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What Happens When You Iterate Gödel’s Theorem?
Let be Peano Arithmetic. Gödel’s Second Incompleteness Theorem says that no consistent theory extending can prove its own consistency. (I’ll write for the statement asserting ‘s consistency; more on...
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An Interesting Puzzle in Propositional Logic
Suppose that you’re translating an ancient text, and in this text you come across three words whose meaning you are unsure of: , , and . So, you head down to the ancient language department of your...
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A Suite of Cool Logic Programs
You may have heard about the Tarski-Seidenberg theorem, which says that the first-order theory of the reals is decidable, that the first-order theory of the complex numbers is similarly decidable, or...
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Functions with Very Low Symmetry and the Continuum Hypothesis
A function from to is called even if for all , . We might call it even about the point if, for all , . Conversely, we can call a function strongly non-even if for all , , . Finding strongly non-even...
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Quantish Physics: A Discrete Model of Quantum Physics
In the book Good and Real, author Gary Drescher, who received his PhD from MIT’s AI lab, defends the view that determinism is a consistent and coherent view of the world. In doing so, he enters many...
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Two Interesting Observations about Voting I Hadn’t Seen Until Recently
By “voting”, I mean the following general problem: Suppose there are candidates and voters. Each voter produces a total ordering of all candidates. A voting procedure is a function which takes as...
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Topology and First-Order Modal Logic
The normal square root function can be considered to be multi-valued. Let’s momentarily accept the heresy of saying that the square root of a negative number is , so that our function will be total....
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The Spectrum From Logic to Probability
Let be the set of propositions considered by some rational logician (call her Sue). Further, suppose that is closed under the propositional connectives , , . Here are two related but different...
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A Logical Interpretation of Some Bits of Topology
Edit: These ideas are also discussed here and here (thanks to Qiaochu Yuan: I found out about those links by him linking back to this post). Although topology is usually motivated as a study of spatial...
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Mathematica and Quantifier Elimination
In 1931, Alfred Tarski proved that the real ordered field allows quantifier elimination: i.e., every first-order formula is equivalent to one with no quantifiers. This is implemented in Mathematica’s...
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Generating Functions as Cardinality of Set Maps
There is a class of all cardinalities , and it has elements , and operations , , and so forth defined on it. Furthermore, there is a map which takes sets to cardinalities such that (and so on)....
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Complexity to Simplicity and Back Again
Generalizing a problem can make the solution simpler or more complicated, and it’s often hard to predict which beforehand. Here’s a mini-example of a puzzle and four generalizations which alternately...
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Why is the derivative of a generating function meaningful?
A generating function is a formal power series where the sequence of coefficients is the object of interest. Usually the point of using them is that operations on the power series (like addition,...
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A Good Definition of Randomness
Most mathy people have a pretty good mental model of what a random process is (for example, generating a sequence of 20 independent bits). I think most mathy people also have the intuition that there’s...
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Two Constants: Khinchin and Chaitin
Take a real number, . Write out its continued fraction: It’s an intriguing fact that if you look at the sequence of geometric means this approaches a single constant, called Khinchin’s constant, which...
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A Complexity-Theoretic Account of The Strong Law of Small Numbers
The Strong Law of Small Numbers (see also Wikipedia) says that “There aren’t enough small numbers to meet the many demands made of them.” It means that when you look at small numbers, it’s easy to see...
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YouTube Physics Explanations Shouldn’t Use the Right-Hand Rule
Popular explanations of physical phenomena like gyroscopes or magnetic fields often end up having to explain the right-hand rule to explain how rotational quantities add (say, by using the right-hand...
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The Arithmetic Hierarchy Meets the Real World
Mathematical logic has a categorization of sentences in terms of increasing complexity called the Arithmetic Hierarchy. This hierarchy defines sets of sentences and for all nonnegative integers . The...
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Gravity is Stronger Than I Thought
I’m not a physicist, and I’d always supposed that, while the Earth has a significant gravitational pull because it’s so massive, the gravitational pull between everyday objects must be completely...
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Thermodynamics is Easier Than I Thought
Actually, thermodynamics is hard and I don’t understand it. But even without totally understanding thermodynamics, it turns out its possible to do a surprising number of useful calculations with just...
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