# Can truths about the natural numbers vary across possible worlds?

The truths of logic are the same in all possible worlds. However, what about truths about natural numbers? Like, for instance, is there a world where there are only finitely many primes, or a world where Fermat's Last Theorem is false, or a world where Goldbach's conjecture is true and another world where it is false? I would be very interested if there are philosophers who have thought about these things.

• Modal logic and metaphysics aren't really the same thing. (You've tagged only with the former.) One can happily use various modal logics for non-metaphysical purposes. – Fizz Apr 4 at 21:50
• What you talk about here seems to be necessitism (or its denial) ndpr.nd.edu/reviews/modal-logic-as-metaphysics – Fizz Apr 4 at 21:56
• Well, no, if they use same definitions and axioms about natural numbers. For example, speed of light is very important fundamental constant in an universe, but it does not change set theory (for example). However, it is quite possible that inhabitants of other worlds may need different definitions than we have, and could limit prime numbers to let's say those under 1 million etc .. – rs.29 Apr 4 at 22:21
• Although you've accepted an answer, the Q isn't too clear in retrospect. By "truth" do you mean provable truth from the same axioms? Or model truth that isn't provable. Your two examples ("Fermat's last" theorem) and Goldbach's conjecture are actually of different kinds in that regard. FLT is formally proved in our world now, Goldbach's conjecture is suspected true via failing to find countexamples in the intended/standard semantics/model for integers. – Fizz Apr 5 at 0:32
• No. Truths about sets, numbers and other mathematical objects are traditionally taken to be necessary, i.e. the same in all possible worlds, see Yli‐Vakkuri, Hawthorne, The Necessity of Mathematics:"It is a commonplace that statements of pure mathematics are necessarily true if true at all." There are some generalizations (impossible worlds) where they vary, but those are exotic. – Conifold Apr 5 at 6:06

## 2 Answers

Accounts of impossible worlds come in many flavors, but none seems to require that natural numbers obey the same laws in all possible worlds unless these laws count as "logical" to some (required) extent. Or one might posit numbers as dwelling in their own world(s), wherefore the question of variation becomes the question of something like the set-theoretic multiverse.

Note, for example, that Zermelo format (2 = {{0}}) yields slightly different fundamentals for numbers as sets, compared to von Neumann format (2 = {0, 1}). I.e. here, in Zf, 0 is not directly an element of the set corresponding to 2, but it is an element of vNf 2. Supposedly there are infinitely many variations on these themes.

Or consider questions about the existence of zero sharp. Granted, this goes beyond the "natural numbers" but it is not too far from them in character, I think. So imagine one world where zero sharp exists and one where it does not...

With respect to at least one of your specific examples, suppose some kind of finitism was possibly true (metaphysically, not just epistemically). Then there would be a possible world with only finitely many natural numbers, including only finitely many primes.

EDIT: Saharon Shelah poses the question of applying forcing to Peano Arithmetic in "Reflecting on Logical Dreams," but my cursory glance at the paper didn't show me an example of this application. If you could find a "reasonable arithmetical statement" that you could force answers to, I think that would count as an arithmetical statement that might vary across possible worlds.

• Although the OP accepted this answer, it's not clear to me they were talking about impossible worlds.. According to the SEP page you linked impossible worlds violate (classical) logic, but he OP seems to be excluding these from their notion of possible worlds. – Fizz Apr 5 at 0:21
• I meant to imply that there seem to be no impossible worlds defined as ones where arithmetic fails, so by some sort of contraposition it would be that there were possible worlds where arithmetic fails/is different. I did read the article more, though, and found that this seeming did not quite pan out... – Kristian Berry Apr 5 at 0:29
• The other thing is it's not clear to me if Benacerraf’s identification problem (your middle para) says anything about numbers in the sense of it affecting any conjecture of the kind that the OP asking about (twin primes, Goldbach's etc.) I'd wager that it doesn't. – Fizz Apr 5 at 1:15
• The OP referred to those as instances of a general question. Idk enough about those instances to say, so I just went at the general question of "truths about natural numbers" (supposing correspondence with sets in various formats counts as such truths). – Kristian Berry Apr 5 at 2:08
• The truths of logic and mathematics are taken to be necessary on the standard conception, i.e. the same across all possible worlds, see e.g. SEP, and Yli‐Vakkuri, Hawthorne, The Necessity of Mathematics for extended discussion. So it seems to me the primary answer should be "no". Then you can discuss generalizations that involve impossible worlds. – Conifold Apr 5 at 4:15

Here's a mad idea. If you had an unnatural number of eyes you might for example count earth 3 as alien 2. Then alien 2 x 9 is not the earth 2 x 9. I suppose, in other words that the truths about natural numbers would vary if the algorithm for counting varied and I suppose it could. But then you could say that the earth way of counting is the one your talking about. It ain't modal logic but that's neither here nor there.