Infinity as a Skolem FunctionDate: 10/28/2000 at 00:58:23 From: Todd Zaba Subject: Is infinity an absolute concept, relative, or both? Could one reasonably say that infinity is relative? The following discussion came up today: There is an infinite set of odd numbers (1, 3, 5, 7, 9, ...) There is an infinite set of even numbers (2, 4, 6, 8, 10,...) So, each set is infinite, yet 1/2 of another infinite set... Does this make infinity relative? Or does this get back to the "infinity is a concept, not a number" that I've run across on your site so much? Thank you for your time. Date: 11/03/2000 at 18:41:57 From: Doctor Ian Subject: Re: Is infinity an absolute concept, relative, or both? Hi Todd, I'm not sure that what you mean by 'relative' is any clearer than what mathematicians mean by 'infinity'. Certainly some infinities (the size of the set of real numbers) are larger than others (the size of the set of integers). But there are exactly as many integers as there are even integers. Personally, I think 'infinity' is a skolem function. What do I mean by that? In logic, you can set up relations between entities, such as: is( alice, mother-of( bob ) ) And you can use these relations to discuss entities whose identities you don't actually know: is( ?, mother-of( bob ) ) That is, we know that Bob has a mother, even if we don't know who it is. The function mother-of is called a skolem function. Skolem functions are really useful, for the reason just described - sometimes we want to talk about something, but we aren't yet able to identify it: the winner of the next lottery drawing, or the candidate who was ahead in the polls yesterday, or... But sometimes they can cause problems, as when they are used to refer to things that don't actually exist, and may never exist: the cure for cancer, the cause of war, the solution to the education crisis, the sexiest man alive, the creator of the universe... In other words, skolem functions let people talk nonsense without realizing it. So, what does this have to do with infinity? Well, when we talk about sets, we refer to attributes of the sets: the first element, the last element, the size, and so on. So size-of is a skolem function: equals( 435, size-of( House-of-Representatives ) ) equals( 88, size-of( piano-keyboard ) ) equals( ?, size-of( employees-of( IBM ) ) ) It just so happens that when we talk about sets that don't have finite numbers of elements, we can glibly use the skolem function size-of to talk about something that doesn't really exist, e.g., the 'size of' the set of integers: equals( 'infinity', size-of( integers ) ) Actually, this only became a problem when Cantor showed that equals( 'infinity[1]', size-of( integers ) ) equals( 'infinity[2]', size-of( real-numbers ) ) not-equals( 'infinity[1]', 'infinity[2]' ); Infinity is a process, an algorithm, a mapping, a limit, a singularity, a destination... infinity is a lot of things, but it's not a number, even though you can assign it a symbol that makes it look like a number. The confusion starts when you try to treat it like one. I'm not sure that I answered your question. Write back if you'd like to discuss this further, or if you have any other questions. - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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