Peano, sidestepped the question of existence by taking a leaf out of Euclids Book. Peano exists wrote down a list of axioms for whole numbers. The main features were:
There exists a number 0
Every number n has a successor, s(n).( which we think of as n+1
If p (n) is a property of numbers, such that P (0) is true, and whenever P (n) is true then P (s (n)) is true, then p (n) is true for every n (Principle of Mathematical Induction)
1= s (0)
2= s (s(s(0)).
it pins down exactly what we have to prove if we want to show, by some means or other, that whole numbers exist.
Transfinite numbers: different sizes of infinity. There is no largest whole number- because adding one always produces a larger number still- so there are infinitely many whole numbers.
Godel: if you proved that mathematics is logically consistent, then it is impossible to prove that. Not just that he could not find a proof, but that no proof exists. Unsolved problems have no solution at all- neither true or false, but in the limbo of undecidability.
It is better to be aware of our limitations than to live in a fool's paradise.
There exists a number 0
Every number n has a successor, s(n).( which we think of as n+1
If p (n) is a property of numbers, such that P (0) is true, and whenever P (n) is true then P (s (n)) is true, then p (n) is true for every n (Principle of Mathematical Induction)
1= s (0)
2= s (s(s(0)).
it pins down exactly what we have to prove if we want to show, by some means or other, that whole numbers exist.
Transfinite numbers: different sizes of infinity. There is no largest whole number- because adding one always produces a larger number still- so there are infinitely many whole numbers.
Godel: if you proved that mathematics is logically consistent, then it is impossible to prove that. Not just that he could not find a proof, but that no proof exists. Unsolved problems have no solution at all- neither true or false, but in the limbo of undecidability.
It is better to be aware of our limitations than to live in a fool's paradise.