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Stanford Encyclopedia of Philosophy
Zermelo’s Axiomatization of Set Theory
First published Tue Jul 2, 2013
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3.2.1 Representing Ordinary Mathematics
The first obvious question concerns the representation of the ordinary number systems.
The natural numbers are represented by Zermelo as by Φ, {Φ}, {{Φ}}, …, and the Axiom of Infinity gives us a set of these.
Moreover, it seems that, since both the set of natural numbers and the power set axiom are available, there are enough sets to represent the rationals and the reals, functions on reals etc.
What are missing, though, are the details: how exactly does one represent the right equivalence classes, sequences etc.?
And assuming that one could define the real numbers, how does one characterise the field operations on them?
In addition, as mentioned previously, Zermelo has no natural way of representing either the general notions of relation or of function.
This means that his presentation of set theory has no natural way of representing those parts of mathematics (like real analysis) in which the general notion of function plays a fundamental part.
3.2.2 Ordinality
Zermelo's idea (1908a) was pursued by Kuratowski in the 1920s, thereby generalising and systematising work, not just of Zermelo, but of Hessenberg and Hausdorff too, giving a simple set of necessary and sufficient conditions for a subset ordering to represent a linear ordering.
He also argues forcefully that it is in fact undesirable for set theory to go beyond this and present a general theory of ordinal numbers:
(引用終り)