the Inhaler]

extro, that's what I had. A very neat problem.

[CrystyB]

Chuck, yes they ARE made up, but they're axiomatised. I just thought i'd ask you not to use the 'set of all sets' in a math related discussion - it is a proven thing that such a notion would be meaningless.
BTW, Bicho, i think you missunderstood me - i never sustained that notion.

Extro, the problem was very enjoyable - i think i would've come up with a demonstration like that in two or three months maybe!
But the actual math definition of smallest is 'has the studied property and for any other entity with the same property, the first is smaller than the latter'. I was trying to get you say that smaller as in 'inclusion' smaller, and not 'cardinality' smaller. Though 'cardinality' would have been easier. And suppose (just suppose) there are two sets which satisfy your 'a)' & 'b)'. By definition, neither would be included in the other. How would you compare these? ((bah, i just realised i am nitpicking))

Just another question about it: is the mapping S->N recurrent? It seems so to me...

[Chuck]

I don't think that infinite sets belong in a mathematical discussion since they lead to meaningless notions.

[Bicho the Inhaler]

Oops Sorry, Crysty

But Chuck...they're so much fun

[Bicho the Inhaler]

As for the "smallest" issue, would it make more sense to speak of "the intersection of all sets that contain all their finite subsets"? I realize that this notion is not generally equivalent to "smallest," but in this case it works.

[CrystyB]

Yeah Chuck, and they're already set up! Why avoid them if you can be friends?

[Chuck]

No! If I can't have a set of all sets then I don't want any!

[Bicho the Inhaler]

Maybe the answer to the Continuum Hypothesis problem will fall out of some other axiom we've neglected; neither the Continuum Hypothesis nor its negation make very appealing axioms, in my opinion. I vote we don't accept either as an axiom and search elsewhere.

[CrystyB]

Great! You do that! In the mean time, i'll just look elsewhere, period, (not necessarily math :p) having lost interest in the Continuum Hypothesis when it became unsolveable.

[MTGAP]

I think what we need to do is prove some axioms. We will prove them using NO assumptions except about what logical arguments we're allowed to make. It will be revolutionary. From that, all else will follow.
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[Jack_Ian]