5 ms·

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 enc

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.

simonh 3y ago
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.
- trabant00 3y ago
  > 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.
  - simonh 3y ago
    Sure, personally I see it as a language for expressing relationships and processes. I expounded on this in detail in another comment.
skissane 3y ago
> 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/ https://plato.stanford.edu/entries/mathematics-inconsistent/ and https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/12633603_Real%20Analysis%20in%20Paraconsistent%20Logic_a.pdf https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...