Some remarks on axiomatized set theory

is a talk on occasion of the Fith Congress of Scandinavian Mathematicians in which some major ideas of set theory appear and the idea of set theoretic foundation is discussed.


Axiomatic set theory at the stage of this talk is not a formal theory, i.e. not conceived as being a manipulation of strings. Instead the axioms are considered to describe properties of a certain universe or range of things BB (German “Bereich”). This BB would be called underlying set of a model in modern terminology.

The peculiar fact that, in order to treat “sets”, we must begin with “domains” that are constituted in a certain way

First, Skolem points out the problem of how to think of BB itself: if we conceive BB as a set itself, we can not do so as a thing of the theory we are developing. Second, it is pointed out that BB is not uniquely determined by the set of axioms.

A definition, much to be desired, that makes Zermelo’s notion “definite proposition” precise

In modern language a (well-formed) formula.

The fact that in every thoroughgoing axiomatization set-theoretic notions are unavoidably relative

Downward version of Löwenheim-Skolem theorem is proved, i.e. existence of countable models. Skolem's paradox is explained.

The fact that Zermelo’s system is not sufficient to provide a foundation for ordinary set theory

Axioms of replacement is suggested.

The difficulties caused by nonpredicative stipulations when one wants to prove the consistency of the axioms

Skolem argues that the consistency of set theory can not be proved (note that at that time Gödel’s incompleteness theorem was still unknown). Such a proof is in Skolem’s opinion not possible as in set theory sets are formed in a “non-predicative” way, that is to say they are formed from all sets of BB and not just a finite number of sets, e.g. the intersection of all sets with some property.

Skolem remarks that a the “non-predicative requirement of reproduction of sets” also occurs in Russell's type theory in form of the axiom of reducibility?.

The nonuniqueness [Mehrdeutigkeit] of the domain BB

In modern language: Based on a given model of set theory, a new model is constructed by extending the underlying set by a new element. In a footnote Skolem speculates that the continuum hypothesis can not decided in the set theory of his time.

The fact that mathematical induction is necessary for the logical investigation of abstractly given systems of axioms

A skeptic view on Hilbert's program, that was ongoing at that point, is explained. In modern language the argument amounts to saying that in the metalanguage of mathematical logic the induction axiom is indispensable.

A remark on the principle of choice

Skolem criticizes opponents of the axiom of choice by rooting their opposition to a “non-axiomatic” understanding of sets: sets are for them not things subject to some axioms but just a collection on explicitely given things.

Concluding remark

… I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics …

… glaubte ich, dass es so klar sei, dass diese Mengenaxiomatik keine befriedigende letzte Grundlage der Mathematik wäre, dass die Mathematiker grösstenteils sich nicht so sehr darum kümmern würden. In der letzten Zeit habe ich aber zu meinem Erstauenen gesehen, dass sehr viele Mathematiker diese Axiome der Mengenlehre als ideale Begründung der Mathematik betrachten; …


Originally published in German

English translation

category: reference