The set theoretical foundations of nonstandard analysis

• Published in: The Journal of symbolic logic 1973-06, Vol.38 (2), p.189-192
• Main Author: Phillips, N. C. K.
• Format: Article
• Language: English
• Subjects: Mathematical functions; Mathematical relations; Mathematical set theory; Nonstandard analysis; Superstructures; Ultrapowers
• ISSN: 0022-4812; 1943-5886
• DOI: 10.2307/2272054

Nonstandard analysis was developed in [1] within a logic which has a language with finite types. In [2] and [3] the logic is first order and the language is that of set theory. The set theoretical approach can be described in the following setting. Let be a relational structure ( A , ∈) where A is a nonempty set and ∈ is a restriction of the elementhood relation of set theory. Let Ext be the wf (∀ x )[(∃ w )[ w ∈ x ] → [(∀ y )(∀ z ) [ z ∈ x ↔ z ∈ y ] → x = y ]]. Call a fragment of set theory if ⊧ Ext. By forming a nontrivial ultrapower of one obtains a structure which, after canonically embedding in , becomes a proper elementary extension of . Let j embed canonically in . Let be the substructure ( B ′, ∈′) of where B ′ is the ∈′-closure of j(A) in B . That is, B ′ is the smallest subset of B containing j(A) such that if b ∈ B , b ′ ∈ B ′ and ⊧ b ∈ b ′ then b ∈ B ′.