N is infinite (a proof)

Question

In the book Introduction to Set Theory (Third Edition, CRC Press, 1999), by Karel Hrbacek and Thomas Jeck, the following appears on page 70:

2.2 Lemma. -- If n ∈ N , then there is no one-to-one mapping of n onto a proper subset X ⊂ n. --

Soon after, comes the Corollary (2.3), which states:

(c) N is infinite.

Proof:

(c) by Exercise 2.3 in Chapter 3, the successor function is a one-to-one mapping of N onto its proper subset N - {0}.

Well... I don't understand that suggestion. The book does not previously present the theorem "If a set X is equipotent to the set Y and Y is a proper subset of X, then X is infinite".

Answer 1

It's a proof by contradiction. Suppose $\mathbb N$ is not infinite. Then it is finite. By (2.3), it can admit no 1-1 map to a proper subset. But the successor function is such a map. This is a contradiction. The contradiction establishes the fact that $\mathbb N$ is infinite. I have no idea how to prove this in intuitionistic logic.

Answer 2

By Exercise 2.3 in Chapter 3, the successor function is a one-to-one mapping of N onto its proper subset N - {0}.

In other words, N is equipotent to N - {0}.

If N were finite, then, by definition, N would be equipotent to a natural number n.

Thus:

n would be equipotent to N - {0} (transitivity of the equipotency relation) and, therefore, n U {n} would be equipotent to N. (*)

Thus, n would be equipotent to n U {n}, which contradicts Lemma 2.2.

(*) Here, I am resorting to a theorem not explicitly presented in the book:

 If A ~ B, a ∉ A, b ∉ B, then (A ∪ {a}) ~ (B ∪ {b}).

(A ~ B stands for A is equipotent to B.)