# Can Internal Set Theory provide a complete system of arithmetic?

Internal Set Theory (IST) is a conservative extension of ZFC that adds three axioms that serve to define a predicate standard such that all numbers are either standard or not. There are finitely many standard numbers and numbers larger than these are said to be illimited or i-large. Informally speaking, illimited numbers are just too big to grasp so they don't matter.

Suppose we define a formal language in the usual way by choosing an alphabet and setting out the recursive syntactical rules that define a well-formed formula (wff). I have in mind the language of Peano arithmetic. But now we impose an additional criterion that a string needs to satisfy in order to qualify as a wff: its Gödel number must be a standard number. The intuitive justification for doing this is that strings with non-standard Gödel numbers are too large to be accessible to us and so should not matter.

Once we have the language defined, we form a theory using the usual recursive axioms for first order Peano arithmetic, closed under classical logic. We have not imposed any specific upper bound on the length of a formula, so the theory is adequate for the purposes of basic arithmetic. And we have not imposed any specific upper bound on the length of a proof, so we do not have to worry that a given sentence is unprovable just because it is too long.

Since there are finitely many standard numbers, this language contains finitely many wffs. Since the language is finite, any theory defined within it is finite also. Ordinarily, a theory that has a finite model cannot be essentially incomplete.

So, is there any interesting sense in which such a theory of arithmetic avoids Gödel's first incompleteness theorem?

I can imagine a few possible responses... We've made the definition of wff non-constructive, so we've broken the usual relation between finiteness and completeness. The collection of wffs is not a set, so there isn't a clear way to define incompleteness. We've just transplanted the incompleteness from the theory into the language. This is just a version of ultrafinitism and not particularly interesting. I'd be interested to hear what our logic experts think.

• "illimited numbers are just too big to grasp so they don't matter" : why do they not matter :-? Commented Jul 8 at 11:48
• Not just numbers, incidentally; the predicate "standard" applies to sets, as well. Commented Jul 8 at 11:48
• "Since there are finitely many standard numbers" : this is incorrect. Where did you get such an idea? Commented Jul 8 at 11:50
• Your ref says IST is to clearly express Robin's nonstandard analysis via syntactic enrichment. Since it's conservative extension of ZFC it cannot avoid incompleteness theorems. Besides issues already raised, your 'a theory that has a finite model cannot be essentially incomplete' can be raised subtly, PA's standard model is NOT a finite model viewed within PA? On a more deeper insight of incompleteness, its root cause lies in the multiplication operation even through standard numbers. Thus the extended Standard predicate introduced by IST has no effect on any incompleteness results at all... Commented Jul 8 at 23:33
• @DoubleKnot IST itself is incomplete for sure. But that is not the point. We can use set theory to construct theories that are complete. What I am doing is asking whether if we construct a theory of arithmetic in a particular way we can avoid Gödel's first incompleteness theorem in some interesting sense. We could avoid GFIT trivially just by imposing a specific upper bound on Gödel numbers, e.g. 10 ↑ (10 ↑ 10) but that would have unfortunate consequences since there would be cases where φ is a wff but ¬φ is not. Commented Jul 9 at 1:49