# Definability and Infinity

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What does it mean for a collection to be finite? On the one hand, we have our preschool notion that a collection is finite when can be counted with natural numbers in a way that terminates. On the other hand, there is a definition due to Dedekind that a set is finite if and only if it cannot be put in one-to-one correspondence with a proper subset. Intuitively these two notions should be equivalent, but can we prove it? I will argue that to avoid a circular argument, one direction requires more care than one would initially think. Further, the other direction is true only by virtue of the Axiom of Choice. To outline the proof of this fact, we will examine formal notions of definiabilty and the set-theoretic technique of forcing.

# Workshop on Applying to Graduate School in Mathematics

# Hot Career Tips for Math Majors

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# Examining the mathematically rich game of Tchoukaillon

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Tchoukaillon is a member of the sowing family of board games that originated in Africa and Asia, of which mancala is the most commonly known. Tchoukaillon is a solitaire sowing game and sowing occurs when a player picks up stones in a particular bin and distributes them in adjacent bins. We will analyze interesting patterns, uncover multiple ways to find a winning board, apply linear algebra, discuss commutative diagrams, connect this game with Erdos, and conclude with the Chinese Remainder Theorem. You should also be slightly intrigued how game is defined!

Pizza and soda will be served!