# Playing Games in the Transfinite: An Introduction to “Ordinal Chomp”

Chomp is a two-player game which is played as follows: The two players, A and B, start with a “board” which is a chocolate bar divided into small squares. With Player A starting, they take turns choosing a square and eating it together with all squares above and to the right. The catch is that […]

# Avoiding Set-Theoretic Paradoxes using Symmetry

Intuitively, for any property of sets, there should be a set which has as its members all and only those sets such that holds. But this can’t actually work, due to Russell’s Paradox: Let , and then you can derive a contradiction from both and . The standard solution to this is essentially to forbid […]

# The Undecidability of Identities Involving Sine, Exponentiation, and Absolute Value

In the book A=B, the authors point out that while the identity is provable (by a very simple proof!), it’s not possible to prove the truth or falsity of all such identities. This is because Daniel Richardson proved the following: Let denote the class of expressions generated by The rational numbers, and . The variable […]

# A Geometrically Natural Uncomputable Function

There are many functions from to that cannot be computed by any algorithm or computer program. For example, a famous one is the halting problem, defined by if the th Turing machine halts and if the th Turing machine does not halt. Another one in the same spirit is the busy beaver function. We also […]

# Integrability Conditions (Guest Post!)

Please enjoy the following guest post on differential geometry by Tim Goldberg. A symplectic structure on a manifold is a differential -form satisfying two conditions: is non-degenerate, i.e. for each and tangent vector based at , if for all tangent vectors based at , then is the zero vector; is closed, i.e. the exterior derivative […]