Draygomb

CardinalMan
Wouldn't one be twice the size of the other? or did you mean to compare positives just to negatives? <img src="confused.gif" border="0">

Clutch

Nope. That's the idea behind defining an infinite set as one that's the same cardinality of some subset of itself. It's a feature that distinguishes the finite from the infinite case.

Remember, we define "same size" for sets in terms of one-one correspondence: two sets are the same size if you can "match up" the elements of the two sets in such a way that there's no element of either that can't be matched to an element of the other. Our familiarity with the finite case makes it seem intuitive that {0,1,2...} is twice as big as {0,2,4...}. But this is misleading.

You can see that, as you extend the series, there will be, for each next member of one series, exactly one next member of the other to which it can be matched. These two sets are the same size ("aleph nought") -- as are the sets containing the negative integers, and the fractions too. All these sets are countably infinite, not meaning that you could complete the task of counting them, but that they contain no element to which you could not count by a finite process.

But if you took the powerset of the natural numbers -- that is, the set of all the combinations you can make of natural numbers -- then you'd have a bigger infinite set. That set contains elements that can't be numbered by the naturals; the Real numbers are of this cardinality (aleph one). There are real numbers to which one could not count via any finite process; the reals are uncountable.

I hope that's all right; in any case, now you know as much as I do...

jpbrooks

The Cantorian observations (expounded above) about the cardinality of sets are correct.
However, the question of whether the universe is infinite in size (or volume) seems somewhat unrelated to these observations. The universe can be a "continuum" (with a cardinality of "aleph 1"), but still be a "bounded" one.

Perhaps this whole issue actually centers on whether (for example) quantum phenomena, which underlies matter and energy, extend out into what we refer to as "space" without end. If actual infinities (such as an infinite number of quantum events) cannot exist in the real world then, in this sense, the universe cannot be said to be infinite in expanse.

[ February 01, 2002: Message edited by: jpbrooks ]</p>

CardinalMan

Clutch gave a nice description about why the two sets {0,1,2,...} and {...,-2,-1,0,1,2,...} have the same size and so are assigned the same cardinal number. I'm in the middle of writing a web page about the different sizes of infinity, so if you're interested check out <a href="http://www.math.uiuc.edu/~mileti/infinite.html" target="_blank">The Different Sizes of Infinity</a>. I hope that the playful presentation doesn't come across as too silly (my target audience is more broad). I'll add more to it (and clarify the last few paragraphs) soon.

Aleph_1 is the next infinite cardinal after aleph_0. The question of whether or not the real numbers have this cardinality is known as the Continuum Hypothesis and is one of the most famous problems in mathematics. It turns out that the Continuum Hypothesis is independent of the normal axioms of set theory, which means that working in usual axiom system that mathematicians use, you can neither prove nor disprove the Continuum Hypothesis. This is a very deep result in set theory and the foundations of mathematics. The philosophical implications of this are far from understood.

CardinalMan

Draygomb

If when I add 1 to the first set I also add 1 and -1 to the second set doesn't the second set grow twice as fast?

jpbrooks

If I'm not mistaken, adding rational numbers to each element of infinite sets of rational numbers only shifts the numbers in the sets around without actually adding any new numbers to them. So the two sets above still have the same cardinality after the addition operations have been performed on their elements that they had originally. The only way to get a set of numbers with a greater cardinality is to add in all of the (remaining) irrationals.

[ February 02, 2002: Message edited by: jpbrooks ]</p>

jpbrooks

That's an interesting explanation of the sizes of infinity, but I'm not familiar with any work by Smullyan that contains it.
(I enjoyed his book, The Tao Is Silent, however.)