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> logicians have always said: if it doesn't abide by the rules, we can ignore it: "I hypothesize that only computable and decidable (in Godel's sense) structure
by cscurmudgeon 3y ago
> logicians have always said: if it doesn't abide by the rules, we can ignore it: "I hypothesize that only computable and decidable (in Godel's sense) structures exist"
No, thats far from what they say. Gödel's theorems (based on the first incompleteness theorem) are famous for saying the opposite of your last sentence.
- Sharlin 3y agoHow so? Certainly Gödel doesn't say anything about the ontological status of any kind of structures, not in the Tegmark sense! Or, more accurately, Gödel forbids the existence (in both logical and Tegmarkian senses) mathematical structures that are (a) powerful enough, (b) complete, and (c) self-consistent. But Tegmark here seems to make a much stronger claim, namely that, in his ontology, structures that are (a) do not exist and thus everything ontologically real is both (b) and (c) – finite in the sense that neither undecidable nor semidecidable problems exist.
- cscurmudgeon 3y ago> hypothesize that only computable and decidable (in Godel's sense) structures exist The arithmetic hierarchy is a study of non-computable/decidable sets. https://en.wikipedia.org/wiki/Arithmetical_hierarchy https://en.wikipedia.org/wiki/Arithmetical_hierarchy > Or, more accurately, Gödel forbids the existence (in both logical and Tegmarkian senses) mathematical structures that are (a) powerful enough, (b) complete, and (c) self-consistent. But Tegmark here seems to make a much stronger claim, namely that, in his ontology, structures that are (a) do not exist and thus everything ontologically real is both (b) and (c) – finite in the sense that neither undecidable nor semidecidable problems exist. Ys, but is different from claiming that "only computable and decidable exist"