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Mar
8
awarded  Popular Question
Dec
18
awarded  Yearling
Nov
9
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Mar
19
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3
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Dec
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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 $f(x + h) - f(x)$?