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• I have added at HomePage in the section Discussion a new sentence with a new link:

If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page Welcome to the nForum (nlabmeta). I re-doscivered it only after my recent related comment here.

• making this a disambiguation page

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• More than half of this list is devoted to listing various proof assistants and formalization projects. Does this topic really warrant such an oversized representation in an article with a generic title “mathematics”?

Also, Categories and Sheaves, Sheaves in Geometry and Logic, Higher Topos Theory are good books, but do they really deserve such a prominent placement on top of the article? I suggest removing them.

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Arthur Sard was a mathematician at Queens College.

He got his PhD in 1936 from Harvard University, advised by Marston Morse, in which he proved the Morse–Sard theorem, named after him and Anthony P. Morse (no relation to Marston Morse).

## Selected writings

On the Morse–Sard theorem:

• Arthur Sard, The measure of the critical values of differentiable maps, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, doi.
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Not to be confused with Marston Morse.

Anthony Perry Morse was a mathematician at the University of California, Berkeley, primarily working in geometric measure theory.

He got his PhD degree in 1937 from Brown University, advised by Clarence Raymond Adams.

## Selected writings

On the Morse–Kelley set theory:

• Anthony P. Morse, A theory of sets, Pure and Applied Mathematics XVIII, Academic Press (1965), xxxi+130 pp. Second Edition, Pure and Applied Mathematics 108, Academic Press (1986), xxxii+179 pp. ISBN: 0-12-507952-4

On the Morse–Sard lemma:

• Anthony P. Morse, The Behavior of a Function on Its Critical Set, Annals of Mathematics 40:1 (1939), 62–70. doi.

## References

A definitive source (by one of the authors of the theory) is

• Anthony P. Morse, A theory of sets, Pure and Applied Mathematics XVIII, Academic Press (1965), xxxi+130 pp. Second Edition, Pure and Applied Mathematics 108, Academic Press (1986), xxxii+179 pp. ISBN: 0-12-507952-4
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• starting discussion page here

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• Checking what we have on free coproduct completion I was redirected to this old entry here. I think it will be easier to have a standalone entry on free coproduct completion, and I will create that now and change the redirects.

But while I was here, I added some missing hyperlinks here and there. Such as to free cocompletion

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• Added a few additional descriptions of $\Box_{\leq 1}$, which is the same as $\Delta_{\leq 1}$.

• adding info to this stub article

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• added to identity type a mentioning of the alternative definition in terms of inductive types (paths).

• created some minimum at Cardy condition.

Back then some kind soul provided these cobordism pictures at Frobenius algebra. Is that somebody still around and might easily provide also the picture for the Cardy condition?

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• Added a concrete example: inserters of categories.

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• I pasted in something Mike wrote on sketches and accessible models to sketch. But now it needs tidying up, and I’m wondering if it might have been better placed at accessible category. Alternatively we start a new page on sketch-theoretic model theory. Ideas?

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• I did a bit of reorganization and added more examples at paraconsistent logic, including some comments about linear logic.

Here I copy an old discussion from that page:

I made a big change here; I would argue that the failure of $\bot \vdash B$ means that ‘$\bot$’ simply doesn't mean $\bot$; but in any case, I've always seen the definition given in terms of negation. In particular, dual-intuitionistic logic has $\bot \vdash B$ (just as intuitionistic logic has $B \vdash \top$) but is still considered paraconsistent.

Finn: Hmm. (I presume you meant the ’)’ to come before ’but is still…’; as it stands that last statement is false.) The definitions I’ve seen correspond to what I wrote, but you make a good point – if we want to think of $LJ^{op}$ as paraconsistent then the definition by means of ex falso quodlibet does seem wrong. If this is the standard definition, then I certainly won’t object – I’ll just avoid relying on philosophers for information about logic in future.

Toby: Yeah, I'm sure about the standard; our links agree with me too. (And thanks for catching my parenthesis.) As for philosophers, they're not always as precise as mathematicians; that may be the problem. Not to mention, there's a tendency not to include $\bot$ as a logical constant but instead to simply define it as $A \wedge \neg{A}$ (after proving that these are all equivalent, but ignoring the possibility of an empty model), which leads to conflating the two versions. (In a paraconsistent logic where $A \wedge \neg{A} \equiv B \wedge \neg{B}$ need not hold, one ought to catch the inapplicability of this definition, but maybe not.)

Finn: I think you’ve hit the nail on the head there: I think ’ex falso quodlibet’ means $\bot\vdash A$ (as you said above, this says that ’$\bot$’ means $\bot$, the least truth-value), but it seems my sources may have meant $A\wedge\neg A\vdash B$, without saying so. This is why I never cared much for what’s called ’philosophical logic’ – even though I’m a philosophy graduate studying logic (albeit in a computer science department), it was always too fuzzy for my tastes.

Thanks for clearing this up. Obviously your version should stand.

Toby: You're welcome. But your fancy Latin reminds me that what we have here is technically really the combination of the law of non-contradiction (any contradiction is false: $A \wedge \neg{A} \equiv \bot$) and ex falso quodlibet (anything follows from falsehood: $\bot \vdash B$), so I changed the name above.

Finn: Right – I was going to mention that (no, really), but forgot. Perhaps we should incorporate some of this discussion into the article proper, to help resolve any imprecision in other sources. I’ll do that, if you’d rather not, but not now – it’s way past bedtime here in GMT-land.

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