First, some context. In one of the comments to an answer to the recent question Why not adopt the constructibility axiom V=L? I was directed to some papers of Nik Weaver at this link, on conceptualism. Many of the ideas in those papers appeal to me, especially the idea (put in my own words, but hopefully accurate) that the power set of the natural numbers is a work in progress and not a completed infinity like $\mathbb{N}$.
In some of those papers the idea of a supertask is used to argue for the existence of the completed natural numbers. One could think of performing a supertask as building a machine that does an infinite computation in a finite amount of time and space, say by doing the $n$th step, and then building a machine of half the size that will work twice as fast to do the $(n+1)$th step and also recurse. (We will suppose that the concept of a supertask machine is not unreasonable, although I think this point can definitely be argued.)
The way I'm picturing such a machine is that it would be a $\Sigma_1$ oracle, able to answer certain questions about the natural numbers. I suppose we would also have machines that do "super-supertasks", and so forth, yielding higher order oracles.
To help motivate my question, suppose that beings from outer space came to earth and taught us how to build such machines. I suppose that some of us would start checking the validity of our work as it appears in the literature. Others would turn to the big questions: P vs. NP, RH, Goldbach, twin primes. With sufficient iterations of "super" we could even use the machines to start writing our proofs for us. Some would stop bothering.
Others would want to do quality control to check that the machines were working as intended. Suppose that the machine came back with: "Con(PA) is false." We would go to our space alien friends and say, "Something is wrong. The machines say that PA is not consistent." The aliens respond, "They are only saying that Con(PA) is false."
We start experimenting and discover that the machines also tell us that the shortest proof that "Con(PA) is false" is larger than BB(1000). It is larger than BB(BB(BB(1000))), and so forth. Thus, there would be no hope that we could ever verify by hand (or even realize in our own universe with atoms) a proof that $0=1$.
One possibility would be that the machines were not working as intended. Another possibility, that we could simply never rule out (but could perhaps verify to our satisfaction if we had access to many more atoms), is that these machines were giving evidence that PA is inconsistent. But a third, important possibility would be that they were doing supertasks on a nonstandard model of PA. We would then have the option of defining natural numbers as those things "counted" by these supertask machines. And indeed, suppose our alien friends did just that–their natural numbers were those expressed by the supertask machines. From our point of view, with the standard model in mind, we might say that there were these "extra" natural numbers that the machines had to pass through in order to finish their computations–something vaguely similar to those extra compact dimensions that many versions of string theory posit. But from the aliens' perspective, these extra numbers were not extra–they were just as actual to reality as the (very) small numbers we encounter in everyday life.
So, here (finally!) come my questions.
Question 1: How would we communicate to these aliens what we mean, precisely, by "the standard model"?
The one way I know to define the standard model is via second order quantification over subsets. But we know that the axiom of the power set leads to all sorts of different models for set theory. Does this fact affect the claim that the standard model is "unique"? More to the point:
Question 2: To assert the existence of a "standard model" we have to go well beyond assuming PA (and Con(PA)). Is that extra part really expressible?
Best Answer
These are fundamental questions. We know that any computable set of axioms which holds of the natural numbers must also have nonstandard models. But, paraphrasing Hilary Putnam, if axioms cannot capture the "intuitive notion of a natural number", what possibly could?
As I see it, there are two possible positions on this. One is that we do know what the natural numbers are, and the fact that axioms cannot capture them shows some limitation in the axiomatic method, not in the concept of number itself. The other is that our inability to capture the natural numbers axiomatically shows that we do not actually have a definite conception of them.
I hold the first view, but I admit that the second has its appeal. Asking how we could communicate the idea of a standard model to aliens brings home the difficulty of affirming that we have a clear conception of something while admitting that we are unable to communicate it through language (in particular, axioms).
This leads to deep philosophical questions about how we can communicate anything through language. Cf. Wittgenstein's "private language" argument and his ideas about rule-following.
Here are some things I would say in defense of the view that our conception of $\mathbb{N}$ really is definite, despite the fact that we cannot capture it with (first order) axioms:
Skepticism about the natural numbers can be ramped up. What do you say to someone who denies that we have a clear conception of $10^{100}$? There are serious people who would say we don't. Frankly, I think I have a clearer conception of $\mathbb{N}$ than of $10^{100}$.
My sense is that everyone accepts that our conception of the natural numbers is definite until they learn the incompleteness theorems, but some people are so impressed by these results that they abandon the idea that there even is a definite set of natural numbers. But Wittgenstein's rule-following paradox shows that even axioms may lack the definite character we ascribe to them. So why would we take them as the be-all end-all?
Taking the view that anything meaningful is captured by axioms, and thus that $\mathbb{N}$ is indefinite, has some unpleasant consequences. Say you take the view that there is no distinguished "standard" model of PA: all that matters is what statements can be proven from the Peano axioms. Then you have to accept that "what statements can be proven from the Peano axioms" is itself indefinite. Because the length of a valid proof in PA is a natural number, so if we don't know what the natural numbers are then we don't know what are the possible lengths of proofs. There could be "proofs in PA" which are valid on one version of $\mathbb{N}$ but not on another. Can you really swallow this?
I think the strongest argument in favor of the definiteness of $\mathbb{N}$, and against the idea that PA, or any other axiomatization, is the be-all end-all, is the evident fact that we have an open-ended ability to go beyond any computable set of axioms, for instance by affirming their consistency. If you accept PA you should accept Con(PA), and the process doesn't stop there: you can then accept Con(PA + Con(PA)), and so on. This goes on to transfinite levels. If our understanding of $\mathbb{N}$ really were fully captured by some particular set of axioms then we would not feel we had a right to strengthen those axioms in any special way; the fact that we do feel we have this right shows that our understanding is not captured by any particular set of axioms.
This is my view.