Carl Mummert

Marshall University

science.marshall.edu/mummertc

Age: 36

I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.
1d
comment A subspace of a metric space is normal
The Hausdorff axiom I know says we can find disjoint neighborhoods of two points - it doesn't say anything about starting with two closed sets or starting with two open sets. A separate issue is that not all closed sets are closed balls; some closed sets have empty interior, even.
1d
comment A subspace of a metric space is normal
How does the Hausdorff axiom allow you to pick disjoint open neighborhoods of arbitrary closed balls (let alone arbitrary closed sets)?
1d
answered Why Mendelson axiom schemas are true?
1d
comment Intuitionistically, are these inequivalent? $P \rightarrow Q,\; \neg Q \rightarrow \neg P,\; P \wedge \neg Q \rightarrow \bot,\; \neg P \vee Q$
Similarly, if $Q$ is already negative, say $Q = \lnot S$, then it is intuitionistically fine to prove $\lnot Q$ by assuming $P \land S$ and deriving a contradiction.
1d
comment Intuitionistically, are these inequivalent? $P \rightarrow Q,\; \neg Q \rightarrow \neg P,\; P \wedge \neg Q \rightarrow \bot,\; \neg P \vee Q$
Of course, the logic known as "intuitionistic logic" in proof theory does include the scheme $\bot \vdash \psi$, which is a different sort of "proof by contradiction". In fact, this is how one would derive $P \to Q$ from $(P \to \bot) \lor Q$, that is, derive (2) from (1). That derivation would not work in minimal logic.
1d
comment Intuitionistically, are these inequivalent? $P \rightarrow Q,\; \neg Q \rightarrow \neg P,\; P \wedge \neg Q \rightarrow \bot,\; \neg P \vee Q$
There is one slightly subtle point: if $P$ is already negative, say $P = \lnot R$, and $Q$ is already negative, say $Q = \lnot S$, then someone working intuitionistically would try to prove the "stronger" contrapositive $S \to R$, rather than the weaker $\lnot \lnot S \to \lnot \lnot R$. The stronger $S \to R$ does imply $\lnot R \to \lnot S$ intuitionistically, that is, it implies $P \to Q$ as desired. The "two contrapositives" are of course equivalent in classical logic, but someone working in intuitionistic logic has to be a little more careful.
1d
comment Should I vote to reopen a closed post that was closed for the wrong reason
The better way to use the review queue seems to be to open questions in a new tab before voting to close or reopen. This has two advantages. First, you can perform all actions in the new tab; sometimes the review queue inexplicably prevents you from commenting and/or voting. Second, if you only click "skip" in the review tab, you are insulated from the silly "audits".
2d
comment False $\Sigma_1$-sentences consistent with PA
I would use something like $ w = 0$,,, but I am not sure why you are looking at $\psi \to \bot$. What is your formula $\phi$, and why does $\mathbb{N}$ satisfy $(\exists ! w)\phi(w)$?
2d
awarded Nice Answer
May
19
reviewed Close How to use Stokes theorem
May
19
reviewed Close trigonometry: prove that (tanA)(tanB)+(tanB)(tanC)+(tanA)(tanC)=1
May
19
reviewed Close Is p a limit point of the range of {$P_n$} when it converges to p
May
19
reviewed Close Evaluate $ \lim_{n\to\infty} \frac{(n!)^2}{(2n)!} $
May
19
reviewed Close If $A\bar A = B\bar B = I$ why are $A$ and $B$ similar over $\mathbb{C}$ if and only if they are similar over $\mathbb{R}$?
May
18
revised How does PA prove all $\Delta_0$-formulas which are true in the standard model?
added 3 characters in body
May
18
answered How does PA prove all $\Delta_0$-formulas which are true in the standard model?
May
17
comment A complex variable function integrated over an infinitesimal disk
The question has been edited to completely change the answer; the original question said that the contour was a circle. It is better to ask a new question, rather than to edit the question in a way that the correct answers to the old question become obsolete.
May
17
revised Why is the numbering of computable functions significant?
edited tags
May
17
answered Why is the numbering of computable functions significant?
May
16
comment Question 7F From S. Willard, *General Topology*
I have voted to "migrate" this question to our partner site mathematics.stackexchange.com. This site is for research-oriented questions, while that site is for all math questions, and this question is not a research-oriented question. To make the question more acceptable to the other site, you should edit it to indicate what you have attempted and where you ran into difficulty.
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