If the universe contains a finite amount of information, would that disprove the existence of an infinite set? I.e. if the representation of a number contained more information than the amount of information available in the entire universe.
If the universe contains a finite amount of information, would that disprove the existence of an infinite set? I.e. if the representation of a number contained more information than the amount of information available in the entire universe.
Not really. Math uses no physical observation, only axioms. Nothing can "prove" or "disprove" axioms. However, if observation supports the axiomatic theory, then we use the theory for physical prediction. If observation doesn't, then we don't use the theory. Does that count as "disproof"?
In practice, infinite sets never exist as enumerations of every element, but as ways to generate more elements along with descriptions for which elements to include. Infinite set theories allow for equivocating a finite description with the infinite enumeration. In contrast, programming languages usually make a distinction between data (always finite) and data generation (possibly infinite). I would think that counts as a "disproof" in a way.
It's a very interesting idea; if you want to learn more about it, look up "ultrafinitism".