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Set Theory and Weather Prediction
Here’s a puzzle: You and Bob are going to play a game which has the following steps. Bob thinks of some function (it’s arbitrary: it doesn’t have to be continuous or anything). You pick an . Bob...
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Almost a Number-Theoretic Miracle
An arithmetic statement is one made up of quantifiers “,” “,” the logical connectives “and,” “or,” “not”, function symbols , , constants , , and variables which are bound by the aforementioned...
View ArticleA Curious Application of Ambiguity with Respect to the Possessive Form
Why did the chicken cross the island on Lost? To get to the Others’ side. (Composed by Tim Goldberg.)
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Non-Rigorous Arguments 1: Two Formulas For e
I’m a big fan of non-rigorous arguments, especially in calculus and analysis. I think there should be a book cataloging all the beautiful, morally-true-but-not-actually-true proofs that mathematicians...
View ArticleLots of Fun Math Papers
In the course of looking up a link for my last blog entry, I discovered the MAA Writing Awards site, which collects many pdfs of articles that have won MAA writing awards. From browsing it a bit, it...
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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...
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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...
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The Undecidability of Identities Involving Sine, Exponentiation, and Absolute...
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...
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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...
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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...
View ArticleA language which does term inference
Many strongly typed languages like OCaml do type inference. That is, even though they’re strongly typed, you don’t have to explicitly say what the type of everything is since a lot of the time the...
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What Would the World Look Like if Everything was Computable?: An Introduction...
Suppose that we wanted to construct a mathematical universe where all objects were computable in some sense. How would we do it? Well, we could certainly allow the set into our universe: natural...
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When are the Real Numbers Necessary?
The natural numbers can all be finitely represented, as can the rational numbers. The real numbers, however, cannot be so represented and require some notion of “infinity” to define. This makes it both...
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Two Puzzles in Recursion Theory: Verbose Sets and Terse Sets
Let be the set of all such that the th Turing machine halts. (For these puzzles, we will assume that Turing machines are always run on a blank initial state, i.e., they take no input.) Recall that is...
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Another Puzzle in Recursion Theory: n-Enumerable Sets
We can think of a computably enumerable (or c.e.) set as a bag which some computer program puts more and more numbers into over time. The set then consists of all numbers which are in the bag from some...
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How to Show that Games are Hard
Peg Solitaire is a pretty popular game, often found in restaurants (including Cracker Barrel, if I remember correctly). It’s also NP-complete (by which I mean determining a winning strategy given the...
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Games Which are Impossible to Analyze
In the last post, I mentioned the computational complexity of various games. To be explicit, we consider each “game” to actually be a sequence of games for . For example, would be checkers played on an...
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Doing Calculus on the Rationals (with the help of Nonstandard Analysis)
Nonstandard Analysis is usually used to introduce infinitesimals into the real numbers in an attempt to make arguments in analysis more intuitive. The idea is that you construct a superset which...
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A Simple Introduction to Quantum Groups
In the course of reading some background material for an article by James Worthington on using bialgebraic structures in automata theory, I was led to finally reading up on what a Hopf algebra...
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Trigonometric Series and the Beginnings of Set Theory
Let be a -periodic function. It may or may not have a representation as a trigonometric series A natural question to ask is whether or not the representation of as a trigonometric series is unique, if...
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