I was watching the following lecture on model theory online: https://www.youtube.com/watch?v=xNJHw8E_36g&t=1055s Around 24:24, the speaker makes the claim that Euclidean Geometry is incomplete because it can can model Peano arithmetic (which is incomplete). As I understand, however, E Geometry is consistent, complete, and decidable; so is there a generous way to interpret what the speaker is saying.... or is this possibly why comments were disabled on the video?
A second (closely related) question, when he says that 'Euclidean Geometry can model Peano' does this mean that `given the axioms of EG, one can derive the axioms of PA'? If so, it seems that EG would be incomplete; if not, how can it be said to model PA?
UPDATE:
I can't find the precise moment he says that comment but it is stated clearly on the powerpoint and cites Gödel's Incompleteness Theorem as the reason...
I didn't find Patryshev saying what you said he said, so I won't try to comment on where he's right or wrong; I'll just discuss the relevant results.
"Euclidean geometry", at least if you mean e.g. Tarski's axiomatisation, is complete. (Euclid unknowingly used more assumptions than the five he listed, so more recent axiomatisations have been more carefully formulated.) The theory doesn't describe enough arithmetic to satisfy the antecedent of Gödel's first incompleteness theorem. Equivalently, the theory of $\Bbb R$ as a field can't define which values are natural numbers.