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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 $f(x + h) - f(x)$?
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
18
asked Does the Cartesian product of an infinite family have all the elements we expect?
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
17
accepted Equivalence between “mathematical induction” and “transfinite induction” for natural numbers?
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.