Version: v1, Published online: 1998

Retrieved September 21, 2020, from https://www.rep.routledge.com/articles/thematic/paradoxes-of-set-and-property/v-1

## 8. Zermelo: solving the paradoxes by axiomatizing set theory

In 1908 Zermelo published his axioms for set theory, without reformulating logic or mentioning the semantic paradoxes. To understand Zermelo’s motivation, we must consider two articles he published that year in the same journal, finished within two weeks of each other, and which referred to each other. The second article contained his axiomatization, while the first defended his theorem that every set can be well-ordered. The first also gave a new proof of his theorem, based explicitly on axioms from his second article. The two articles form a single unity, revealing that his axiomatization was motivated as much by a concern to secure his theorem as by the need to solve the paradoxes (see Moore 1978).

The key to Zermelo’s axiomatization was his axiom of separation, which restricted the principle of comprehension to subsets of any previously existing set. To preserve set theory, he included axioms which generated sets from previously existing sets, especially the power set and union axioms (see Set theory, different systems of).

But Zermelo’s approach seemed arbitrary until he introduced the cumulative type hierarchy of sets in 1930. This hierarchy began with the empty set as the lowest level, and any level *S* was followed by a next level, the power set of *S*. The levels were indexed by ordinals, and when the index was a limit ordinal, the level was the union of all previous levels. The hierarchy of levels was cumulative since a member of any level belonged to all following levels, by contrast with Russell’s types, which were disjoint.

Today Zermelo’s hierarchy, as embodied in the Zermelo–Fraenkel axioms formulated in first-order logic, is generally accepted by mathematicians as the proper solution to the logical paradoxes. Among philosophers, the semantic paradoxes still lack such a generally accepted solution.

Moore, Gregory H.. Zermelo: solving the paradoxes by axiomatizing set theory. Paradoxes of set and property, 1998, doi:10.4324/9780415249126-Y024-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/paradoxes-of-set-and-property/v-1/sections/zermelo-solving-the-paradoxes-by-axiomatizing-set-theory.

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