Could my infinity be bigger than your infinity?

[Achernar]

Perhaps we can at least describe what a powerset is. This should be fairly easy to grasp. Consider a set S of cardinality 3:

S = { A, B, C }

Now consider all subsets of S, starting with the null, or empty, set:

{}, { A }, { B }, { C }, { A, B }, { A, C }, { B, C}, { A, B, C }

As you can see, there are 8 subsets. It’s no coincidence that it’s 2[sup]3[/sup]. Anyway, the set of all subsets of S is called the powerset of S:

P(S) = { {}, { A }, { B }, { C }, { A, B }, { A, C }, { B, C}, { A, B, C } }

In general, if a set has N elements, its powerset will have 2[sup]N[/sup] elements. If N is finite, then it’s obvious that 2[sup]N[/sup] > N. However, for infinite N’s, we’ve seen trickery before, but in this case, 2[sup]N[/sup] > N works for infinite N’s as well. As DrMatrix said before: “The power set is always larger than the set. Given any set, you can construct a set with a larger cardinality, by using the power set.”

[Chronos]

As for the relationship between the number of points on a line and the number of integers:

Consider a line segment of length 1. We can represent any number on this segment with a number, between 0 and 1. Let’s do this in binary. For instance, one point might be 0.001100111000101001110110… There is one place after the decimal point for each positive integer (i.e., there’s the first place after the point, the second place, the 654873214893219th place, etc.), so there are Aleph[sub]0[/sub] different places. Each of those places can hold either a 0 or a 1, two possibilities. Hence the number of real numbers between 0 and 1 is two to the Aleph[sub]0[/sub] power.

For the mathematicians: I know I glossed over the issue of multiple representations for the same number, but this is the gist of it.

What’s also interesting is that there is the same number of points on a short line segment as there are in a line of infinite length, or in a three-dimensional cube, or even in an n-dimensional infinite “plane”.

[ultrafilter]

Notice that if you took the reals in decimal, you’d get that the number of reals is ten to the Aleph[sub]0[/sub] power. The argument’s valid either way, so those two numbers must be equal.

[bryanmcc]

Originally posted by DrMatrix**

Originally posted by bryanmcc**
You can’t define a way to match [the real numbers and the integers] all up so that no points are missed. This is because not only are there an infinite number of points, but there are an infinite number of points between any other two points.**

You are correct that the set of reals has a larger cardinality than the integers, but not for the reason you gave. The rational numbers can be put into a one-to-one correspondence with the integers. I’d give Cantor’s diagonal proof, but this post is already long enough.**

Originally posted by DrMatrix**
The set of real numbers cannot be put into a 1-to-1 correspondence with the integers.**

There seems to be some kind of inconsistancy here. Could you clarify your position?

I agree it seems intuitively likely that you could match up the integers to the rational numbers (though I don’t know how, and would love to see the proof). My point in my original post, however, was that I don’t believe you could match the integers up to the real numbers, which you seem to agree with. How was the reason I gave incorrect? The real numbers are all of the points on the number line. You cannot pair them up with the integers in a one-to-one correspondence, despite what you can do with the rationals. I may be wrong, however, and would love to see how to do it, if it is possible.

Originally posted by Sputnik**
And I think I get the difference between Aleph-Null and the “continuum” type of infinity (unclear to me whether that’s called Aleph-One or not).**

It’s not known. There is a problem with something called the Continuum Hypothesis and with something else called Godel’s Incompleteness Theorem and the question is actually “undecidable”. What this boils down to is that yes the reals are a higher cardinality than the integers, but it’s not known (and is not able to be known) if it’s the next higher cardinality. This is all way over my head, but I think that pretty much sums up the “For Dummies” version.

-b

[ultrafilter]

*Originally posted by bryanmcc *
I agree it seems intuitively likely that you could match up the integers to the rational numbers (though I don’t know how, and would love to see the proof). My point in my original post, however, was that I don’t believe you could match the integers up to the real numbers, which you seem to agree with. How was the reason I gave incorrect? The real numbers are all of the points on the number line. You cannot pair them up with the integers in a one-to-one correspondence, despite what you can do with the rationals. I may be wrong, however, and would love to see how to do it, if it is possible.

