Not OP, feel free to correct me, but Godel proved that any formal system of logic can be shown to be logically inconsistent at at least one point.
"Formal system of logic" maps well onto Turing machines, and Turing machines map onto "any computer system" if you are very abstract about it.
Now, people are wrong about mathematical facts, but seemingly not forever. We puzzle it over, come back to it, someone makes a breakthrough. It doesn't seem like there is a fixed blind spot where our logic breaks down.
So what are people doing that is not captured by a formal system of logic?
Options:
1. The mapping of Turing machines to computers isn't airtight - they are doing something "more" than a TM with infinite time + space is capable of
2. People are not "more than" an infinite TM either, we just romantically believe we are, and ignore our own flaws
3. People are doing something special that is not captured by our theory of computation
> Godel proved that any formal system of logic can be shown to be logically inconsistent at at least one point.
He proved they can be inconsistent, or incomplete. The ones that mathematicians work with are incomplete and assumed to be consistent[0], namely that there are statements which are true, but not provable, within that system. Consistent-but-incomplete systems don't have any contradictions or logical holes; they just can't determine the truth of every possible statement.
From an incomplete system you can build a "more powerful" system- one with another axiom that makes more things provable without contradicting anything in the original one.
In inconsistent systems, everything is provable, even statements' own negations, so they aren't very useful.
[0] one statement that you can't prove within a (sufficiently powerful) system is "this system is consistent."
"Formal system of logic" maps well onto Turing machines, and Turing machines map onto "any computer system" if you are very abstract about it.
Now, people are wrong about mathematical facts, but seemingly not forever. We puzzle it over, come back to it, someone makes a breakthrough. It doesn't seem like there is a fixed blind spot where our logic breaks down.
So what are people doing that is not captured by a formal system of logic?
Options:
1. The mapping of Turing machines to computers isn't airtight - they are doing something "more" than a TM with infinite time + space is capable of
2. People are not "more than" an infinite TM either, we just romantically believe we are, and ignore our own flaws
3. People are doing something special that is not captured by our theory of computation