The other day, Carl mentioned that Euclidean Geometry was decidable. I thought that was impossible because I thought it would have an isomorphic copy of Peano arithmetic which is not decidable. Later he pointed me toward Tarski’s axioms. Here’s a quote from the Wikipedia page “This fact allowed Tarski to prove that Euclidean geometry is decidable: there exists an algorithm which can determine the truth or falsity of any sentence.” I found the whole article to be pretty cool because I had never really dug into geometry as a first order predicate calculus.
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Interesting. I just saw this.
I used Tarski’s result in my thesis to show that the question of whether an abstract simplicial complex has an embedding in R^d is decidable. I had not realized that it extends to all of geometry.
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