Truth-value realisms about arithmetic

Arithmetical truth-value realists hold that any proposition in the language of arithmetic has a fully determined truth value. Arithmetical truth-value necessists add that this truth value is necessary rather than merely contingent. Although we know from the incompleteness theorems that there are alternate non-standard natural number structures, with different truth values (e.g., there is a non-standard natural number structure according to which the Peano Axioms are inconsistent), the realist and necessist hold that when we engage in arithmetical language, we aren’t talking about these structures. (I am assuming either first-order arithmetic or second-order with Henkin semantics.)

Start by assuming arithmetical truth-value necessitism.

There is an interesting decision point for truth-value necessitism about arithmetic: Are these necessary truths twin-earthable? I.e., could there be a world whose denizens who talk arithmetically like we do, and function physically like we do, but whose arithmetical sentences express different propositions, with different and necessary truth values? This would be akin to a world where instead of water there is XYZ, a world whose denizens would be saying something false if they said “Water has hydrogen in it”.

Here is a theory on which we have twin-earthability. Suppose that the correct semantics of natural number talk works as follows. Our universe has an infinite future sequence of days, and the truth-values of arithmetical language are fixed by requiring the Peano Axioms (or just the Robinson Axioms) together with the thesis that the natural number ordering is order-isomorphic to our universe’s infinite future sequence of days, and then are rigidified by rigid reference to the actual world’s sequence of future days. But in another world—and perhaps even in another universe in our multiverse if we live in a multiverse—the infinite future sequence of days is different (presumably longer!), and hence the denizens of that world end up rigidifying a different future sequence of days to define the truth values of their arithmetical language. Their propositions expressed by arithmetical sentences sometimes have different truth values from ours, but that’s because they are different propositions—and they’re still as necessary as ours. (This kind of a theory will violate causal finitism.)

One may think of a twin-earthable necessitism about arithmetic as a kind of cheaper version of necessitism.

Should a necessitist go cheap and allow for such twin-earthing?

Here is a reason not to. On such a twin-earthable necessitism, there are possible universes for whose denizens the sentence “The Peano Axioms are consistent” expresses a necessary falsehood and there are possible universes for whose denizens the sentence expresses a necessary truth. Now, in fact, pretty much everybody with great confidence thinks that the sentence “The Peano Axioms are consistent” expresses a truth. But it is difficult to hold on to this confidence on twin-earthable necessitism. Why should we think that the universes the non-standard future sequences of days are less likely?

Here is the only way I can think of answering this question. The standard naturals embed into the non-standard naturals. There is a sense in which they are the simplest possible natural number structure. Simplicity is a guide to truth, and so the universes with simpler future sequences of days are more likely.

But this answer does not lead to a stable view. For if we grant that what I just said makes sense—that the simplest future sequences of days are the ones that correspond to the standard naturals—then we have a non-twin-earthable way of fixing the meaning of arithmetical language: assuming S5, we fix it by the shortest possible future sequence of days that can be made to satisfy the requisite axioms by adding appropriate addition and multiplication operations. And this seems a superior way to fix the meaning of arithmetical language, because it better fits with common intuitions about the “absoluteness” of arithmetical language. Thus it it provides a better theory than twin-earthable necessitism did.

I think the skepticism-based argument against twin-earthable necessitism about arithmetic also applies to non-necessitist truth-value realism about arithmetic. On non-necessitist truth-value realism, why should we think we are so lucky as to live in a world where the Peano Axioms are consistent?

Putting the above together, I think we get an argument like this:

1. Twin-earthable truth-value necessitism about arithmetic leads to skepticism about the consistency of arithmetic or is unstable.

2. Non-necessitist truth-value realism about arithmetic leads to skepticism about the consistency of arithmetic.

3. Thus, probably, if truth-value realism about arithmetic is true, non-twin-earthable truth-value necessitism about arithmetic is true.

The resulting realist view holds arithmetical truth to be fixed along both dimensions of Chalmers’ two-dimensional semantics.

(In the argument I assumed that there is no tenable way to be a truth-value realist only about Σ₁⁰ claims like “Peano Arithmetic is consistent” while resisting realism about higher levels of the hierarchy. If I am wrong about that, then in the above argument and conclusions “truth-value” should be replaced by “Σ₁⁰-truth-value”.)