Item — HN Reader

Ok, 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

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

2 comments

I 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.

IIRC, incompleteness theorems applies to arithmetic not geometry, the difference being subtle right?

The 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.

Oh, 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.

Good to know thanks