[PartiallyTyped]

You shouldn’t see mathematics as consistent because they are not.

A system can be complete, meaning all true statements can be built based on axioms, but it cannot be sound leading to contradictions.

Alternatively a system may be sound, but not all true statements could be derived.

And finally there are statements impossible to prove because the proof is undecidable.

[simonh]

You’re quite right, which is why I described it as highly rather than completely or perfectly consistent.

It’s an interesting question whether the physical world is perfectly or merely highly consistent. If it’s made of mathematics, it may be that it cannot be perfectly consistent. So if we ever find that it is perfectly consistent, that might be evidence that it isn’t made of mathematics.

Most likely we’ll never be able to tell, but we’ll see. Or at least maybe our descendants will.

[staunton]

> whether the physical world is perfectly or merely highly consistent

What does that even mean? The physical world is not a formal system in any obvious sense.

Nor does "consistent" apply to mathematics, by the way, only to formal systems used/studied by mathematics. You cannot mathematically prove or disprove that mathematics is consistent or define precisely what that would mean because mathematics itself isn't rigorously defined. If you define it as "whatever mathematicians are doing", I guess it's in some sense inconsistent, since mathematicians often disagree.