>>667
つづき

これらを順次自然数 0, 1, 2, . . . に対応させることで、無（空集合）から自
然数が構成できるとした。が、しかし、これは、落ち着いて考えてみると、Φ の記号を取り囲む括弧の数を数
えているに過ぎないのであって、当然といえば当然のことである。

https://en.wikipedia.org/wiki/Axiom_of_infinity
Axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo?Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1]
Formal statement
In the formal language of the Zermelo?Fraenkel axioms, the axiom reads:
∃ Ｉ (Φ ∈ Ｉ ∧ ∀x∈ Ｉ ((x∪{x})∈ Ｉ )).
In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I. Such a set is sometimes called an inductive set.
This axiom asserts that there is a set I that contains 0 and is closed under the operation of taking the successor; that is, for each element of I, the successor of that element is also in I.
Interpretation and consequences
This axiom is closely related to the von Neumann construction of the natural numbers in set theory, in which the successor of x is defined as x ∪ {x}. If x is a set, then it follows from the other axioms of set theory that this successor is also a uniquely defined set. Successors are used to define the usual set-theoretic encoding of the natural numbers. In this encoding, zero is the empty set:
0 = {}.
The number 1 is the successor of 0:
1 = 0 ∪ {0} = {} ∪ {0} = {0} = {{}}.
Likewise, 2 is the successor of 1:
2 = 1 ∪ {1} = {0} ∪ {1} = {0,1} = { {}, {{}} },

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