Quote: "A set S is said to be countable if there exists a bijection from S to the set of natural numbers N
Let S = N2 and N = N1
Clearly, S here has a cardinality that is greater than N and there exists no bijection from S to N
Therefore the set of natural numbers N2 is uncountable if we assume that N1 is the set of natural numbers
But the set of natural numbers N2 is supposed to be countable and infinite."
And yeah. There are certain theorems which are difficult to disprove.
But I'm the never-say-die type.
Let the set of natural numbers be N1 = {1,2,3...}
Assume that for any set S we can find a function f1 from S to N1 such that it is a bijection.
Now let the set of natural numbers be N2 = {0,1,2,3...}
It would be necessary to find another function f2 from S to N2 such that it is a bijection.
If you cannot find such a function, the set S is uncountable.
Clearly, the concept of countability is related to the definition of the set of natural numbers.
You would therefore have to prove for any set S, that there is a bijection not only from S to N1, but from S to N2, in order to show that S is countable.
If any sets have been shown to be countable using N1 as the set of natural numbers, you would now have to show that those same sets are countable w.r.t. N2, and vice versa.
If there exists even one set S that is countable w.r.t. N1 but not countable w.r.t. N2 Cantor's theorem is in jeopardy.
Unless you can arrive at a general consensus on which of N1 or N2 is the set of natural numbers. But clearly, this is not clear.
And if tomorrow a third set of natural numbers was declared as universally accepted by mathematicians, all sets Si which were shown to be countable/uncountable would be in jeopardy.
That is how solid Cantor's theorem on countability is.