The rationals are also have the property that between any two points there’s another one, but you can’t match them up with the reals. A dense set must be infinite, but it can be countable. I suspect that there are no countable complete sets, but I’m not sure.

[bryanmcc]

Ah! Now I see what you are saying. Tricky little hangup, that. Crap, and that was my favorite simple demonstration that some infinities could be larger than others. Hopefully it is still salvagable if I just assert that the integers cannot be mapped onto the reals, and leave out the bit about “because there are an infinite number of points between any two points”?

-b

[ultrafilter]

Use Cantor’s diagonalization argument.

[mipsman]

The great book “one, two, three … infinity” described the various infinities very simply:
Aleph sub zero - number of counting numbers
Aleph sub one - number of points on a line
Aleph sub two - number of curves that could be drawn through points on a line
There is (or, at that time, was) no Aleph sub three.

[John_Kentzel-Griffin]

mipsman Aleph-three is the power set of aleph-two. (Assuming the Generalized Continuum Hypothesis.) There is no end to the alephs. Even after all the aleph-n’s, you have aleph-aleph-null. If you’re gonna get picky it’s aleph-omega-null. And it keeps going. (“It keeps going.” is an understatement.)

bryanmcc
Yeah, it was the “because there are an infinite number of points between any two points” that bothered me.
To map the positive rational numbers to the natural numbers:
Note that every rational number can be expressed as i/j, i,j natural numbers; j non-zero. We can put them in sequence as follows:

1. Take all the numbers i/j where i + j = 1. OK, all here means one – the number 0/1. Map it to 0.
2. Take all the numbers i/j where i + j = 2. {0/2, 1/1}. Order them by increasing values of i. Skipping over any we’ve mapped before, like 0/2, map the remaining to the next integer. 1/1 maps to 1.
   n. Take all the numbers i/j where i + j = n. {0/n, 1/(n-1), …} Order by increasing … Well, you get the idea.

ultrafilter,
The beauty of using binary notation is that you have a correspondence between binary representations and elements of the power set of positive integers. Take a subset of the integers. Its representation will have a one in the n[sup]th[/sup] position when n is in the subset, and a zero if n is not in the subset. (Glossing over the fact that some reals have two representations.)

[Sputnik]

One, Two, Three … Infinity is already on my Amazon wishlist, as are Gamow’s Mr. Tompkins books.

now if I can only get the she-Sput to surf there…

[ultrafilter]

*Originally posted by DrMatrix *
**ultrafilter,
The beauty of using binary notation is that you have a correspondence between binary representations and elements of the power set of positive integers. Take a subset of the integers. Its representation will have a one in the n[sup]th[/sup] position when n is in the subset, and a zero if n is not in the subset. (Glossing over the fact that some reals have two representations.) **

Sure, I realized that. Using base n, you’ve got a 1-1 correspondence between R and n[sup]Z[/sup], the powerset of Z when you’re using n different truth values. But it is kinda strange that 2[sup]Aleph-Null[/sup] = 10[sup]Aleph-Null[/sup] if you’re not used to it. That’s what I was trying to point out.

[fixed coding - DrM]

[DarrenS]

*Originally posted by DrMatrix *
**mipsman Aleph-three is the power set of aleph-two. (Assuming the Generalized Continuum Hypothesis.) There is no end to the alephs. Even after all the aleph-n’s, you have aleph-aleph-null. If you’re gonna get picky it’s aleph-omega-null. And it keeps going. **

DrMatrix - does contemporary math posit the eixstence of any “ultimate” Aleph? An “infinite infinity”, the big bad daddy Aleph, in the same way Aleph-null stands to the natural numbers? In the book I’m reading (“Infinity and the Mind”) Rucker calls this “Omega” but skirts around explaining what it is, and seems reluctant to assert that it exists. He also gets pretty mystical and seems to compare Omega to God (whatever that means).

