Consistency and Platonism

1 02 2005

I noticed an odd thing this evening about set theoretic platonists. They seem to accept a proposed axiom as true just in case we can never prove it consistent with the previously accepted axioms, and they reject the ones (like V=L) that are equiconsistent with the previous system. This sounds odd at first, because of course we’d like to believe only things that actually are consistent with the axioms that we already have, so it seems that axioms of the latter sort would be better candidates than the former. But this doesn’t seem to be the practice of most set theorists.

If I understand John Steel right, this is because the equiconsistent axioms add no new interesting mathematical structures, while the others do. After all, if one is interested in studying the structure of ZF+V=L, these all exist inside structures of (say) ZF+”there exists a measurable”, because each of the latter structures gives a structure for the former by the relative consistency proof. However, structures of ZF+V=L don’t give rise to structures that satisfy some of these large cardinal axioms, so we lose that interesting area of mathematical research.

In addition, there seems to be another problem with consistency for platonists, in that whatever system of axioms is “true” (according to the platonists) may turn out not to be consistent. For instance, given a universe satisfying ZFC, either it contains a model of ZFC (in which case we can take the smallest one) or else it contains no model of ZFC. Now, based on Gödel’s Completeness Theorem, there is a model for a theory iff that theory is consistent, so if the universe contains no model of ZFC (as it mustn’t, if we “choose” the universe to be the smallest model of ZFC inside the “actual” universe), then ZFC is inconsistent. If the universe satisfies some large cardinal axioms, then these can be added to the picture as well.

What exactly this inconsistency means is a bit confusing to me. After all, it doesn’t seem like it could mean that there’s “actually” a proof of “0=1” from ZFC, but rather just that the universe “thinks” that some sequence of formulas actually is such a proof. If the model of ZFC that thinks ZFC is inconsistent is a set model within the actual universe, then it’s clear that this proof must “actually” either be infinitely long or use some steps or axioms that the model mistakenly thinks are valid but “actually” aren’t. But for the “actual” universe, I’m not sure what this could mean. Thus, I can see why a platonist who believes that ZFC is “actually true” might be forced to believe as a result that Con(ZFC) is “actually true” as well, and thus Con(ZFC+Con(ZFC)), and so on. This sequence of claims is I believe weaker even than the claim that there exists even a weakly inaccessible cardinal, but I can see how the analogy might hold. The set theorist is forced to accept a sequence of claims, each of which she knows can never be shown to be consistent merely given the (relatively) uncontroversial claims of ZFC. Thus, higher consistency strength, together with a lack of an obvious disproof, can be seen as a good guide to truth, allowing the set theorist to ascend the hierarchy of large cardinals. The fact that the large cardinals are linearly ordered only makes this seem more convincing.

However, I think if one doesn’t take the platonist view about ZFC, then one isn’t forced into this chain of reasoning. For the fictionalist (or formalist, or intuitionist, or whatever), it doesn’t make sense to say that ZFC is “actually” true, and so Gödel’s Completeness Theorem doesn’t force us into any awkward positions for denying Con(ZFC) – especially since we’re withholding judgement on whether Con(ZFC) itself is “actually true” also. If “ZFC is consistent” is interpreted in its natural language sense, we can believe it to be literally true while remaining non-committal about the set-theoretic claim Con(ZFC). Thus, we are never forced to take even one step up the hierarchy of consistency strengths.

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