I'm sorry, from time to time, it happens to me that I don't understand sarcasm (maybe it is due to the fact English isn't my native language)
Anyway mine was a friendly argumentation :)

I totally agree with you about my property C(n), I knew that someone could have noticed this logical flaw! But, as I told you, it was the most "natural" (credible) idea I got.

Oh, yes. I know that not all mathematicians like induction because it can't be demonstrated (in fact it is a principle) and the same they think about Numbers in general!
I know that many mathematicians tried to demonstrate the existence of Natural numbers (because you can demonstrate the existence of other bigger sets (Z,Q...) only assuming that N exists) but it wasn't impossible, so all the existence of all other sets of number isn't demonstrated.
So mathematics needs axioms in order to demonstrate everything.
All the logic of math, is based on something that is supposed to exist but we won't never if it actually exist...
To extend the concept:
Do we exist? Are we talking now? XD