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It is obvious that the statement that asserts consistency of euclidean geometry (in fact its formalized versions due to Hilbert and Tarski) is a $\Pi_1$ sentence in the language of $PA$ (Peano arithmetic). Is this sentence provable in $PA$ ?

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    $\begingroup$ I think that's what David Hilbert proved. $\endgroup$ – Shubhashish Mar 21 '18 at 18:44
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    $\begingroup$ Hilbert's axioms aren't first-order. However, Tarski's are first-order and have he proved they have a decidable and complete theory. $\endgroup$ – Not Mike Mar 21 '18 at 20:32

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