Towards a History of Model Theory

Philosophy of Science Colloquium
The Institute Vienna Circle holds a Philosophy of Science Colloquium with talks by our present fellows.

Date: 22/02/2024

Time: 15h00

Venue: New Institute Building (NIG), Universitätsstraße 7, 1010 Wien, HS 3D

Abstract:

My current research centers on the question of how to adjudicate an appropriate background logic for axiomatizations of mathematics. Currently, first-order logics are the default background logic. They have the advantage of being simple yet expressive, deductively complete, but has two main disadvantages. The first is that it is not categorical, which means that there are infinitely many non-isomorphic models for e.g., the natural number sequence 0, 1, 2, 3, 4, . . . — most including strange entities like “non-standard” numbers capable of being greater than all natural numbers, and in which addition and multiplication aren’t computable! The second disadvantage is that making first-order logics expressive enough to axiomatize the natural numbers renders it incomplete! In many ways, first-order logics simply are not up to the task. Second-order logics are deductively incomplete, but have the advantage of being categorical, which makes them far more suitable to talk with precision about a mathematical structure. As such, I intend to use the history of model theory as of scientific progress and decision-making with respect to our background theories when the advantages and disadvantages aren’t themselves decisive. But first, this history must be generated, and the early history of model theory is what I will focus on for this talk.