Sunday, October 05, 2008

Finite Sets Prove Mathematical Induction Correct

In this post, I asked two questions:

1. Where does our knowledge about nonhalting programs come from?

2. What grounds the mathematical concept of "set"?

One thing I didn't mention is that an answer to the second question would probably provide an answer to the first. This is because practically all of our knowledge about nonhalting programs can be proven in axiomatic set theory, and it is (somewhat) reasonable to assume that a "perfect" set theory (one that really did capture everything worth capturing about the human intuition of sets) would capture all of it.

Now, I should reserve myself a bit: explaining where the grounding comes from does not necessarily explain where the knowledge comes from, as illustrated with my logic that (in my opinion) has a grounded notion of "programs" and "computation", but which does not necessarily have any additional knowledge to accompany that grounding. But this is beside my point, which is...

Finite sets prove mathematical induction correct!

I mentioned in that post that most (if not all) knowledge about halting can be provided by the proof method called mathematical induction (no relation to the use of "induction" meaning "learning"). It is not difficult to prove the correctness of mathematical induction from set theory; but, I did not consider this route before, because full set theory is harder to justify then induction. However, I came across a paper by George Boolos called "The Justification of Mathematical Induction" which shows how this can, after all, be a useful approach.

The paper justifies mathematical induction from only an extremely minimal set theory, plus the transitivity of less-than (if a is less then b and b is less than c, a is less then c). The set-theoretic fact needed is approximately this:

Pick any number, n, and formula, F. There exists a set X at level n, containing all and only the those things existing at lower levels and satisfying F.

This is an axiom schema, because we can fill in any formula F to get a specific statement. (First-order logic does not actually have a way of saying "for all formulas".) It establishes levels of existence, one for each natural number. It is "extremely weak" because without more principles, it only requires the existence of each possible finite set! There is no grounding problem, because each set that is forced to exist can be explicitly manipulated. (Note: the principle does not rule out the existence of more complicated sets.) I think this is quite satisfying.

On the other hand, I don't have a good feel for where the extra knowledge "comes from"... I can read through the proof, but it is surprising that merely adding the ability to collect entities together into sets gives us so much power to prove important truths! It is dangerous, too. Dropping the notion of levels gives this schema:

Pick any formula F. There exists a set X, containing all and only the those things satisfying F.

This is the naive set comprehension, which (as is well known) leads to a contradiction known as Russel's paradox. Russel's paradox cannot be derived from the version I gave that is restricted to levels. however, perhaps Russel's paradox can still serve as a lesson that, more often than not, an axiom schema will yield unexpected results.

Remaining Questions:

How can talk of larger infinities be grounded, and how can their mathematical existence be proved? How can the way human mathematicians talk about them be justified? Or, is it all actually nonsense? How much of human mathematical knowledge does an AI need to be given before it can learn the rest? (Remember, all facts about halting are in principle learnable; the reason I'm worried about proving some of them is that humans seem to be able to. Also, humans seem capable of coming to know facts about convergence of processes, which are not in principle learnable or provable, unless we really can prove facts about halting.)

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