Comment by burrows
4 years ago
> But what if there is some universe (in the math sense), which is actually complete and decidable, it's just that the moment you take discrete subsets of it, you also remove the connections that make it consistent or complete.
Presumably this hypothetical universe implements arithmetic, so it’s not complete.
You missed my point.
Incompleteness means that there are true statments for which there are no proofs.
But that doesn't preclude proofs that are beyond the proof system that you proofed incompleteness for.
A non discete/symbolic proof might exist after all.
It would therefore not be the existence of the natural number subset that causes undecidability, but the missing parts of the non-natural superset required to talk about the proofs that cause undecidability.
Doesn’t the incompleteness theorem say that any consistent proof system which implements arithmetic must be incomplete?
Then the system you are proposing, does it implement arithmetic or not?
The incompleteness proof comes to a russel paradox like contradiction, caused by a sentence of the form "I am not proovable." encoded via goedel numbering onto peano atrithmetic.
But proof by contradiction itself is problematic, because it relies on the law of the excluded middle, which only holds in classical two valued logic.
If you construct math from the top down rather than from the bottom up, then it is natural that it also has a multi valued logic.
In fact, it also would have infinitely valued logic. Infinite sentences, infinite theorems.
Such math is non expressible for us, because we rely on discrete descriptions.
But that doesn't preclude its existence.
Why should it though? What evidence do we have, if we can't express it or fit it into our existing mathematical framework.
To illustrate, think of Goedelsz approach and turn it backwards for a second. Instead of taking predicate logic and assigning each sentence a natural number, imagine that predicate/classical logic is a different view on the natural numbers. Now that means that there might also be logical interpretation of the reals, the hyperreals and so on. (We could do the same with Alephs, in fact they might be the more fundamental objects.)
Indeed! I suspect/realize that the incompleteness proofs are in fact artifacts of our discrete symbolization - a discrete symbolization that though has been incredibly effective about reasoning about continuous 'phenomena'.
Perhaps that is exactly what mathematics is? and all it can be? Or perhaps there is a 'higher mathematics' that we cannot reach yet? (or ever?)
Yeah, it's frustratingly difficult to investigate these avenues of thought, because they are almost by definition out of our grasp. Even worse the barrier to esoteric pseudo-science is very thin.