I’ve been thinking about the set theory multiverse and its philosophical implications, particularly regarding mathematical truth. While I understand the pragmatic benefits of the multiverse view, I’m struggling with its philosophical implications.
The multiverse view suggests that statements like the Continuum Hypothesis aren’t absolutely true or false, but rather true in some set-theoretic universes and false in others. We have:
Gödel’s Constructible Universe (L) where CH is true
Forcing extensions where CH is false
Various universes with different large cardinal axioms
However, I find myself appealing to basic logical principles like the law of non-contradiction. Even if we can’t currently prove certain axioms, doesn’t this just reflect our epistemological limitations rather than implying all axioms are equally “true”?
To make an analogy: physical theories being underdetermined by evidence doesn’t mean reality itself is underdetermined. Similarly, our inability to prove CH doesn’t necessarily mean it lacks a definite truth value.
Questions I’m wrestling with:
What makes certain axioms “true” beyond mere consistency?
Is there a meaningful distinction between mathematical existence and consistency?
Can we maintain mathematical realism while acknowledging the practical utility of the multiverse approach?
How do we reconcile Platonism with independence results?
I’m leaning towards a view that maintains objective mathematical truth while explaining why we need to work with multiple models pragmatically. But I’m very interested in hearing other perspectives, especially from those who work in set theory or mathematical logic.
The law of non contradiction isn’t true in all “universes” , either. It’s not true in paraconsistent logic, specifically.
Arguably, “basic logical principles” are those that are true in natural language. Otherwise nothing stops us from considering absurd logical systems where “true and true” is false, or the like. Likewise, “one plus one is two” seems to be a “basic mathematical principle” in natural language. Any axiomatization which produces “one plus one is three” can be dismissed on grounds of contradicting the meanings of terms like “one” or “plus” in natural language.
The trouble with set theory is that, unlike logic or arithmetic, it often doesn’t involve strong intuitions from natural language. Sets are a fairly artificial concept compared to natural language collections (empty sets, for example, can produce arbitrary nestings), especially when it comes to infinite sets.
[note: I dabble, at best. This is likely wrong in some ways, so I look forward to corrections. ]
It’s REALLY hard to distinguish between “unprovable” and “unknown truth value”. In fact, this is recursively hard—there are lots of things that are not proven, but it’s not known if they’re provable. And so on.
Mathematical truth is very much about provability from axioms.
“true” is hard to apply to axioms. There’s the common-sense version of “can’t find a counterexample, and have REALLY tried”, which is unsatisfying but pretty effective for practical use. The formal version is just not to use “true”, but “chosen” for axioms. Some are more USEFUL than others. Some are more easily justified than others. It’s not clear how to know which (if any) are true, but that doesn’t make them equally true.
I have spent a long time looking in vain for any reason to think ZFC is consistent, other than that it holds in the One True Universe of Sets (OTUS, henceforth). So far I haven’t found anything compelling, and I am quite doubtful at this point that any such justification exists.
Just believing in the OTUS seems to provide a perfectly satisfactory account of independence and nonstandard models, though: They are just epiphenomenal shadows of the OTUS, which we have deduced from our axioms about the OTUS. They may be interesting and useful (I rather like nonstandard analysis), but they don’t have any foundational significance except as a counterexample showing the limitations of what formal systems can express. I take it that this is more or less what you have in mind when you say
It’s disappointing that we apparently can’t answer some natural questions about the OTUS, like the continuum hypothesis, but Gödel showed that our knowledge of the OTUS was always bound to be incomplete 🤷♂️.
Having said that, I still don’t find the Platonist view entirely satisfactory. How do humans come to have knowledge that the OTUS exists and satisfies the ZFC axioms? Supposing that we do have such knowledge, what is it that distinguishes mathematical propositions whose truth we can directly perceive (which we call axioms) from other mathematical propositions (which we call conjectures, theorems, etc.)?
An objection more specific to set theory, as opposed to Platonism more generally, would be, given a supposed “universe” V of “all” sets, the proper classes of V are set-like objects, so why can’t we extend the cumulative hierarchy another level higher to include them, and continue that process transfinitely? Or, if we can do that, then we can’t claim to ever really be quantifying over all sets. But if that’s so, then why should we believe that the power set axiom holds, i.e. that any of these partial universes of sets that we can quantify over is ever large enough to contain all subsets of N?
But every alternative to Platonism seems to entail skepticism about the consistency of ZFC (or even much weaker foundational theories), which is pragmatically inconvenient, and philosophically unsatisfactory, inasmuch as the ZFC axioms do seem intuitively pretty compelling. So I’m just left with an uneasy agnosticism about the nature of mathematical knowledge.
Getting back to the question of the multiverse view, my take on it is that it all seems to presuppose the consistency of ZFC, and realism about the OTUS is the only good reason to make that presupposition. In his writings on the multiverse (e.g. here), Joel Hamkins seems to be expressing skepticism that there is even a unique (up to isomorphism) standard model of N that embeds into all the nonstandard ones. I would say that if he thinks that, he should first of all be skeptical that the Peano axioms are consistent, to say nothing of ZFC, because the induction principle rests on the assumption that “well-founded” means what we think it means and is a property possessed by N. I have never seen an answer to this objection from Hamkins or another multiverse advocate, but if anyone claims to have one I’d be interested to see it.