Stardate 20030512.1541

(On Screen): Aziz Poonawalla writes:

Take a look at this.

I disagree with it at a number of points, but I am also curious what you think.

Good grief; where did these idiots study mathematics?

Their first premise is "An actual infinite cannot exist." They need this premise because they need to prove that time doesn't extend infinitely far into the past. They need there to be a concrete point where the universe begins, because they want to try to argue that a creation implies a creator. If there's no point of creation, there's no logical need of a creator. So they claim that an "actual infinite" can't exist so that they can argue that time is bounded in the past. But that doesn't mean it's true. Who says?

Well, they've got a snappy appeal to authority in the form of a quote from David Hilbert, an eminent mathematician who died in 1943. The problem is that I don't buy Hilbert's argument. He demonstrates one case where an "actual infinite" is impossible, but doesn't prove that it's overall impossible.

It's possible to define a set consisting of all the points on a line between "0" and "1" and prove that this set is infinite (aleph-one) in size, even though it's based on a finite source (a bounded line segment). By the same token, there may well be sets in the universe which are infinite without requiring the universe to be infinite. For instance, there's no particular reason to believe that space is granular (i.e. that "position" is quantized) and if so then the set of "all possible positions between San Diego and Los Angeles" is infinite. (If position were quantized, that would imply the existence of a fixed frame of reference for the universe, which would contradict Relativity.)

In contemporary set theory, an actual infinite is a collection of things with an infinite number of members, for example, a library with an actually infinite set of books or a museum with an actually infinite set of paintings. One of the unique traits of an actual infinite is that part of an actually infinite set is equal to whole set. For example, in an actually infinite set of numbers, the number of even numbers in the set is equal to all of the numbers in the set. This follows because an infinite set of numbers contains an infinite number of even numbers as well as an infinite number of all numbers; hence a part of the set is equal to the whole of the set. Another trait of the actual infinite is that nothing can be added to it. Not one book can be added to an actually infinite library or one painting to an actually infinite museum.

They were doing fine until the last two sentences, where they just sort of slipped in something that they need for their argument.

Well, actually, they weren't. It's true that in some cases it can be demonstrated that a part of an infinite set can be shown to be the same size as the whole set. But, for instance, I can take the infinite set of natural numbers and create a subset consisting of the numbers 5 and 6, and that subset is not infinite. But even if the subset is infinite in size, that doesn't mean it's equal to the parent set. There's more to equality than size and the set of even numbers is not equal to the set of natural numbers, even though they're both infinite sets.

But it's that last point which is critical, because it's essential to their argument, and because it's false.

It is not true that nothing can be added to a set which is "actually infinite" in size. What they think they're saying is that you can't add anything because it's already in there.

Russell's Paradox shows that it is impossible to create a set which contains everything; no matter how you define it, you can never be certain that you didn't leave something out.

Even if such a set existed, that doesn't preclude putting things in it. There's nothing intrinsically wrong with having duplicates as members; the choice to permit or not permit duplicates belongs to the person who defines any given set. Set Theory as such does not impose any such requirement, and it's completely permissible to create a set consisting of an infinite number of instances of "1".

It is wrong to even state that an infinite set must necessarily already contain everything there is. It's possible to define infinite sets which are not all-inclusive, and the set of natural numbers is both infinite (aleph-null) and not all-inclusive. Their examples were, too: the infinite library may contain all books but contains no paintings and the infinite museum in turn contains no books, and neither of them contains any magazines. Why can't I add books to the infinite museum?

Once they've formed their "actually infinite" set of natural numbers, I can add "blue" or "popcorn" or "Steven Den Beste" to it without duplication. (After all, I'm not a natural number.)

For that matter, I can add "one tenth" (a non-whole rational number), or "the square root of two" (an irrational number) or "the square root of minus seven" (an imaginary number existing in 2 dimensions) or "multiplication" (an arithmetical operation), or "{3,7,2,9,20}" (a five-dimensional vector) or "torus" (a topological figure) to it without duplicating anything already there, even if I'm confined to mathematical constructs