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Something is only unprovable wrt an axiom set. You can add new axioms to an existing axiom set that can be semi-decidably proved inconsistent given the rest of
by emmab 9y ago
Something is only unprovable wrt an axiom set.
You can add new axioms to an existing axiom set that can be semi-decidably proved inconsistent given the rest of the axiom set.
https://en.wikipedia.org/wiki/Large_cardinal https://en.wikipedia.org/wiki/Large_cardinal
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_...
For a theory of how to choose an axiom set probabilistically, see:
https://intelligence.org/2016/09/12/new-paper-logical-induction/ https://intelligence.org/2016/09/12/new-paper-logical-induct...
- Houshalter 9y agoYou can't just add arbitrary axioms. It's entirely possible to have a statement that's unprovable with any axioms mathematicians would consider reasonable.
- emmab 9y agoAn axiom can be wrong wrt the standard model for some first-order theory: http://lesswrong.com/lw/g0i/standard_and_nonstandard_numbers/ http://lesswrong.com/lw/g0i/standard_and_nonstandard_numbers... But, demonstrating an axiom is inconsistent with the standard model of peano arithmetic or ZFC is in the general case uncomputable. For example, an axiom that claims some program halts when it does not actually halt cannot in general be proved to be inconsistent wrt the standard model, because otherwise one could decide the halting problem.