18.03 Serdecznie zapraszamy na wystąpienie prof. Daniela Isaacsona
Serdecznie zapraszamy na wystąpienie prof. Daniela Isaacsona (Uniwersytet Oksfordzki) zatytułowane „On the notion of implicit commitment and the nature of Peano Arithmetic”. Referat będzie wygłoszony w języku angielskim w ramach poniedziałkowego seminarium logicznego na Wydziale Filozofii.
Data: 18 marzec br., godzina 16:45
Miejsce: sala nr 109, budynek Wydziału Filozofii UW
Abstrakt
In “Arithmetical truth and hidden higher-order concepts” I claimed that first-order Peano Arithmetic is complete and sound with respect to a notion of arithmetical truth based on its relation to the categorical characterization of the structure of the natural numbers established by Richard Dedekind using full second-order logic in “Was sind und was sollen die Zahlen?”. Gödel’s incompleteness theorems show that first-order Peano Arithmetic is incomplete with respect to truths expressed in its language, but I consider that truths in the language of arithmetic not provable in PA depend for their truth on higher-order concepts beyond those higher-order concepts needed to give a categorical characterization of the structure of the natural numbers. In this sense, Peano Arithmetic delimits an intrinsically stable proper subset of the truths expressible in the first-order language of arithmetic. This view has sometimes been called Isaacson’s thesis.
Starting with work by Turing, Kreisel, and Feferman, there has been intensive study of sequences of extensions of PA in which true sentences in the language of arithmetic not provable in PA become provable. These added axioms are related, one way or another, to the soundness or consistency of the systems to which they are added. A natural idea in considering such extensions is that if we make use of an axiomatic theory, we must consider it to be consistent, since otherwise proofs in it tell us nothing, and it’s very natural we should consider that the system is sound. This implies that in using an axiom system, we are committed, on pain of irrationality, to its consistency and probably also its soundness, and so to its extension by those axioms. This view has been labelled the Implicit Commitment Thesis. This thesis and the one above seem to pull in opposite directions. I want in this talk to consider if, and if so in what way, they are in tension with each other.