It is often said that set theory is the de facto foundation of mathematics. Regardless of the truth of this claim, this seems to be the story told to students (and mathematicians) who poke their elders about foundations.

Before I proceed, let me explicitly state that, in this MO question, I **am not** interested in

- whether set theory is indeed the foundation of mathematics, or
- whether set theory should serve as the foundation of mathematics, or
- whether set theory is superior or inferior to other possible foundations such as type theory etc.

What I **am** interested in is the historical reasons of why set theory has been seen as the unifying foundational framework in the first place. My questions are uncontroversial ones with (hopefully) definite answers:

Who proposed first that mathematics can/should be based on set-theoretic foundations? How did the mathematical community come to accept this?

Let me now explain why I am interested in this question and then list my two findings.

Mathematicians who are not logicians often consider set theory only as a foundational framework. This point of view seems to be somewhat irrelevant to the point of view of a uniformly chosen set theorist who usually sees set theory as the study of the transfinite and the structure of hierarchy of sets. Clearly, such investigations may have foundational implications and therefore, may be of importance even if one only adopts the first point of view.

Nevertheless, as far as I can tell, the development of set theory does not seem to be fuelled by its foundational role. For example, in this article by Kanamori, there are several places where he alludes to this:

Set theory had its beginnings not as some abstract foundation for mathematics but rather as a setting for the articulation and solution of the Continuum Problem: to determine whether there are more than two powers embedded in the continuum.

With ordinals and replacement, set theory continued its shift away from pretensions of a general foundation to a more specific theory of the transfinite, a process fueled by the incorporation of well-foundedness.

From Skolem relativism to Cohen relativism the role of set theory for mathematics would become even more evidently one of an open-ended framework rather than an elucidating foundation.

Assuming that all these claims hold, it seems surprising (and *almost contradictory*) that mathematicians of a certain era, most of whom are presumably not even knowledgeable about set theory, decided to play along and accept set-theoretic foundations. This is why I would like to know about the history of this process. Here are what I learned through a Twitter discussion with Kameryn Williams:

In a 1949 ASL address, Bourbaki writes the following on Page 7:

As every one knows, all mathematical theories can be considered as extensions of the general theory of sets, so that, in order to clarify my position as to the foundations of mathematics, it only remains for me to state the axioms which I use for that theory.

It seems that theory of sets being able to code all mathematical theories was a "well-known fact" by 1949. In this SEP article, at the beginning of Section 3, José Ferreirós stated (without reference, but echoing chapter III, section 4 of his 1999 book) that

In the late nineteenth century, it was a widespread idea that pure mathematics is nothing but an elaborate form of arithmetic. Thus it was usual to talk about the “arithmetisation” of mathematics, and how it had brought about the highest standards of rigor.

With Dedekind and Hilbert, this viewpoint led to the idea of grounding all of pure mathematics in set theory. The most difficult steps in bringing forth this viewpoint had been the establishment of a theory of the real numbers, and a set-theoretic reduction of the natural numbers. Both problems had been solved by the work of Cantor and Dedekind.

Thus the earliest proposal of set-theoretic foundations may even date back to pre-ZFC era. Unfortunately, since I do not know any German, I couldn't track down the aforementioned work of Dedekind and Hilbert. According to this SEP article, they seem to be the prime suspects but I have no other sources.

canbe based on set-theoretic foundations?” The version with “should” is both less justifiable and less popular — and the word “should” does not appear in any of the quotes in the post. $\endgroup$canbe derived from a single set of axioms (set-theoretic or otherwise) was a major advance that dawned on the mathematical community gradually over time. It arose in large part because of the success of set theory "locally" in providing foundations for various areas of mathematics. We take this major advance for granted nowadays, like the air we breathe, and underestimate what a conceptual breakthrough it was. $\endgroup$8more comments