I remember there were some speculations by Yurii Manin that our arithmetic is impect because it distinguishes between huge numbers that differ by 1 which should for all practical purposes be the same. I am not sure I follow Manin here but anyway he did make such an argument, though for the moment I can't find it in MathSciNet. Perhaps this did not reach MSN, and perhaps in the context of physics this actually makes sense. At any rate this seems like an intriguing question, though in my view the answer is negative: you can have an infinite prime number $H$ which is unlike $H+1$.