Wikipedia mentioned the limitation of Gödel's theorems. According to it, Euclidean geometry doesn't satisfy the hypotheses of Gödel's incompleteness theorems, that is, it cannot define natural numbers. Why is that? Isn't the number of points, lines, or polygons natural number? I thought it would be easy to define natural numbers in Euclidean geometry.
Logic – Why Euclidean Geometry Cannot Be Proved Incomplete by Gödel’s Incompleteness Theorems
euclidean-geometryincompletenesslogic
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Hilbert's axioms predate the development (or at least the wide adoption) of fully symbolic logic, so they are expressed in partially informal language -- though Hilbert strove to make them as precise as he could.
They include the Axiom of Archimedes, formulated in language that presupposes that the natural numbers are already known. As such, if we want to formalize the system in a modern sense, we'd need to add in some machinery for talking about integers, and it would be somewhat hard to avoid making that machinery powerful enough that Gödel's theorem would apply to it.
Furthermore, there's also an Axiom of Completeness which attempts to claim directly that the structure we're speaking about is maximal in a certain technical sense. In the modern view of formal theories, this hardly qualifies as an "axiom" at all, because it doesn't assert the truth of any formula interpreted inside the language of the theory itself. In modern terms, it appears to try to say that every existential formula of a certain shape that is true in some model of the other axioms must itself be elevated to axiomatic status.
However, it is not clear that "formulas that are true in some model of the other axioms" are even recursively enumerable -- which means that Hilbert's axioms, interpreted in this way, are not effective. In other words there's no systematic way to check whether a purported proof is valid or not. Therefore Gödel's incompleteness theorem does not apply to Hilbert's axioms. It seems at least plausible that if we interpret them inside set theory in the above sense, they do have $\mathbb R^3$ as their only model up to isomorphism. (That is, whatever the set theory in question considers $\mathbb R^3$ to be).
Tarski's axioms for geometry were created after formal logic was better developed. They form a genuine, effectively axiomatized, first-order theory, with no ad-hoc appeals to integers, sets, or model theory. They manage to be a complete theory because they are not strong enough to express or simulate arithmetic.
The price paid for the completeness in Tarski's case is that the language the axioms are formulated in is not expressive enough to speak of even finite sets of points, or for example general polygons. Every theorem has to be proved separately for triangles, quadrilaterals, pentagons, and so forth.
This restriction is unavoidable for a complete theory, because as soon as we extend the language with a way to speak of finite sets of points and lines in a reasonable way, we can use those sets as proxies for natural numbers (each set representing the number of points in it) and begin to speak about enough arithmetic that Gödel's incompleteness theorem will apply to it.
Later, Tarski gave his own axiomatization of Euclidean geometry that is entirely in first order logic so by Gödel's completeness theorem, it can demonstrate its consistency, it is decidable and complete.
As mentioned in the comments, this is a complete misunderstanding of what the completeness theorem says. Tarski's geometry cannot even speak about its own consistency, much less prove it.
See Gödel's Incompleteness Theorems.
"Preliminary" formulation :
First incompleteness theorem : Any consistent formal system $F$ within which a "certain amount of elementary arithmetic" can be carried out is incomplete; i.e., there are statements of the language of $F$ which can neither be proved nor disproved in $F$.
Precise formulation :
Gödel's First Incompleteness Theorem : Assume $F$ is a formalized system which contains Robinson arithmetic $\mathsf Q$. Then a sentence $G_F$ of the language of $F$ can be mechanically constructed from $F$ such that:
If $F$ is consistent, then $F ⊬ G_F$.
If $F$ is $1$-consistent, then $F ⊬ ¬G_F$.
$\mathsf {ZFC}$ is precisely such a formalized system $F$ (i.e. "a formal system within which a 'certain amount of elementary arithmetic' can be carried out").
Gödel's original proof in his Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems I") does not use $\mathsf {ZFC}$ nor $\mathsf {PA}$.
His proof is relative to a "variant" of Principia Mathematica system; but his proof - in addition to esatblish the result now know as Gödel's Incompleteness Theorems - provides also a method applicable to many formal systems, provided that they satisfy some "initial conditions".
All the systems known as $\mathsf Q, \mathsf {PA}, \mathsf {ZFC}$ satisfy these conditions.
Relevant to your question, you can see :
- Melvin Fitting, Incompleteness in the Land of Sets (2007).
It can be interesting to note that Melvin Fitting's thesis advisor was Raymond Smullyan.
Best Answer
No, the natural numbers cannot be defined in Euclidean geometry.
In what sense is the number of points, lines, or polygons a natural number? In Euclidean geometry there are infinitely many such objects.
Tarski proved that Euclidean geometry (or rather, his axiomatization of Euclidean geometry) is decidable, while Goedel showed that no reasonable theory of natural numbers is decidable. This alone shows that the natural numbers cannot be defined in geometric terms.
It might help to say something more about what definability means in this situation. Euclidean geometry is in some precise sense just the theory of real closed fields, that is, the theory of the real numbers (as an ordered field).
And by theory I mean the set of all first order statements in the language of ordered fields that are true in $\mathbb R$. Now, you might think that the natural numbers can be defined in $\mathbb R$ by saying that the natural numbers are all numbers that can be obtained by iteratively adding 1's. Or you might say that the natural numbers are the smallest subset of $\mathbb R$ that contains $1$ and is closed under addition.
The problem with these two "definitions" is that none of them can be expressed in first order logic, where you can only quantify over elements of the structure (in this case $\mathbb R$) and not over subsets.
I hope this clarifies things a bit.