I recently stumbled across this fascinating discussion on FOM. The question at stake: why should we believe that ZFC doesn’t prove any false statements about numbers? That is, while of course we should believe ZFC is consistent and -consistent, that is no reason to expect it not to prove false things: perhaps even false things about numbers that we could, in some sense, verify.
Of course – the “in some sense” is important, as Harvey Friedman stressed in one of the later posts. After all ZFC can prove everything PA can, so whatever the false consequences of ZFC are, we couldn’t prove them from PA. There were a number of interesting suggestions. For example it might prove the negation of something we have lots of evidence for (e.g. something like Goldbach’s conjecture where we have verified lots of its instances – except unlike GC it can’t be .) Or perhaps it would prove there was some Turing machine that would halt, but which never would if we were to make it. There’s a clear sense that it’s false that the TM halts, even though we can’t verify it conclusively.
Anyway, while reading all this I became quite a lot less sure about some things I used to be pretty certain about. In particular, a view I had never thought worth serious consideration: the view that there isn’t a determinate notion of being ‘arithmetically sound’. Or more transparently, the view that there’s no such thing as *the* standard model of arithmetic, i.e. there are lots of equally good candidate structures for the natural numbers, and that there’s no determinate notion of true and false for arithmetical statements. Now I have given it fair consideration I’m actually beginning to be swayed by it. (Note: this is not to say I don’t think there’s a matter of fact about statements concerning certain physical things like the ordering of yearly events in time, or whether a physical Turing machine will eventually halt. It’s just I think this could turn out to be contingent. It’ll depend, I’m guessing, on the structure of time and the structure of space in which the machine tape is embedded. Thus, on this view, arithmetic is like geometry – there is no determinate notion of true-for-geometry, but there is is a determinate notion of true of the geometry of our spacetime, which actually turns out to be a weird geometry.)
Something that would greatly increase my credence in this view would be if we could find a pair of “mysterious axioms”, (MA1) and (MA2), which had the following properties. (a) they are like the continuum hypothesis, (CH), in that they are independent of our currently accepted set theory, say ZFC plus large cardinals, and, like (CH), it is unclear how things would have to be for it to be true or false. (b) unlike (CH) and its negation, (MA1) and (MA2) its negation disagree about some arithmetical statement.
Let me first say a bit more about (a). On some days of the week I doubt there are any sets, or that there are as many things as there would need to be for there to be sets. However I believe in plural quantification, and believe that if there *were* enough things, then we could generate models for ZFC just by considering pluralities of ordered pairs. But even given all that I don’t think I know what things would have to be like for (CH) to be true. If there is a plurality of ordered pairs that satisfies ZF(C), then there is one that satisfies ZFC+CH, namely Gödel’s constructible universe, and also one that doesn’t satisfy CH. So even given we have all these objects, it is not clear which relation should represent membership between them. I can only think of two reasons to think there is a preferred relation: (1) if there were a perfectly natural relation, membership, between these objects which somehow set theorists are able latch onto and intuit things about from their armchair or (2) there is only one such relation (up to isomorphism anyway) compatible with the linguistic practices of set theorists. Neither of these seem particularly plausible to me.
Now let me say a bit about (b). Note firstly that Con(ZFC) is an arithmetical statement independent of ZFC. However it is not like (CH) in that we have good reason to believe its negation is false. And more to the point, its negation is inconsistent with there being any inaccessibles. (MA) is going to have to be subtler than that.
It is also instructive to consider the following argument that ZFC *is* arithmetically sound. Suppose it’s determinate that there’s an inaccessible (a reasonable assumption, if we grant there are enough things, and that the truth of these claims are partially fixed by the practices of set theorists.) Let be the first one. Then is a model for ZFC which models every true arithmetical statement (because the natural numbers are an initial segment of [edit: and arithmetical statements are absolute].) So ZFC cannot prove any false arithmetical statement. That is, determinately, ZFC is arithmetically sound. And all we’ve assumed is that it’s determinate that there’s an inaccessible.
Now I find this argument convincing. But clearly this doesn’t prove that every arithmetic statement is determinate. All it shows is that arithmetic is determinate if ZFC is. But (CH) has already brought the antecedent into doubt! So although determinately decides every arithmetical statement correctly, it is still indeterminate what makes true. That is, both (MA1) and (MA2) disagree not only over some arithmetical statement, but also over whether makes that statement true.
Now maybe there isn’t anything like (MA1/2). Maybe we will always be able to find a clear reason to accept or reject any set theoretic statement that has consequences for arithmetic. But I see absolutely no good reason to think that there won’t be anything like (MA1/2). To make it more vivid, there are these really really weird results from Harvey Friedman showing that simple combinatorial principles about numbers imply all kinds of immensely strong things about large cardinals. While these simple principles about numbers look determinate they imply highly sophisticated principles that are independent of ZFC. I see no reason why someone might not find a simple number theoretic principle that implies another continuum hypothesis type statement. And in the absence of face value platonism – *a lot* of objects, and a uniquely preferred membership (perhaps natural) relation between them – it is hard to think how these statements could be determinate.