Syllabus Week One: Infinity (Zeno’s Paradox, Galileo’s Paradox, very basic set theory, infinite sets).

Week Two: Truth (Tarski’s theory of truth, recursive definitions, complete induction over sentences, Liar Paradox).

Week Three: Rational Belief (propositions as sets of possible worlds, rational all-or-nothing belief, rational degrees of belief, bets, Lottery Paradox).

Week Four: If-then (indicative vs subjunctive conditionals, conditionals in mathematics, conditional rational degrees of belief, beliefs in conditionals vs conditional beliefs).

Week Five: Confirmation (the underdetermination thesis, the Monty Hall Problem, Bayesian confirmation theory).

Week Six: Decision (decision making under risk, maximizing expected utility, von Neumann Morgenstern axioms and representation theorem, Allais Paradox, Ellsberg Paradox).

Week Seven: Voting (Condorcet Paradox, Arrows Theorem, Condorcet Jury Theorem, Judgment Aggregation).

Week Eight: Quantum Logic and Probability (statistical correlations, the CHSH inequality, Boolean and non-Boolean algebras, violation of distributivity)

Infinity

absolute infinite

Infinite Cardinalities

The Higher Infinite

Omega-Sequence Paradoxes

Decisions, Probabilities and Measures

Time Travel

Newcomb’s Problem

Probability

Non-Measurable Sets

The Banach-Tarski Theorem

“The Banach-Tarski theorem is an example of a result in mathematics that is surprising and counterintuitive, and which has important implications for our understanding of the nature of mathematics itself. From a philosophical perspective, the theorem raises several questions about the relationship between mathematics and the physical world, and about the nature of infinity and infinite sets.

One way to approach the Banach-Tarski theorem from a philosophical perspective is to consider the idea of “constructibility” in mathematics. In traditional geometry, it is generally assumed that all objects that are studied can be constructed using a finite number of steps, using only a straightedge and compass. This idea of constructibility is closely related to the concept of “realizability,” which is the idea that any mathematical object can be represented by some physical object or process in the real world.

The Banach-Tarski theorem shows that this idea of constructibility does not hold for all objects in mathematics. Specifically, it shows that it is possible to decompose a three-dimensional ball into a finite number of pieces, and then reassemble those pieces to form two identical balls, even though the pieces used in the decomposition are highly irregular and have infinite volume. This means that it is possible to have mathematical objects that cannot be represented by any physical object or process, and which cannot be constructed using a finite number of steps with a straightedge and compass.

This raises questions about the nature of mathematics and its relationship to the physical world. It suggests that there may be some mathematical objects that exist “beyond” the realm of physical reality, and which are not subject to the same constraints as physical objects. This has led some philosophers to argue that mathematics is a purely abstract, self-contained discipline, with no necessary connection to the physical world.

At the same time, the Banach-Tarski theorem also highlights the role of infinity in mathematics. The use of infinite sets and infinite processes is essential to the proof of the theorem, and it suggests that infinity is a fundamental concept in mathematics. This has led to debates among philosophers about the nature of infinity and how it should be understood. Is infinity a real, objective concept, or is it simply a useful mathematical construct? These are just a few of the philosophical questions raised by the Banach-Tarski theorem.”

Computability and Gödel’s Theorem

Computability

Gödel’s Theorem