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 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 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 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 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 $f(x + h) - f(x)$? Aug 18 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". Aug 18 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")? Aug 17 comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? There was definitely induction involved in the proof that $\mathbb{N}$ is well-ordered--I'm not sure how to resolve this issue either. However, I'm okay with leaving the issue alone, and I'm happy with your answer--my main goal here was to get help with formulating the equivalence sensibly. Aug 15 comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? Thanks Thomas, those were oversights on my part when I typed the question. Aug 15 comment Equivalence between “mathematical induction” and “transfinite induction” for natural numbers? Thanks, fixed my mistake. Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Yup, this suffices. Let $S$ be the set of natural numbers for which every transitive subset is a natural number. $0 \in S$ since the only possible subset of $0$ is just $0$. Suppose $n \in S$, and let $x$ be a transitive subset of $n^+$. If $n$ is not an element of $x$, then $x \subset n$, and we just use the inductive hypothesis. Otherwise, $n \in x$, and transitivity implies $n \subset x$, so in fact $n^+ \subset x$, and $n^+ = x$. Either way, $x$ is a natural number, and the proof is complete. Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? That said, all this discussion has been immensely helpful anyway! I ended up doing something like this: (1) prove by induction that if X is a non-empty subset of a natural number, then the intersection of X is a natural number; (2) find a set F such that the intersection of F is the same as the intersection of E, but F is a subset of a natural number Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Martin, I'm still not fully convinced, since to me it looks like you're using the assumption that $a \in n$ if and only if $a$ is a proper subset of $n$ before you've proved that $a$ is a natural number. Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Asaf, does math.stackexchange.com have a prediction market for estimates like this (e.g. one where you can wager reputation points)? :) Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Martin, this is indeed how the natural numbers are defined in the text I'm referencing (Halmos - Naive Set Theory). Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? I feel like this should suffice, but I still need to convince myself that "a transitive set of natural numbers which is a subset of some natural number is itself a natural number". Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Thanks Martin! I agree that $a \subset n$ for each $n \in E$, but how did you then conclude that $a \in n$ for each $n \in E$? Aug 13 comment Intersection of a non-empty set of natural numbers (set-theoretic definition) is a natural number? Unfortunately I need to use this to prove that E has a smallest entry in the first place, so I can't just say "take the smallest entry of E".