If we currently think there is no largest cardinal (or are they ordinals? aargh! :smack: ), aren’t we guilty of the same restricted thinking as pre-Cantorian mathematicians who denied the existence of Aleph-null?

[ultrafilter]

There is no largest Aleph. There are also cardinals which are larger than any aleph, and there’s no largest one of those. And there are cardinals which are larger than any of those…

[DarrenS]

ultrafilter - why doesn’t the same thinking that led Cantor to postulate the existence of Aleph-null, lead us to consider the largest cardinal? Before Cantor, people were probably equally convinced that there was no “largest natural number”. I’m not sure whether Aleph-null even qualifies as “the largest natural number”, but why couldn’t the same reasoning apply to the Alephs?

And, has anyone come up with a browser-independent platform-independent manner for rendering “Aleph” on this board?

[Achernar]

DarrenS, as DrMatrix mentioned before, whenever you have a set of some given cardinality, you can construct a set of a larger cardinality by taking its power set. So there can’t be a biggest Aleph.

[DarrenS]

BTW - I’m not refuting the proof of it (e.g. see here ) but do you see my point? Before Cantor, any mathematician would have told you there is no “largest” natural number - and would have proved it by saying that given any natural number, he could construct a larger one (e.g. n + 1). Cantor sort of side-stepped the issue (as far as I can tell) and said, I don’t care, I’m going to define Aleph-null as the cardinality of the natural numbers. Isn’t it possible that some similar maneuver will provide a “largest” cardinal some day?

[John_Kentzel-Griffin]

DarrenS,
You bring up an interesting point. Before Cantor, infinity was allowed as a potential, but not as an actual. For example, in standard calculus, x and f(x) can become “arbitrarily large”, but they are never actually infinite. In trans-finite analysis, you have numbers that are actually infinite. But you still cannot have a largest cardinal. If you did, its power set would be larger. So there still is a potential infinity that cannot be captured in a set.

ultrafilter,
n valued logic? Is that like: Truth, lies, damn lies, statistics, damn statistics, benchmarks, . . .?

[Sputnik]

I think I’ve got it now:

1. If I were to check on each of my two kids an infinite number of times - that would be 2 x Aleph[sub]0[/sub] = Aleph[sub]0[/sub].

2. If I were to check on an infinite number of children (disregarding daycare issues here) an infinite number of times - that would be Aleph[sub]0[/sub][sup]2[/sup] = Aleph[sub]0[/sub].

3. If I were to attempt to count every combination of my infinite offspring as they grouped and regrouped to play, say, Yahtzee - that would be 2[sup]Aleph[sub]0[/sub][/sup] = Aleph[sub]1[/sub].

4. If I were to attempt to count every combination of dice rolls (they are of course using an infinite number of dice in each game) in each of the Yahtzee games - that would be Aleph[sub]2[/sub].

If each die harbored an infinite colony of square-dancing bacteria…

[bryanmcc]

Except that 2[sup]Aleph[sub]0[/sub][/sup] = Aleph[sub]1[/sub] cannot be proven. 2[sup]Aleph[sub]0[/sub][/sup] equals something of a higher cardinality than Aleph[sub]0[/sub], but it may not be Aleph[sub]1[/sub].

And I don’t think the conditions in item four would produce Aleph[sub]2[/sub], but I’ll leave that to someone more qualified to cover.

-b

PS- DrMatrix, thanks for the correction. I hate impressing my friends with things that are wrong.

[heresiarch]

*Originally posted by Sputnik *
**One, Two, Three … Infinity is already on my Amazon wishlist, as are Gamow’s Mr. Tompkins books.

now if I can only get the she-Sput to surf there… **

[Hijack]Do you share a computer with her? Is it possible to set her browser’s homepage to be your Amazon wish list?

Not exactly subtle, but might be kinda funny.[/hijack]