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Preamble: My knowledge on mathematical logic is very limited but I think this question deals with some concepts in logic.

Questions:

  1. It seems the Peano axioms and axioms for real numbers using Dedekind cut uniquely characterize the natural and real numbers. The question is that does Hilbert axioms for Euclidean geometry do the same for the Euclidean geometry? If yes how.

  2. What is the mathematical importance of having such an axiomatic system (an axiomatic system that uniquely characterizes the model)?

  3. Is there a term used in mathematical logic for axiomatic systems that uniquely characterize the model? I need to know the terminology to help me find the right references on the topic.

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    $\begingroup$ For (3) the word you're looking for is categorical. $\endgroup$ – Henning Makholm May 14 '17 at 1:05
  • $\begingroup$ download the article from maa.org/programs/maa-awards/writing-awards/… $\endgroup$ – Will Jagy May 14 '17 at 1:09
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    $\begingroup$ At first I thought you were using the term "field" in the abstract algebra sense, but reading more of your Question leads me to suspect that "model" is a word closer to your intended meaning. $\endgroup$ – hardmath May 14 '17 at 1:27
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Yes, the axioms of Hilbert uniquely characterize the model, the axiom system is said to be categorical as Henning pointed.

The proof can be found for example in Hartshorne's book, Euclid and Beyond, here is what Hartshorne says on page 70:

Citation from Hartshorne, Geometry Euclid and Beyond

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