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Inspired from the fact that the postulates of Euclid describe uniquely the Euclidean space as a riemannian manifold (in the next sense: consider the "multiverse" arisen by consider the the geometries that fulfill the first 4 postulates, then the unique geometries that too fulfill the fifth postulate are the euclidean planes with other assuptions as the possibility or impossibility of the quadrature of the circle.), I question the follows: Suppose we have a theory, then there exist a theory for each object that characterize it (i.e. describes uniquely the object, absolutely, i.e. in the totality of the "multiverse" and not only as in the motivation; in certain level of the "multiverse")? . For example if we have the theory of topological spaces, there is a theory that only talks about the circle, there is a theory that only talks abut the Klein bottle, and so on?. And we can characterize absolutely the euclidean plane?

In the other hand the euclidean space have a rich amount of structures that make it a space, and there is another question (independent of the above question), there is a notion that captures the intuitive idea of being able to "accept" different structures?

If the answer of my first question is yes, Does the collection of these theories have a structure?

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OK, I'll tell you something --- I don't have a clue what you are asking. –  Gerry Myerson Feb 9 '13 at 5:03
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Vagueness is not a problem --- unless, of course, you actually have a question to which you want an answer. In that case, it's better to write in such a way that people other than yourself can understand you. –  Gerry Myerson Feb 9 '13 at 22:38
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closed as not a real question by Andres Caicedo, Henry T. Horton, Alexander Gruber, Micah, Davide Giraudo Feb 11 '13 at 10:03

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1 Answer

The question is somewhat confusing, because Euclid's postulates don't describe the Euclidean plane uniquely.

The familiar Euclidean plane $\mathbb R^2$ (with the usual definitions of lines, circles, and right angles) is one model of Euclid's postulates. But it is not the only one -- a different model would consist of the subset of $\mathbb R^2$ consisting of points with real algebraic coordinates, plus all lines and circles defined by two such points. This model also satisfies all of Euclid's postulates.

However, the two models are not isomorphic. In particular it is true in $\mathbb R^2$ but not in the model of algebraic points that circles can be rectified -- more precisely,

For every circle $C$ there exists a line segment $AB$ such that for every $D$ strictly between $A$ and $B$ and every finite set of points on $C$, this set of points is among the corners of some simple inscribed polygon whose circumference equals $AE$ for some $E$ between $D$ and $B$.

So Euclid's postulates do not determine whether or not circle-squaring is possible in the model.


It is a general fact about first-order theories that if they allow model with an infinite universe at all, then they have non-isomorphic models. (For example, by the upward Löwenheim-Skolem theorem).

Some theories, such as Tarski's axiomatizations of elementary geometry, do manage the property that their models are all elementarily equivalent, that is, every first-order sentence in the language of the theory is either true in all models (and thus provable) or false in all models (and thus disprovable).

This does not apply to the circle-squaring property above, because it speaks about polygons with an indefinite number of segments, which cannot be expressed in first-order language. So it is conceivable that Tarski's geometry has a model where some circles cannot be squared -- though I don't know offhand whether there's something else that would prevent it. It does, however, mean that Tarski's geometry proves that angles can be trisected and cubes can be doubled (i.e., in plane language, line segments can be divided in the ratio $1:\sqrt[3]2$).

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