Models, Completeness, and Skolem's Paradox
Ahmet Çevik · 2021
Chapter 11 is another section in the book which is particularly of special interest to mathematicians. We study the notion of models, truth in a model, and prove some fundamental theorems of mathematical logic such as the compactness theorem, Gödel&s;s Completeness Theorem , and the Löwenheim-Skolem Theorem . In a nutshell, the completeness of first-order logic states that if a statement is true in every model then it is provable. The compactness theorem says that if every finite subset of a first-order theory is consistent, then the theory itself is also consistent. In the final part we give Skolem&s;s paradox, an interesting antinomy of the Löwenheim-Skolem Theorem, and we talk about its philosophical relevance.