Wednesday, September 24, 2008

Some Comments on the Previous Post

One thing that is very interesting about what I talked about last time-- it provides a surprisingly practical purpose for detailed talk about infinities. The bigger infinities you know how to talk about, the more logical systems you can prove consistent, giving you more information about which Turing machines never halt. But this is very strange! These Turing machines are not being run for so long that they enter into the realm of higher infinities. They either halt in finite time, or keep looping for one infinity (the smallest infinity, sometimes called omega). Why, then, does talk of greater and greater infinities become useful?

The straightforward answer is to recount how that information is used to prove consistency (and therefore non-halting-ness) of a system. First, we assign a mathematical semantics to a system; a meaning for the system-states. Second, we show that the initial meanings are true. Third, we show that if one state is true, then the transition rules guarantee that the next state is true as well; in other words, we show that all manipulations are truth-preserving. This is the step where infinities come in. Although no infinities are involved in the actual states and transitions, the semantics may include infinities, so our reasoning may rely on facts about infinities. This sounds silly, but it creeps in more readily then you might expect. If the logic that we are proving consistent uses no infinity, then we must use a single infinity (omega) when proving it consistent; if it uses a small infinity, we must use a larger one to prove it consistent; and so on.

Utterly strange! Why does this method of proving consistency work? We're somehow making the proof easier by treating facts about finite stuff as if they were facts about infinite, unreachable stuffs.

So, it seems that self-reference (trying to prove oneself consistent) would give grounding for as many infinities as one might desire. OK, but, how do we actually get them? A normal logic is stuck with just the infinities it has, even when it starts to reason about itself.

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