Wednesday, April 05, 2006

Proof of the Existence of Infinity

In doing research for my paper, I came across a proof in Bertrand Russell’s “Essays in Analysis” which I think is particularly important in the discussions we had before on whether it is logically possible to perform a super-task. One of the problems that was brought up was that infinity as well as infinitely small things are defined in terms of limiting operations. However, Russell manages to prove that infinity exists without ever making use of any limiting operation what so ever. He wrote the proof as a response to (quote) “Professor Keyser’s very interesting article ‘On the Axiom of Infinity’” (end quote), which claimed that all accounts of infinity essentially involve some sort of limiting operation, which, Keyser claims, is not sound. Keyser also claimed that all arguments involving infinity are circular arguments, which Russell refutes as well.

Russell’s proof goes something like this:

Definitions and assumptions:

1. Every set has some perfectly well-defined number of terms.

2. Zero is defined to be the number of things fulfilling any condition which nothing fulfils. So, for instance, the number of statements which are both true and false, a condition which nothing fulfils, is zero.

3. Zero exists.

4. The number 1 is the number of members of a set such that there is a member of that set such that when that member is removed the resulting set is identical to zero (i.e. 1-1=0.) A set which contains only zero will have only one member; therefore, the number 1 exists.

5. The set containing 0 and 1 has two members; thus, 2 exists.

6. n is any finite integer.

Russell then moves onto a proof by mathematical induction. The principle of mathematical induction states that any property possessed by the number 0, and possessed by n+1 when it is possessed by n, is possessed by all finite numbers (by “numbers” Russell means integers – i.e. the natural numbers. What he’s really refering to, and he says so later on in the article, are cardinal numbers. But for our purposes we can simply assume that he means integers.)

7. If n is any finite integer, the number of integers from 0 to n, both inclusive, is n+1.

8. If (7), then if n exists, so does n+1.

9. n exists.

10. Therefore, n+1 exists.

11. If (10) and (3) then all finite integers exist.

12. Therefore, all finite integers exist.

13. If m and n are two finite natural numbers (integers), other than 0, then m+n is not identical (i.e. not equal) to m or n.

14. If (13) then if n is any finite number, n is not the number of finite numbers, for the number of numbers from 0 to n is n+1 and (due to (12)) n+1 is not identical to n.

15. If n is any finite number, then n is not the number of finite numbers, for the number of numbers from 0 to n is n+1 and (due to (12)) n+1 is not identical to n.

16. n is any finite number (true by definition.)

17. Therefore, n is not the number of finite numbers.

18. The definition of cardinal numbers (what I’ve been referring to as integers) requires the existence of a number which is the number of the finite numbers (again, what I’ve been referring to as the integers.) This is basically identical to premise (1).

19. If (17) and (18) then at least one non-finite number exists.

20. Therefore, at least one non-finite number exists.

21. If at least one non-finite number exists, then at least one infinite number exists.

22. Therefore, at least one infinite number exists.

I’ve tried to quote most of the argument above straight from Russell; so if there are any premises that I don’t use, it’s because I was trying to stick with the form of Russell’s argument which he presents in his paper. One of the biggest parts where I strayed away from Russell’s original argument was in his use of the word “number” to mean the cardinal numbers. I simply replaced just about everywhere Russell uses the word “number” with the word integer, but this is because Russell’s cardinal numbers are restricted to describing the integers; they cannot describe irrational numbers, for instance, or complex numbers (and thus when Russell says “numbers” he isn’t really referring to all numbers, but rather a subset of the numbers.) But I believe that Russell’s proof still works despite this minor change in wording, namely because there is no change in the meaning of the argument.

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