WLD2022 - Haifa
Paths in Logic

Philosophical Puzzles: Logic to the rescue! by Lavinia Picollo

Logic as the study of valid argument, logical truth, and other logical concepts is in itself a subdiscipline of philosophy. But, as such, Logic is also a tool of which philosophy, mostly the analytic tradition, makes heavy use. As famously promoted by Wittgenstein in his early work, the Tractatus Logic-Philosphicus, one can elucidate metaphysical and other typically philosophical matters by the application of modern logic to analyse the language we deploy in thinking about these matters. In my talk I will illustrate this point by going over a number of famous philosophical puzzles in which logic shines the most.

Sets, truth, and "paradoxes" of infinity by Michał Godziszewski

What is infinity? How is it different from finiteness? Are there different infinities, or, as some of us were taught in school, "infinities you mustn't compare"? And why is it important? Imagine a hotel with an infinite number of (single) rooms, numbered with positive integers 1,2,3, etc. and assume all rooms are occupied. Such a hotel is usually called the Hilbert Hotel (Hilbert was an outstanding mathematician of the turn of the 19th and 20th centuries) for not entirely clear reasons. Can you still check in a new guest who reports to the reception desk - which would be impossible, if there were, say, 73 rooms? It turns out you can! Moreover, if an infinite number of new guests come to the hotel, as long as they can be numbered with natural numbers, we will be able to comfortably accommodate them in our fully occupied hotel. This is possible because the set of all guests (including new arrivals) is equinumerous to the set of all hotel rooms. The concept of equinumerosity, introduced into mathematics by Georg Cantor in the 1870s, allows, inter alia, for comparing infinite sets according to their size (the so-called power, or cardinality). During the meeting, we will learn that not all infinities are created equal, and explain why the Hilbert Hotel cannot accommodate new guests if they are "numbered" with all real numbers. This follows from the fact that the set of real numbers is not equal to the set of natural numbers - the latter is "smaller". We will also try to explain how it then follows that there can be no no set of all sets, and how it is related to important reasons for which some barber services are impossible to be delivered. Time-permitting, we will discuss how this is, in turn, related to the famous Liar Paradox and its infinite counterpart (the so-called Yablo Paradox) and what it tells us about the concept of truth. This will be entirely whiteboard-talk with no slides whatsoever.

Logic, AI, and the Theory of Everything by Stephan Schulz

Logic is the language of mathematics, and mathematics is the language of science. By automatizing logical reasoning, we can support scientific discovery in a similar way in which the automatization of simple computation has supported scientific progress in the past. I will discuss how the integration of logic-based deductive systems with machine learning and and other AI techniques can be used to tackle the problem of unifying quantum theory and general relativity, possibly resulting in a unified field theory, or "Theory of Everything" in the next 20-30 years.

Taming the Monster of Independence by Menachem Magidor
One of the most fundamental issues in the foundation of Mathematics is the phenomena of independence. Gödel’s famous incompleteness theorem claims that every mathematical theory (with effective axiom system) which is strong enough to express some basic facts about the natural numbers, contains a sentence which is not decided by the given theory. Curiously, while the theorem which was published in 1931, had a deep impact on the foundation of Mathematics, it drew little attention by the "Working Mathematician" . The feeling was that the independent statements introduced by Gödel’s theorem are contrived and esoteric and have little to do with the Mathematical practice.
The situation changed when in 1963 Cohen invented the method of forcing in order to show the independence of the Continuum Hypothesis from the accepted axioms of Set Theory. Following Cohen the method of forcing was used to show that many problems that are were considered to be central in different Mathematical fields, can not be decided on the basis of the accepted axioms.
So, it seems that the "Monster on independence" (a term used by Erdös) can not be ignored. How do we cope with the challenge of independence? In the talk we shall discuss some the main approaches of dealing with this challenge. This will raise some very deep and intriguing Philosophical and Mathematical problems.

Sets versus Types by Thorsten Altenkirch
Modern Mathematics is based on set theory but there is an alternative: type theory. What is the difference between types and sets? What happens when we base Mathematics on types instead of sets? I will try to answer these questions in my talk.