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JoshuaZ comments on Exponentiation goes wrong first - Less Wrong
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Yes, the basic math is correct. That's not the point. Nelson's objection is to say that the axiom of induction is not supported in the same way as the others. That's not a math issue, that's an issue of what intuition and evidence tell us. In that sense, the evidence is overwhelming that induction is fine. Almost anyone who thinks for even a short period of time will be ok with it, unless, like Nelson, they have an external motive to be uncomfortable with it.
Incidentally, if one started with the assumption that PA has a contradiction, and asked me given that assumption who would I think is most likely to find the contradiction, Nelson would be very high up on the list simply given his work in foundations. The fact that he has a special motive to find such and still hasn't found it is additional evidence that the system is really consistent.
The only real upshot of this essay is that a contradiction in PA might not result in a contradiction in a sufficiently weakened form of PA that we can still do most forms of useful arithmetic.
Thinking about motives and contradictions, you might find this interesting:
http://video.ias.edu/voevodsky-80th
I hope you are referring to my essay and not Nelson's, which is a gem.
No, I was talking about Nelson's essay. There's nothing in that essay that wouldn't be covered in a basic course in foundations, aside from his ramblings on the nature of monotheistic faith and the meaning of "I AM WHO I AM" (capitalization in the original). The point that there's a distinction between how addition and multiplication behave and how exponentiation and higher analogs behave is not new either, although some sections of the essay might be worth explaining some concepts if one removes the theology.
I'm open to the possibility that PA might be inconsistent, although I assign this claim a very low probability. If one asked the question about some broader useful foundational system, such as ZF, I'd assign a much higher but still low probability.
If you think that either of these claims is wrong, I'd be happy to discuss making some form of wager over the likelyhood of an inconsistency being found within some fixed timespan (say 5 years or a decade?).
I've heard that Nelson has a standing bet with a colleague: he pays one dollar each year until an inconsistency is found in ZF, is paid one hundred dollars each year after an inconsistency is found.
I might take a more domino-oriented bet.
Do you mean one that focuses on PA? Do you care to suggest a specific bet? (For what matters, I'd estimate around a 10^-6 chance that PA is inconsistent.)
I'm teasing. I don't think your domino argument can be used to support induction.
Ok. So joking aside, do you want to make a bet on an inconsistency being found in PA in the next five years? 10 years?
I am fairly certain that no inconsistency will be found in the next 10 years.
Incidentally, I'm a little put off by your bringing up betting so early in the conversation. Isn't it clear that I'm interested in talking about this stuff? That should be enough, especially before you've even located a place where you and I disagree.
I'll mention that if PA is inconsistent, then a consistent prior probability distribution must have P(PA is inconsistent) = 1. (This might not be true if PA is consistent.) Developing a formalism for handling uncertainty about mathematical truths is a line of research in the same ballpark as developing mathematics without induction.
Hmm, I'm not sure why that would be off-putting. There must be some theory of mind issue I'm missing (possibly some sort of implication by suggesting bets that has some sort of negative connotation that I'm not picking up).
Yes, hence making time-based estimates makes more sense.
Ok. So what is your reasoning behind this conclusion?
That you don't think the other party is capable of marshaling arguments to make you update, or of updating on your arguments.
An estimate of how much progress is made on this sort of problem per man-hour, and of how many man-hours will be devoted to the problem in the next ten years. But I am simply agnostic about whether or not a contradiction can be found "in principle."