Relative theory of internal sets
The author proposes a new set theory called RIST. The theory is developed in Nelson style. An undefined two-place predicate SR is used. \(x\text{SR}y\) is the abbreviation of ``\(x\) est standard relativement à \(y\)'', which is also called ``\(x\) is \(y\)-standard''. We assume that SR is an equivalence relation. For a set \(\alpha\) and a formula \(F\), the following notations are introduced: \[ \forall^ \alpha x F\quad\text{for }\forall x (x\text{SR}\alpha\Rightarrow F), \quad \forall^{\sim\alpha}x F \quad \text{for }\forall x((\sim(x\text{SR}\alpha))\Rightarrow F), \] \[ \forall^{\alpha\text{ fin}}x F\quad \text{for }\forall x (x\text{ finite}\Rightarrow F), \] \[ \exists^ \alpha x F\quad \text{for }\exists x (x\text{SR}\alpha\cap F),\quad \exists^{\alpha\text{ fin}}x F\quad \text{for }\exists^ \alpha x (x\text{ finite}\cap F). \] The axiom system of RIST is ZFC with three axiom schemes as follows: 1) Transfer: For an internal formula \(F(x,t_ 1,\dots,t_ n)\) with free variables \(x\), \(t_ 1,\dots,t_ n\), \[ \forall^ \alpha t_ 1 \forall^ \alpha t_ 2\cdots \forall^ \alpha t_ n (\forall^ \alpha x F(x,t_ 1,\dots,t_ n)\Rightarrow \forall^ \alpha x F(x,t_ 1,\dots,t_ n)). \] 2) Idealization: Let \(F(x,y)\) be an internal formula with \(x\), \(y\) as free variables and possibly other parameters. If \(\beta\), \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ k\) are fixed sets such that \(\beta\) is not \((\alpha_ 1,\alpha_ 2,\dots,\alpha_ k)\)- standard, then \[ \forall^{\alpha_ 1,\text{fin}} z_ 1\cdots\forall ^{\alpha_ k,\text{fin}} z_ k \exists^ \beta y\;\forall x_ 1\in z_ 1\cdots\forall x_ k\in z_ k F(x_ 1,\dots,x_ k,y) \] \[ \Leftrightarrow \exists^ \beta y\;\forall^{\alpha_ 1}x_ 1\cdots \forall^{\alpha_ k}x_ k F(x_ 1,\dots,x_ k,y). \] 3) Standardization: If \(\alpha\) is a set, and \(F(x,\dots)\) is an \(\alpha\)- external formula, then \[ \forall^ \alpha y \exists^ \alpha z \forall^ \alpha t\;(t\in z\Leftrightarrow (t\in y\cap F(t,\dots)). \] Some direct consequences obtained from RIST with these axioms are stated and proved. If \(\beta\) is not \(\alpha\)-standard, then there is a finite \(\beta\) set containing all \(\alpha\)-standard sets. As simple applications, uniform convergence and compactness in the set of all real numbers are described in RIST terminology. The possibility of the RIST reconstruction of calculus is shown through these applications. The construction of superstructures by successive extensions gives a model of RIST showing a close connection with Keisler's good ultrafilters. The final object is to prove that every internal theorem of RIST is a theorem of ZFC.
scientific article; zbMATH DE number 219873