Introduction to Axiomatic Set Theory pp 175-195 | Cite as

# The Arithmetization of Model Theory

## Abstract

In Section 12 we developed a theory of internal models and gave meaning to the idea of a class M being a standard transitive model of ZF. For đť“ś to be a standard model of ZF means in particular that đť“ś is a model of Axiom 5, the Axiom Schema of Replacement. Since Axiom 5 is a schema â€śM is a standard model of ZFâ€ť is a meta-statement asserting that a certain infinite collection of sentences of ZF hold. Can this metastatement be formalized in ZF, that is, can â€śđť“ś is a standard model of ZFâ€ť be expressed as a single sentence in ZF? This question is as yet unresolved. It can be resolved however if đť“ś is a set, that is, â€śm is a standard model of ZFâ€ť can be expressed as a single sentence in ZF. The basic objective of this section is to produce such a sentence. Our approach is to assign GĂ¶del numbers to the well formed formulas of our language. This assignment will be made by the mapping *J* of Definition 15.2.

## Keywords

Induction HypothesisÂ Internal ModelÂ Formation RuleÂ Axiom SchemaÂ Intuitive NotionÂ## Preview

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