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 Mar 8 awarded Popular Question Dec 18 awarded Yearling Nov 9 awarded Popular Question Mar 19 awarded Popular Question Jan 3 awarded Notable Question Dec 18 awarded Yearling Oct 21 awarded Popular Question Sep 19 accepted Why isn't there a first-order theory of well order? Sep 19 comment Why is it that $\left(\bigcup_{\alpha \in A} K_{\alpha} \right)^c = \bigcap_{\alpha \in A} K_{\alpha}^c$ Are you familiar with why (a) the complement of an open set is closed, and (b) the intersection of arbitrarily many closed sets is closed? If not, what are the definitions of "open" and "closed" that your teacher gave you? Sep 19 asked Why isn't there a first-order theory of well order? Sep 6 comment Showing that a certain recursive set cannot exist? Fixed LaTeX issues. And thanks for catching (1) on your first comment--I forgot that we can only guarantee $\textbf{Q}$ proves a correct sentence if it is $\exists$-rudimentary. Sep 6 revised Showing that a certain recursive set cannot exist? added 11 characters in body Sep 6 revised Showing that a certain recursive set cannot exist? added 11 characters in body Sep 5 answered Showing that a certain recursive set cannot exist? Sep 5 comment Showing that a certain recursive set cannot exist? $\textbf{g}$ is the Godel number of the sentence $G$, also written $\ulcorner G \urcorner$ (such a $G$ exists by the diagonal lemma) Sep 5 comment Showing that a certain recursive set cannot exist? Thanks Asaf! Does this argument work? Let $\phi(x)$ be the formula that represents $R$, and let $G$ be the sentence such that $T \vdash G \iff \sim\phi(\textbf{g})$. Suppose $G$ is true in the standard interpretation; then since $T$ extends $\textbf{Q}$, $T \vdash G$, so $T \vdash \sim\phi(\textbf{g})$, so $G \not\in R$. But this contradicts that $T \vdash G$. Suppose $G$ is not true in the standard interpretation; then $T \vdash \sim G$, so $T \vdash \phi(\textbf{g})$, and $G \in R$, which contradicts that $T \vdash \sim G$. Sep 5 asked Showing that a certain recursive set cannot exist? Sep 4 comment Prove that transcendental numbers exist: Are there less paniful ways of doing it? This problem also appeared in Rudin's Principles of Mathematical Analysis. (1) Rudin defined algebraic numbers as solutions of integer polynomials (I think this is equivalent to the definition above), and (2) Rudin gave the following hint: how many integer polynomials $c_0 + c_1 x + \ldots + c_d x^d$ are there such that $|c_0| + |c_1| + \ldots + |c_d| = n$, for each positive integer $n$? Aug 19 comment Indirect proof , odd and even numbers Are you familiar with the idea of "contrapositive"? I.e. showing "if A then B" by showing "if not B, then not A"? Aug 19 comment Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there? Are you sure it said $f(x + h)^2 - f(x)^2$, as opposed to $f(x + h) - f(x)$?