True Approximations and Models of Arithmetic

Summary

Introduction

All structures here have recursive language and recursive universe–we could take the universe to be ω or a finite initial segment. A structure A is identified with its atomic diagram D(A). Thus, the Turing degree of A is the degree of D(A). Non-standard models of PA are among the most self-aware of all mathematical structures, but they do not know that they are nonstandard. This results in some interesting properties, both model-theoretic and recursion-theoretic.

The self-awareness properties are related to the definability of satisfaction, for formulas of bounded complexity, coupled with a coding of definable sets in divisibility types. For an arbitrary structure A, T_n(A)= Th(A)∩Σ_n is, uniformly in A and n, for n≥1. Thanks to the self-awareness properties, we have Feferman's observation that if A is a non-standard model of PA, then T_n(A)≤_TA for all n.

The fact that a non-standard model of PA does not know that it is non-standard means that we can carry out certain constructions and operations internally (i.e., definably), which would be impossible externally. In particular, we can determine a total function on the model by giving the value at 0 and saying how the value at x+1 is determined from the value at x – never mind that the order type of the model is ω + (ω^* + ω)·η.