# Some proofs in first-order logic

I had the fortune to study classical logic from László Csirmaz at the Eötvös Loránd University, Budapest. Although I was not officially enrolled in the course, he was kind enough to mark my weekly homework regardless of my lack of student status. These were originally written in Hungarian, and I translated a few of them into English.

# A non-standard model of Robinson arithmetics

Give a model that fulfills every axiom of the Robinson arithmetics, and which contains contains two elements that are neither greater than or equal to, nor smaller than or equal to one another; or prove that such a model doesn’t exist.

# A two-formula version of the diagonal lemma

Let $\Gamma$ be a theorem which can represent every recursive function. Prove that for every pair of formulae $\Phi(x)$ and $\Psi(x)$ with one free variable, there exist closed formulae $\eta$ and $\theta$ such that $\Gamma \proves \eta \,\leftrightarrow\, \Phi(\Godel{\theta})$ and $\Gamma \proves \theta \,\leftrightarrow\, \Psi(\Godel{\eta})$.

# Final steps of the proof of Gödel’s completeness theorem

When proving Gödel’s completeness theorem during the lectures, I was missing a crucial step from the proof, so I proved it myself.