Why can't this logical statement be proved?

Question

In the following, I don't see how the second statement is different from the fourth one.

I think that we can prove 21 is a natural number in the same manner that we can prove 2 is.

Would you explain why the second statement can be proved and the fourth cannot or how they are different? Thank you.

---

The following English statements are logical statements:

• 0 is a natural number
• 2 is a natural number
• For all x, if x is a natural number, then so is the successor of x.
• 21 is a natural number

Predicate calculus:

 natural(0).
 natural(2).
 For all x, natural(x) → natural(successor(x))
 natural(21).

Among these logical statements, the first and third can be viewed as axioms for the natural numbers: statements that are assumed to be true and from which all true statements about natural numbers can be proved. The second statement can be proved:

2 = successor(successor(0)) and natural(0) → natural(sucessor(0)) → natural(successor(successor(0))).

The fourth statement, on the other hand, cannot be proved from the axioms and so can assumed to be false.

Answer 1

Based upon some comments I searched for the errata which says natural(21) should be natural(-1). So it was indeed a typo.