We prove that induction "works" by the use of the fact the positive integers are well ordered with respect .
That is, ‘every non-empty subset of positive integers contains a least element’.
But the real numbers are well ordered with respect .
The real number set contains no least element.
What Plato said - is little more in a formal language with deeper concepts. If you are new to this maybe thinking like this will help
Why does induction work -
a. You prove it for 1.
b. You assume it for some 'n' and based on this assumption prove it will be true for 'n+1'
Now let's combine a and b above. Put n=1, by 'a' this is true for 1. Then by 'b' it is true for n+1(=2). Apply 'b' again and again. You see it is true for 1,2,3,4,5,...... You will 'generate' all the natural numbers by this procedure. Thus statement is true for all natural number.
Now think, will this work for real numbers? or even rationals? Can by repeatedly adding 1 (or any number) will you be able to list all the real numbers?
Hope it helps