Nelson Principles Of Mathematics 10

[Sourn Sanneh]

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Brandon Banes is an Assistant Professor of Mathematics. Formerly a high school mathematics teacher at Lebanon High School, Banes earned his B.S. in Mathematics and Mathematics Education from Lipscomb University...

Born in Searcy, AR. Grew up in Houston, TX. Harding College: B.S, University of Houston: M.S., and Illinois State University: D.A, all in Mathematics. Teaching Assistant, University of Houston (September,...

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The outline begins with a formalist critique of finitism, making the case that there are tacit infinitary assumptions underlying finitism. Then the outline describes how inconsistency will be proved. It concludes with remarks on how to do modern mathematics within a consistent theory.

and similarly for primitive-recursive functions in general. For the schema of primitive recursion to be coherent, it is necessary that the values of the functions always reduce to a numeral, since they are defined only for 0 and iterated successors of numbers for which they have been previously defined.Finitists believe that primitive recursions always terminate; for example, that applying (15)-(18)

While a number is being constructed by recursion, it is only potential, and when the recursion is complete, it is actual. What Bellantoni and Cook, and Leivant, do is restrictrecursions so that they occur only on actual numbers. Then the theorem is that the class of functions computable in this way is precisely the class of polynomial-time functions. This is an astonishing result, since the characterization is qualitative and conceptually motivated, saying nothing whatever about polynomials, Bellantoni, Cook, and Leivant have revealed a profound difference between polynomial-time recursions and all other recursions. The recursions constructed by the BCL schema enjoy a different ontological status from recursions in general. In the former, recursions are performed only on objects that have already been constructed. In the latter, for example in a superexponential recursion, one counts chickens before they arehatched (and the chicks that they produce as well), all in the fond hope that this willeventually take place in the completed infinity by and by.

(One can try to outrun this issue by arithmetising using the full strength of Q 0 *\mathrmQ_0^*, rather than a restricted version of this language in which the rank and level are bounded; but then one would need the consistency of Q 0 *\mathrmQ_0^* to be provable inside Q 0 *\mathrmQ_0^*, which is not possible by the second incompleteness theorem.)

I work in a fixed theory Q_0^*. As Tao remarks, this theory cannot prove its own consistency, by the second incompleteness theorem. But this is not necessary. The virtue of the Kritchman-Raz proof of that theorem is that one needs only consider proofs of fixed rank and level, and finitary reasoning leads to a contradiction.

Let me try to clarify one last time. The one-line summary is that your assertion that one is working in a fixed theory Q 0 *Q_0^* is incorrect; the argument works instead in multiple restricted versions of this theory, and this makes a huge difference. More on this below.

(4) If Q 0 *Q_0^* at a given rank ρ\rho and level λ\lambda is consistent, then there does not exist an xx such that Q 0 *Q_0^* at that rank ρ\rho and level λ\lambda can prove K(x)>l(Q 0 *)K(x) \gt l(Q_0^*).

(5) If Q 0 *Q_0^* at a given rank ρ\rho and level λ\lambda is consistent, then there does not exist an xx such that Q 0 *Q_0^* at that rank ρ\rho and level λ\lambda can prove K(x)>l(Q 0 *(ρ,λ))K(x) \gt l(Q_0^*(\rho,\lambda)),

This is because in order to build a proof verifier for Q 0 *Q_0^* at rank ρ\rho and level λ\lambda, one must at some point encode ρ\rho and λ\lambda. (An unrestricted proof verifier for all of Q 0 *Q_0^* is useless for this purpose unless one is allowed to assume that Q 0 *Q_0^* is consistent.) Thus (barring some unusual encoding trick), the length of the proof verifier must at least equal the Kolmogorov complexity of ρ\rho or λ\lambda. But in the argument, one is playing with at least 2 l(Q 0 *)+12^l(Q_0^*)+1 different values of ρ\rho and λ\lambda, so the Kolmogorov complexity of ρ\rho or λ\lambda is guaranteed to exceed l(Q 0 *)l(Q_0^*) for at least one such choice of ρ\rho or λ\lambda. As such, (5) does not necessarily imply (4).

What Nelson is trying to do is to modify the Kritchman-Raz argument using a theory slightly weaker than PA that he calls Q_0^*. Of course, we know that this theory cannot prove its own consistency (if it is consistent), by the second incompleteness theorem. However, the theory is hierarchical, and roughly speaking contains a nested sequence Q_1 \subset Q_2 \subset Q_3 \subset \ldots of increasingly stronger theories, such that each Q_i+1 can prove the consistency of the predecessor Q_i. The idea is to run the Kritchman-Raz argument to show that Q_1 can prove that \mu>1, that Q_2 can prove that \mu>2, and so forth, until one shows that Q_2^l+1+1 (and thus Q_0^* and PA) can prove that \mu > 2^l+1+1, which is absurd.

IIRC, Q^*_0 is close to I\Delta_0, a very weak theory. It can be interpreted in Q. I think for the sake of the argument you can ignore Q^*_0 and replace it with something like I\Delta_0. Stronger theories like S_2 are also interpretable in Q, but they are still quite weak, nothing like I\Delta_0+EXP or PA.

What Nelson means by relativization is that he starts with Q, defines an interpretation of a slightly stronger theory and a formula that defines a cut in each model of Q that would be the model of the stronger theory. The word relativization here comes from the fact that we are relativizing formulas to those cuts (e.g. if the formula defining the cut is C, we replace quantifiers like \forall x with \forall x \in C).

In other words: just because the successor function is total, that does not, to him, imply that ω\omega exists as a completed object. Being always able to take one more step does not mean that the infinite sequence of steps exists: for the next step to exist you have to actually take it.

Personally I find it very discomforting to think that ideology should be a prerequisite to finding a certain kind of mathematical result. The fact that Nelson has this ultrafinitist ideology made initially skeptical of the result, much like one would be skeptical of studies claiming no link between cigarettes and lung cancer produced by scientists employed by tabacco companies.

I, in turn, am uncomfortable with these remarks, which I think go too far. For one thing, Edward Nelson is likely reading these words, since he has already participated in this thread. It might help to bear this in mind, and also bear in mind that he is obviously no fool in logic. (He has done, for example, terrific work in Internal Set Theory.)

It may sound corny, although I really believe it: it takes all types to make the mathematical world go round. In the final analysis, there is no absolutely right way of being, and iconoclasts are good for the overall health of mathematics.

On a digressive personal note: I met Esenin-Volpin back when I was grad student at MIT. He is quite a jolly fellow, despite having been imprisoned by the Soviets three times for his dissident political views.

I know his student Christer (now Catherine) Hennix much better, and had many arguments with him, as he claimed all my mathematical research was founded on sand. At the time, my work involved a lot of analysis, based on principles that he disbelieved and argued against. I found that rather upsetting. I think I could handle it better now.

Another extremist that C.C. Hennix probably knew was Richard Stallman, who also worked in the MIT AI Lab. When most of the lab left to work for private companies, Stallman, a strong advocate of hacker culture and the free sharing of code and ideas, started the free software movement:

I was reminded of Stallman by your remarks about ideologues and extremists pushing on the boundaries of our ideas. I think Stallman is one of these, and while I do not share his level of commitment to free software, it is this very idealism that has made him a positive force in the world.

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