Realism about arithmetical truth

It seems very plausible that for any specific Turing machine M there is a fact of the matter about whether M would halt. We can just imagine running the experiment in an idealized world with an infinite future, and surely either it will halt or it won’t halt. No supertasks are needed.

This commits one to realism about Σ₁ arithmetical propositions: for every proposition expressible in the form ∃nϕ(n) where ϕ(n) has only bounded quantifiers, there is a fact of the matter whether the proposition is true. For there is a Turing machine that halts if and only if ∃nϕ(n).

But now consider a Π₂ proposition, one expressible in the form ∀m∃nϕ(m,n), where again ϕ(m,n) has only bounded quantifiers. For each fixed m, there is a Turing machine M_m whose halting is equivalent to ∃nϕ(m,n). Imagine now a scenario where on day n of an infinite future you build and start M_m. Then there surely will be a fact of the matter whether any of these Turing machines will halt, a fact equivlent to ∀m∃nϕ(m,n).

What about a Σ₃ proposition, one expressible in the form ∃r∀m∃nϕ(r,m,n)? Well, we could imagine for each fixed r running the above experiment starting on day r in the future to determine whether the Π₂ proposition ∀m∃nϕ(r,m,n) is true, and then there surely is a fact of the matter whether at least one of these experiments gives a positive answer.

And so on. Thus there is a fact of the matter whether any statement in the arithmetical hierarchy—and hence any statement in the language of arithmetic—is true or false.

This argument presupposes a realism about deterministic idealized machine counterfactuals: if I were to build such and such a sequence of deterministic idealized machines, they would behave in such and such a way.

The argument also presupposes that we have a concept of the finite and of countable infinity: it is essential that our Turing machines be run for a countable sequence of steps in the future and that the tape begin with a finite number of symbols on it. If we have causal finitism, we can get the concept of the finite out of the metaphysics of the world, and a discrete future-directed causal sequence of steps is guaranteed to be countable.