you can't prove something just by assuming it that would be trivial, if we can build or convince ourselves how to build a wormhole or a time machine then we can believe in Real numbers

What Koshkin says is correct and has already been proven by others. You want to get a book that covers sets, maps and cardinal numbers, and read it carefully if you are interested in this sort of stuff.

I can see that you can symbolically talk about the power set of (0,1) and say it's cardinality is bigger than the set itself (although I haven't studied sets and cardinalities deeply enough) but I can't see what Real numbers offers against the computable Universe hypothesis