ZFC[Ω]: a nonstandard set theory where all sets have nonstandard extensions

Mauro Di Nasso

Let ZFC[Ω] be the first order theory in the language of set theory plus a binary relation symbol Ω, where all axioms of ZFC except regularity are assumed — separation and replacement schemata are considered for all formulas — and the following are postulated. 1) Ω is a functional correspondance whose domain is the class of all sequences (i.e. functions defined on some infinite cardinal κ). 2) Ω(φ) = {Ω(ψ) : ψ(i) ∈ φ(i) for all i}. 3) If ϑ(i) = {φ(i), ψ(i)} for all i, then Ω(ϑ) = {Ω(φ), Ω(ψ)}. 4) If Ω(φ) = Ω(ψ) then Ω(f ∘ φ) = Ω(f ∘ ψ) (provided such compositions are defined). For each set A, define its κ-nonstandard extension J_{κ}(A) = {Ω(φ) : φ : κ → A}, κ any infinite cardinal. 5) For every κ, the κ-nonstandard embedding J_{κ} satisfies κ^{+}-saturation (a set x is internal relative to J_{κ} if x ∈ J_{κ}(A) for some A). By restricting to countable sequences, the above axioms can be arranged so to get a truly elementary presentation of nonstandard analysis, suitable even at the freshman level (see [1]). Theorem. (Transfer principle) For every κ, J_{κ} is a nontrivial bounded elementary embedding of the universe into itself.

In analogy to the von Neumann hierarchy ⋃_{α∈ON} V_{α}(∅) of wellfounded sets, define: W_{0} = ∅; W_{α+1} = P( ⋃_{κ≤α} J_{κ}(W_{α}) ∪ W_{α}); W_{λ} = ⋃_{α<λ} W_{α} if λ limit. As a substitute of regularity, we postulate: 6) Every set is in some W_{α}. In other words, all sets in ZFC[Ω] are obtained by transfinite iterations of power-set and nonstandard extension operations. As a consequence, reflection properties hold. Theorem. ZFC[Ω] is equiconsistent with ZFC.

Symbol Ω is used for historical reasons. In fact, ZFC[Ω] can be presented as a strengthening in the general framework of set theory of the pre-Robinsonian approach to nonstandard analysis by C. Schmieden and D. Laugwitz [2], where Ω denoted a new ”infinite” natural number.

[1] V. Benci and M. Di Nasso, A truly elementary presentation of nonstandard analysis by axiomatic approach, preprint.

[2] C. Schmieden and D. Laugwitz, Eine erweiterung der infinitesimalrechnung, Math. Zeitschr., vol. 69 (1958), pp. 1-39.

Address: Dipartimento di Matematica Applicata, via Bonanno 25/b, 56126 Pisa, Italy

E-mail: dinasso@dma.unipi.it