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I am reading Geometry: Euclid and Beyond by R. Hartshorne and there is a section discussing the possible axiomatizations for planar geometry. In the following paragraph, Hartshorne mentions Gödel's results (p. 71):

Finally, one can ask whether the axiom system is complete, which means, can every statement that is true in every model of the axiom system be proved as a consequence of the axioms? Again, Godel has shown that any axiomatic system of reasonable richness cannot be complete. For a fuller discussion of these questions, see Chapter 51 of Kline (1972) on the foundations of mathematics.

This paragraph is a bit elusive as the author doesn't mention which one of Gödel's results he is referring to.

My impression is that the first sentence uses the term complete in the sense of Gödel's completeness theorem, so I was expecting him to mention this theorem but the second sentence actually mentions an incompleteness result, which sounds more like Gödel incompleteness theorem.

I am under the impression that the author somehow mixed up these two results and gave a false account of Gödel's work (see also Wikipedia on the relationship between Completeness and Incompleteness theorems and how complete does not have the same meaning in both results).

Question: Did the the author give an erroneous account of Gödel's results? If not, how to interpret the paragraph in relation with Gödel's results?

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  • $\begingroup$ Please, avoid the use of images. Here's why. $\endgroup$
    – jjagmath
    Commented Jun 16 at 7:05
  • $\begingroup$ @jjagmath I've replaced the image by a quote. $\endgroup$
    – Weier
    Commented Jun 16 at 7:12
  • $\begingroup$ @Peter in the incompleteness theorem, the term "complete" isn't defined as in the first sentence of the quote: it should be, for every proposition P, either P or not P can be proved within the axiomatic system. $\endgroup$
    – Weier
    Commented Jun 16 at 7:36
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    $\begingroup$ @MauroALLEGRANZA did you read the first sentence of the quote? That is not the definition of "complete" that is used in GIT. $\endgroup$
    – Weier
    Commented Jun 16 at 9:03
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    $\begingroup$ There are axiomatization of Euclidean geometry that are complete. $\endgroup$ Commented Jun 16 at 12:06

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This has been answered in the comments, so I'm making this post CW; I just want to move this question off the unanswered queue.

Yes, Hartshorne got this wrong.

  • The completeness theorem says that for any first-order theory $T$ and any first-order sentence $\varphi$, $\varphi$ is provable in $T$ iff $\varphi$ is true in every model of $T$. (Technically one half of this is the soundness theorem.)

  • The incompleteness theorem says (for example) that any c.e. first-order theory interpreting $\mathsf{Q}$ is incomplete.

The simplest way to edit Hartshorne's statement is to replace

that is, can every statement that is true in every model of the axiom system be proved as a consequence of the axioms

(which is about completeness of the logical apparatus being used)

with

that is, can every statement be either proved or disproved from the axioms

( which is about completeness of the theory in question). I suspect it's this unfortunate double-meaning of "completeness" which tripped up Hartshorne.

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