To prove 2 is FINITE, we need to know that 4 is a NATURAL number?

[tonychanyt]

Using Peano Axioms, Dr. Alexander Pruss says:

We've proved that 2 is a natural number and hence finite.

Peano Axioms does not prove that 2 is a finite number. That requires a different proof. (See Appendix.)

That sounds great until we realize that It has taken us 4 non-axiom steps to do this and we are worried

No worry. You are not trying to prove that 4 is a natural or finite number. You have only proved that 2 is a natural number.

and the reason we started worry about what is finite because we are worried how do we know that we are dealing with finite proofs.

No worry. We know that it is a finite proof when the proof stops as you did.

So it seems like that to prove 2 is finite, we use a proof that has 4 non-axiomatic steps, and to know that that's a good proof, we need to know that 4 is a natural number.

(Bold emphases added)

No, you don't need to prove 4 is a natural number to prove 2 is. You only need to make sure that when you are proving 2, your proof terminates. You are not required to count the number of steps. That's not part of the formal Peano proof. Pruss conflated the definition of a natural number and the definition of a finite number. A natural number and a finite number are two distinct mathematical concepts.

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Appendix 1

In set theory, a finite number corresponds to the cardinality of a finite set—a set that contains a specific, limited number of elements. Formally:

A set S is finite if there exists a bijection (one-to-one correspondence) between S and the set {1,2,3,…,n} for some natural number n. The number n is called the cardinality of the set S, and it is a finite number.

You can prove that 2 is a finite number according to this set-theoretic definition. Using von Neumann ordinal construction:

• 0=∅ (the empty set),
• 1={0}={∅},
• 2={0,1}={∅,{∅}},

Therefore, 2 is a finite number. There is no need to know that 4 is a natural number.

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Appendix 2

In the video, Pruss' son raised the question: Is infinity odd or even?

My answer:

Infinity is not a natural number. The parity property does not apply to infinity. Is π odd or even?