The Arithmetization of Model Theory
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 𝓜 to be a standard model of ZF means in particular that 𝓜 is a model of Axiom 5, the Axiom Schema of Replacement. Since Axiom 5 is a schema “M is a standard model of ZF” 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 “𝓜 is a standard model of ZF” be expressed as a single sentence in ZF? This question is as yet unresolved. It can be resolved however if 𝓜 is a set, that is, “m is a standard model of ZF” 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 Gödel numbers to the well formed formulas of our language. This assignment will be made by the mapping J of Definition 15.2.
KeywordsInduction Hypothesis Internal Model Formation Rule Axiom Schema Intuitive Notion
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