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> ZFC is consistent with itself, which is good enough for math. That's what we believe, but we don't know it. Nor do we have any hope of proving it in some rea
by Tainnor 4y ago
> ZFC is consistent with itself, which is good enough for math.
That's what we believe, but we don't know it. Nor do we have any hope of proving it in some reasonable way (via Gödel's second incompleteness theorem).
- TuringTest 4y agoMath is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. Math is but a language to express your thoughts in a very precise and rational way. When you find out that two of your carefully expressed thoughts contradict each other, you tweak one or both so that they are no longer contradictory. That's why "the barber who shaves those who don't shave themselves" and "the set of all sets that don't contain themselves" are not considered valid sets anymore.
- Tainnor 4y ago> Math is necessarily consistent because, when we find inconsistencies, we remove them, so they are no longer valid math. The former doesn't follow from the latter, because we may simply not yet have found any inconsistency, even though it exists. edit: Not sure if your point is just "if someone finds a contradiction in ZFC, it doesn't break maths, we'll just find a different foundation". Because I agree. But GP's point was about ZFC, a concrete set of axioms, being consistent.