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Stanford Encyclopedia of Philosophy
Zermelo’s Axiomatization of Set Theory
First published Tue Jul 2, 2013
(抜粋)
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