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 Mar19 awarded Popular Question Jan3 awarded Notable Question Dec18 awarded Yearling Oct21 awarded Popular Question Sep19 accepted Why isn't there a first-order theory of well order? Sep19 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? Sep19 asked Why isn't there a first-order theory of well order? Sep6 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. Sep6 revised Showing that a certain recursive set cannot exist? added 11 characters in body Sep6 revised Showing that a certain recursive set cannot exist? added 11 characters in body Sep5 answered Showing that a certain recursive set cannot exist? Sep5 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) Sep5 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 $T \vdash \sim G$. Sep5 asked Showing that a certain recursive set cannot exist? Sep4 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 $n$? Aug19 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"? Aug19 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 $f(x + h) - f(x)$? Aug18 comment Does the Cartesian product of an infinite family have all the elements we expect? Andres, I'm looking for an answer like "yes, that products are nonempty implies that any element satisfying certain conditions appears in the set", or "yes, the size of the resulting set is at least $X$", or "no, you generally can't say anything stronger than the existence of a single element". Aug18 comment Does the Cartesian product of an infinite family have all the elements we expect? Asaf, is it correct to say that the axiom of choice lets you use the fact that "such a set exists" (in addition to the more commonly cited fact that "such a set is non-empty")? Aug18 asked Does the Cartesian product of an infinite family have all the elements we expect?