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Tarski's axioms for geometry are the standard example of a system which is both complete and consistent. It is even decidable - there is an algorithm which you
by eigenket 3y ago
Tarski's axioms for geometry are the standard example of a system which is both complete and consistent. It is even decidable - there is an algorithm which you can feed a statement and it outputs whether it is true or not.
- throwoutway 3y agoOk, how relevant is Tarski's though? Seems academically, they are weak > We partially prove the theorem of [7] that Tarski’s (extremely weak!) plane geometry axioms imply Hilbert’s axioms. https://sciendo.com/article/10.2478/forma-2014-0017 https://sciendo.com/article/10.2478/forma-2014-0017 Also, thats for geometric space but I dont see how that's particularly relevant to this. Elsewhere I find that it doesn't contradict Gödel
- eigenket 3y agoI don't understand your questions. They are relevant for geometry, they aren't really relevant to anything else but they're axioms for geometry so being relevant for geometry is what you'd expect.
- throwoutway 3y agoIIRC, incompleteness theorems applies to arithmetic not geometry, the difference being subtle right?
- eigenket 3y agoThe original incompleteness theorems apply to any logical system which can formulate and prove a certain chunk of arithmetic, yes. It turns out that the amount of arithmetic you actually need is incredibly small, so these results apply to many systems including some formulations of geometry.
- eigenket 3y agoOh, by the way, it matters here that "weak" in the source you quote had a very specific technical meaning. If axioms are weak it means they apply to many things. Stronger axioms make more restrictions and apply to more specific things. So (arguably) axioms being weak is a good thing. Generally you want to work with axioms that are as weak as you can get away with.
- throwoutway 3y agoGood to know thanks