Finitist Axiomatic Truth
Logic and Set Theory Seminar
7th June 2019, 4:00 pm – 5:30 pm
30-32 Tyndall Park Road, Room G2
In the opposite direction to Fujimoto's works on axiomatic truth formulated over set theory, I will consider axiomatic truth over finitist arithmetic. As opposed to first-order arithmetic, finitist arithmetic is formulated over a language without unbounded quantifiers. Over such a weak setting, the main issue is how to treat the valuation function, that assigns the values to (codes of) closed term, because in the exactly same way as the liar paradox the self-applicable valuation function leads to a contradiction.
I show a possible solution, with which I can formulate finitist analogues of popular truth theories: Tarski typed, Friedman-Sheard, Kripke-Feferman and Kripke-Feferman-Burgess. I analyze the strengths of these theories, by identifying the finitist arithmetic over which these truth theories are conservative.