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*ZFC: An axiomatic *approach to nonstandard methods. (English. Abridged French version) Zbl 0906.03066

The author establishes a new foundation of nonstandard analysis in which it is possible to deal with external sets. To do so, a function * defined on the class \(S\) of all standard sets is introduced, namely \[ \biggl\{ \forall x,y,z\;\bigl[^* x=y\wedge {^*x}= z\to(y=z \wedge x\in S)\bigr] \biggr\} \wedge [\forall x\in S\;\exists y\;({}^*x =y)]. \] The class \(I\) of internal sets is defined as the class \(\{y\mid \exists x\in S\) \((y \in{^*x})\}\). Then an external set is a set which is not internal. As axiom, for every \(\in\)-formula \(\varphi\), \(\forall x_1, \dots, x_n \in S(\varphi^S (x_1, \dots, x_n)\to \varphi^I(x_1, \dots, x_n))\) is assumed (transfer principle). \(\varphi^S\), \(\varphi^I\) denote the relativizations to \(S,I\) of \(\varphi\). An axiom of \(\kappa\)-saturation for every definable cardinal \(\kappa\) is: \(\forall F\subset I\) \([| F| <\kappa\wedge \forall F_0 \subset F\) \((F_0\) is finite \(\to\bigcap F_0\) is nonempty)]\(\to\bigcap F\) is nonempty. The author’s system together with the *-function plus five axioms including the above two axioms is considered as \(^*\text{ZFC}\). For every \(\in\)-formula \(\varphi\), \(\text{ZFC} \models \varphi \Leftrightarrow^* \text{ZFC} \models \varphi^S\). \(^*\text{ZFC}\) and ZFC are equiconsistent. Therefore the author obtained a very useful system for the investigation of external sets.
Reviewer: K.Iséki (Osaka)

MSC:

03H05 Nonstandard models in mathematics
03E70 Nonclassical and second-order set theories
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