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Comment by simonh | Hacker News Reader

That's not quite correct. Systems of mathematics cannot be both complete and consistent, but incomplete systems of mathematics can be consistent. For example Presburger arithmetic is provably consistent. There are limits to consistency for sure, but that doesn't mean there's no such thing as mathematical consistency.

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I'm not sure what you mean. Presburger arithmetic is famously complete. What a system can't be is consistent, complete, and strong enough to perform a Godel encoding (which requires something multiplication-like). Drop any of the three requirements and it's possible.

Inconsistent: trivial, from falsehood follows anything.

Incomplete: Peano.

Weak: Presburger.

The comment could be interpreted as meaning that such systems cannot be complete or consistent, I'm just pointing out they can be one or the other. As I understand it, it is possible to consistently prove and decide things in mathematics, just not everything. Godel proved limits to mathematics, not that mathematics doesn't work. That's all.

> Godel proved limits to mathematics, not that mathematics doesn't work

Nobody claimed it doesn't work. It clearly does. The question is if it's a fundamental property of the universe or just an useful but flowed human mental model.

Sure, personally I see it as a language for expressing relationships and processes. I expounded on this in detail in another comment.

> Inconsistent: trivial, from falsehood follows anything.

Only trivial if you accept the principle of explosion (ex falso quodlibet or ex contradictione quodlibet). If you reject it, you end up with paraconsistent logic, from which one can develop nontrivial inconsistent mathematics see https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

> Systems of mathematics cannot be both complete and consistent

No. They can't be at the same times complete, consistent, decidable and powerful enough to express arithmetic. You can do complete, consistent and decidable though.

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