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