I heard somewhere on the internet once something along the following lines:
Any conceivable foundational theory of mathematics (be it ZFC or, if ZFC was found to be inconsistent, some modification of it), assuming it is actually consistent, will fall prey to Gödel's incompleteness theorems inevitably.
Is this statement true or false?
"Conceivable" is not really a technical term, but the technical prerequisites for the incompleteness theorem are hard to wiggle out of:
It must be computable whether a purported proof is actually a valid proof (and if so, what it is it proves).
The system must be consistent (that is, it doesn't prove a contradiction).
The system must be based on something resembling ordinary logic, especially such that we can speak of negations and quantifiers.
The system must be able to express calculation with primitive recursive functions, which is a fairly restricted class of "easily computable functions" from natural numbers to natural numbers.
The last of these assumptions is the one it takes the most work to wrap one's head around, but it is generally accepted that a system that cannot even express primitive recursive computation will not be useful as a foundational theory for most of mathematics.
