Internal Set Theory - Formal Axioms For IST

Formal Axioms For IST

IST is an axiomatic theory in the first-order logic with equality in a language containing a binary predicate symbol ∈ and a unary predicate symbol standard(x). Formulas not involving st (i.e., formulas of the usual language of set theory) are called internal, other formulas are called external. We use the abbreviations

\begin{align}\exists^\mathrm{st}x\,\phi(x)&=\exists x\,(\operatorname{standard}(x)\land\phi(x)),\\ \forall^\mathrm{st}x\,\phi(x)&=\forall x\,(\operatorname{standard}(x)\to\phi(x)).\end{align}

IST includes all axioms of the Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Note that the ZFC schemata of separation and replacement are not extended to the new language, they can only be used with internal formulas. Moreover, IST includes three new axiom schemata – conveniently one for each letter in its name: Idealisation, Standardisation, and Transfer.

Read more about this topic:  Internal Set Theory

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