That's merely an assertion that you can do that for all numbers I could give you. I have a mapping that does that - I start a counter from 1, and for every subsequent different number I'm given, I increment it.

For many irrational numbers (an uncountable infinity of the things, actually), the only way for me to express them would be to recite an infinite string of numbers at you. I mean, sure, there are a few that can handily be expressed as 'Pi' or 'root 2', but if your mapping can only cope with those, then it doesn't work for all numbers. And if your 'proof' of your mapping is to have me tell you a number and you give me the integer for it, then you've restricted yourself to proving the mapping for any numbers small enough for me to be able to give them to you. Which isn't very many, and doesn't prove a thing.

You did say it worked for all numbers. Not just 'all numbers you'd be able to give me'.