Why Euclidean geometry cannot be proved incomplete by Gödel's incompleteness theorems?

Question

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.

Answer 1

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.

Answer 2

Gödel's theorems apply to formal systems. Euclidean geometry in itself is not a formal system. So you have to look to particular formalizations of Euclidean geometry. I suppose there may be many of them. Some will be strong enough that Gödel's theorems apply. But apparently others are not.