In May of 2024, I made a poster for the Ohio Student EXPO about Gödel’s Completeness Theorem, which won 1st place in the mathematics category. While I would simply copy and paste the poster here, it had an interesting feature that allowed me to switch out sub-posters with different levels of explanation for different audiences. For this reason, the poster is not ideal for viewing online. This is why I have included all the content of the poster on this webpage. You can choose the level of explanation you find satisfactory by navigating the bar below.
Abstract
What is First-Order Logic?
Who was Kurt Gödel?
Gödel’s completeness theorem establishes an important relationship between logical conclusions and logical deductions. It states that if a formula holds true within a mathematical system, that formula can be logically deduced from the axioms and rules of the system. An equivalent statement to the theorem is the following: any set of formulas that don’t contradict each other must have a consistent interpretation.
First-order logic consists of statements containing variables, where assigning different truth values to these variables determines whether the statement is true or false. For instance, the statement ‘If P then Q” is true only if both P and Q are true. If P is true but Q is false, the statement is false. First-order logic forms the basis of much of mathematics. It allows us to express general statements and then interpret them within specific mathematical frameworks.
Gödel was born in 1906 in Austria-Hungary and at age 18 began studying at the University of Vienna. In 1929 he proved his completeness theorem in his doctoral dissertation. He later went on to prove his more famous incompleteness theorems, cementing his place as one of the most influential logicians in history Fleeing from Nazi Germany, Gödel ended up at Princeton where he remained the rest of his life. Gödel passed away at Princeton Hospital in 1978 aged 71.
