Here is a question that popped into my head right as I fell asleep last night. 


I was thinking about constructions of irrational numbers, like pi. I was wondering if there are two constructions (any constructions, the pair doesn't have to include pi itself) such that:


1. We have good reason to believe they are different numbers.
2. We have a demonstration that, if they are in fact different, then their difference must be small enough that there is no way we will ever find this difference. By this I mean, the number of calculations it would take to find the difference would take more computational resources than the energy of the Universe would allow.


Heads up, I'm an amateur computer science enthusiast, and am neither a mathematician nor a physicist, but this question involves a mix of all three, so some of the phrasing might be awkward, and the whole question could be conceptually confused. If you think it is, please let me know! 


The inspiration for this question comes from reading about hash functions in software version control. It seems like one of the reasons that numbers are "useful" from a computational standpoint, is that they are easy to compare for equality. This probably has something to do with the ease with which they are placed in order. So the puzzle I was pondering was, if we know that numbers can be ordered by definition, do we also have numbers that we can't, in physical reality, put into an order? 


Is this question reducible to some incompleteness argument? Has it been asked before? Apologies if this is too open-ended a question for Math Overflow. Any and all suggestions would be greatly appreciated. 



Provably undecidable number inequality?
I think standard examples are like this ... Enumerate the cases of Goldbach's conjecture. Let $a\_n = 0$ if the $n$th case is true, and $a\_n = 1$ if false. Consider the number $A = \sum\_{k=1}^\infty a\_k 2^{-k}$ with the binary expansion $(a\_n)$. We may compute $A$ as accurately as we like. But $A=0$ is equivalent to Goldbach's conjecture.


Repeat this with many other undecidable propositions instead of Goldbach.


