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赤ペン先生

無限公理は帰納的集合の存在を主張する。帰納的集合は空集合と後者関数から生成できるすべての元 {},{{}},{{},{{}}},・・・ を持つ。
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無限公理は帰納的無限集合の存在を主張する。帰納的集合は空集合と後者関数から生成できるすべての元 {},{{}},{{},{{}}},・・・（可算無限） を含む。

だな。詳しくは 下記 Set-theoretic definition of natural numbers ご参照
要するに、帰納的定義だけでは、有限の範囲でいくらでも大きな自然数nが作れるが
しかし、無限集合としてのNは構成できない
従って、非構成的に なんらかの無限集合公理を与える必要があるのです■

(参考)
https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
Set-theoretic definition of natural numbers
Definition as von Neumann ordinals
In Zermelo–Fraenkel (ZF) set theory, the natural numbers are defined recursively by letting 0 = {} be the empty set and n + 1 (the successor function) = n ∪ {n} for each n. In this way n = {0, 1, …, n − 1} for each natural number n. This definition has the property that n is a set with n elements. The first few numbers defined this way are: (Goldrei 1996)
略
The set N of natural numbers is defined in this system as the smallest set containing 0 and closed under the successor function S defined by S(n) = n ∪ {n}. The structure ⟨N, 0, S⟩ is a model of the Peano axioms (Goldrei 1996). The existence of the set N is equivalent to the axiom of infinity in ZF set theory.

Frege and Russell
For enabling natural numbers to form a set, equinumerous classes are replaced by special sets, named cardinal. The simplest way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice).[2]