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(参考)
https://en.wikipedia.org/wiki/Von_Neumann_universe
Von Neumann universe
Definition
Finite and low cardinality stages of the hierarchy
The set Vω has the same cardinality as ω.
The set Vω+1 has the same cardinality as the set of real numbers.

The existential status of V
Since the class V may be considered to be the arena for most of mathematics, it is important to establish that it "exists" in some sense. Since existence is a difficult concept, one typically replaces the existence question with the consistency question, that is, whether the concept is free of contradictions. A major obstacle is posed by Godel's incompleteness theorems, which effectively imply the impossibility of proving the consistency of ZF set theory in ZF set theory itself, provided that it is in fact consistent.[12]

The integrity of the von Neumann universe depends fundamentally on the integrity of the ordinal numbers, which act as the rank parameter in the construction, and the integrity of transfinite induction, by which both the ordinal numbers and the von Neumann universe are constructed. The integrity of the ordinal number construction may be said to rest upon von Neumann's 1923 and 1928 papers.[13] The integrity of the construction of V by transfinite induction may be said to have then been established in Zermelo's 1930 paper.[7]
(引用終り)
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