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 Apr19 answered Mathematically, why was the Enigma machine so hard to crack? Apr10 comment Are there any irrational numbers that have a difference of a rational number? It's the greek letter xi, a standard letter for dealing with algebraic irrational numbers. Apr8 comment Are there any irrational numbers that have a difference of a rational number? $1/\phi = \phi - 1$ isn't that special. For example, $1/\sqrt{2}=\sqrt{2}/2$ which is similar. This is going to be the case with all roots $\xi$ of polynomials irreducible over $\mathbb{Q}[X]$: in that case $\mathbb{Q}[\xi]=\mathbb{Q}(\xi)$ meaning that $1/\xi$ has a simple expression as a polynomial of $\xi$. Mar16 comment Examples of Diophantine equations with a large finite number of solutions An easy fix would be: the set of numbers smaller than the max runtime of a machine of size $n$. That set is certainly r.e., and so Diophantine by Matiyasevich's theorem. Mar16 comment Examples of Diophantine equations with a large finite number of solutions The basic idea is sound and gives a good intuition. However it might be useful to note that the number of distinct runtimes for machines of size (exactly) $n$ isn't that large (to a CS guy like me at least). In fact it is something like $O(2^n)$, since there are about that many distinct Turing Machines of size $n$ (and surely there cannot be more finite runtimes than there are machines!). Some of the runtimes are huge, but the size of the set itself is not absurd. Jan30 answered Every infinite set has an infinite countable subset? Jan30 comment An easy example of a non-constructive proof without an obvious “fix”? The explicit computation of the digits is harder, but the proof that it is indeed irrational is much easier! Jan29 comment An easy example of a non-constructive proof without an obvious “fix”? I'm afraid that proof is constructive! Building an explicit enumeration of the algebraic numbers isn't terribly hard, and Cantor's diagonalization argument explicitly gives a process to compute each digit of the non-algebraic number. Jan29 comment An easy example of a non-constructive proof without an obvious “fix”? Note that it is rather easy in intuitionistic logic to prove: for each number $n$, there is a number $k$, such that there exists a digit $d$ that occurs more than $n$ times in the first $k$ digits of $\pi$. It's "passing to the limit" that is difficult, i.e. going from $\forall n\exists d$ to $\exists d\forall n$. Jan23 comment Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa? @GFauxPas: You typically want to require $z>0$ iff whenever $x>y$ then $xz>yz$. But this is only possible only if $0<1$. Jan6 comment How to introduce type theory to newcomer Have you looked at existing introductions to type theory? Dec12 comment Why are integers subset of reals? As @HenningMakholm remarks, there is a lot of work on formalizing the mathematical (or logical) principles of such embeddings, usually called coercions. This is of particular interest in computer assisted mathematical formalization, see e.g. Luo or Wenzel. Sep24 awarded Autobiographer Apr30 awarded Commentator Apr30 comment Mathematicians ahead of their time? Jeezus! Did he also invent LISP? Mar17 comment Why can't you pick socks using coin flips? As a more concrete example, the Hahn-Banach theorem is a powerful and useful theorem of Hilbert spaces which is commonly used in QM and operator theory. Feb27 comment Is there any point in a logician studying $\infty$-categories? Constructive mathematics is actually relevant to mathematicians at large! Who would have thunk it? Also; Topoi are constructive in general: the algebra of open sets of a space is Heyting but not Boolean in general. Feb20 comment Smallest next real number after an integer It's not just you! One possible way to do this (quite naturally) is by constructing the "surreal numbers": en.wikipedia.org/wiki/Surreal_number. The clean cut definitions are a bit technical though, so you might want to look at some of the references at the end of the article. Oct17 awarded Critic May30 comment Why do we believe the Church-Turing Thesis? @NickKidman: Light travels always at the same speed relative to the observer regardless of the situation. However, light traveling from a point near the edge of a black hole may be very hard to detect, as it has been red-shifted to possibly extreme amounts.