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Well, with Godel's Incompleteness Theorems, such behavior appears to be an unavoidable property of any mathematical or computational system that can do anything
by lmkg 9y ago
Well, with Godel's Incompleteness Theorems, such behavior appears to be an unavoidable property of any mathematical or computational system that can do anything of note. We may or may not be able to build systems where such properties are further removed from the 'practical' aspects... although such work would necessarily build upon the work in the linked post, which is finding just how far removed it is in the current system.
- tialaramex 9y agoNot quite "anything of note" it has to be able to do arithmetic. There are very simple systems that are useful but not powerful enough for Gödel's trick and so we think those would be safe.
- davrosthedalek 9y agoGeometry, for example.
- Bromskloss 9y agoHow does one write down those axioms?
- roywiggins 9y agoWell there's Euclid's, but the famous modern one is Hilbert's. https://en.m.wikipedia.org/wiki/Hilbert%27s_axioms?wprov=sfla1 https://en.m.wikipedia.org/wiki/Hilbert%27s_axioms?wprov=sfl...
- yorwba 9y agoIf your axioms of Geometry allow statements about the length of line segments, it's probably possible to implement integer arithmetic.
- roywiggins 9y agoEudidean geometry is decidable. https://math.stackexchange.com/questions/90393/why-euclidean-geometry-cannot-be-proved-incomplete-by-gödels-incompleteness-the https://math.stackexchange.com/questions/90393/why-euclidean...
- lacker 9y agoMaybe I should only believe in the existence of functions that can be implemented by a terminating computer program.