Extending First Order Logic to Quantify Over Formulas

Open Access
Author:
Miller, Justin Davis
Area of Honors:
Mathematics
Degree:
Bachelor of Science
Document Type:
Thesis
Thesis Supervisors:
  • Jason Michael Rute, Thesis Supervisor
  • Kirsten Eisentraeger, Honors Advisor
Keywords:
  • First Order Logic
  • Mathematical Logic
  • Mathematics
  • Foundations of Mathematics
Abstract:
In standard first order logic there is no limit on the cardinality of a set of axioms. All proofs, however, must be finite. This means that for infinite sets of axioms it isn't possible to use them all in a given proof. Given a set G of first order formulas is there a way to modify our logic so that we can have a set G' in our new logic that has finitely many axioms yet can prove the same statements as G? \\ \\ The natural idea here is that of compactness. If G is ``nice enough," can we find a finite set of similar axioms that proves the same things that G proves, e.g. a ``cover" of G? Here we consider the special case where all but finitely many axioms in G do not fall into one of finitely many structures. The obvious example would be the induction scheme on the natural numbers. In first order logic we would need infinitely many axioms to express the induction scheme. We will construct a new type of logic with the means of collapsing these infinitely many formulas into one formula. We will check if the analogous structures in our new language have the same nice properties of first order logic, namely soundness and completeness. We will then investigate which first order statements we can prove with our new logic.