(Zermelo 1908a: 110)
The list of axioms was eventually modified by Zermelo and by the Israeli mathematician Abraham Fraenkel, and the result is usually known as Zermelo-Fraenkel set theory, or ZF, which is now almost universally accepted as the standard form of set theory.
Another was his debate with Zermelo, in 1896–1897.
One example is Russell’s Paradox, also known to Zermelo:
Yet his 1896 paper (Zermelo 1896a) is by no means hostile.
Like Zermelo set theory, NFU has advantages and disadvantages.
Every axiom of Zermelo set theory except Choice is an axiom of naive set theory.
What axioms governing set-existence does Zermelo rely on in Zermelo (1908a)?
., real analysis), which is equal to that of Zermelo–Fraenkel set theory without the Axiom of Power Sets.
A treatment of ordinal number closely related to mine was known to Zermelo in 1916, as I learned subsequently from a personal communication.
However, when Zermelo (1930) introduced the axioms which constitute the modern ZFC axiom system, he formulated the axioms in second-order logic.
An alternative formal logical system for predicative constructive mathematics is Myhill and Aczel’s constructive Zermelo-Fraenkel set theory (CZF).
The study of alternative set theories can dispel a facile identification of “set theory” with “Zermelo-Fraenkel set theory”; they are not the same thing.
(Henceforth we use the standard abbreviations for Zermelo-Fraenkel set theory, ZF, and Zermelo-Fraenkel set theory with the Axiom of Choice, ZFC.)
Constructive and intuitionistic Zermelo-Fraenkel set theories are axiomatic theories of sets in the style of Zermelo-Fraenkel set theory (ZF) which are based on intuitionistic logic.
In the meantime, mathematicians became convinced that the highly impredicative transfinite set theory developed by Cantor and Zermelo was less acutely threatened by Russell’s paradox than previously suspected.
The first axiomatisation of set theory was given by Zermelo in his 1908 paper “Untersuchungen über die Grundlagen der Mengenlehre, I” (Zermelo 1908b), which became the basis for the modern theory of sets.
Looking back at all of these contributions, it is no wonder that Zermelo—who knew the relevant history well—considered the modern theory of sets as having been “created by Cantor and Dedekind” (quoted in Ferreirós 2016b).
With the inclusion of this last, Zermelo explicitly rejects any attempt to prove the existence of an infinite collection from other principles, as we find in Dedekind (1888: §66), or in Frege via the establishment of what is known as ‘Hume's Principle’.
Zermelo gives no obvious way of representing much of ‘ordinary mathematics’, yet it is clear from his opening remarks that he regards the task of the theory of sets to stand as the fundamental theory which should ‘investigate mathematically the fundamental notions “number”, “order”, and “function” ’.
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