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Oct
8
comment Best Mathematical Logic Books the Style of Which is Like a Mathematics Publication rather than a Logic Publication?
@Comeseeconquer: I am afraid I have the same troubles than Carl to follow your question. How do you think it is written "for every $x$, if $x$ is a real number then $x^2 \geq 0$" in a mathematical book?
Oct
8
comment Colimits in the 2-category of partial functions (which is locally posetal)
@Zhen: Why is this true? Given a plain bicomplete category $\mathcal{C}$, there are many ways to look at it as a locally posetal one. Thus, I can imagine it might be the case there is some strange partial order where bicolimits do not exist (while colimits, as a plain category, exist). Is this not possible?
Sep
7
comment Any Computational Number Theory Book, include software programs for key steps of the proofs of major theorem?
Something close (except for what Conifold has said in his comment) can be found in the book "Elementary Number Theory: Primes, Congruences, and Secrets" by William Stein. The book and the software are freely availabe: visit wstein.org/ent and sagemath.org
Sep
2
comment Colimits in the category of “sets with partial mappings”
Thanks for the clarification. I guess you mean "colimits" (since limits are not preserved).
May
30
comment How did Kurt Gödel's Incompleteness Theorem affect the mathematical world?
To start take a look at the papers published in this volume of the Notices AMS ams.org/notices/200604 (in particular I point out the paper by Solomon Feferman). Another related paper worth looking at it is one by Macintyre (take a look at books.google.es/books?id=Tg0WXU5_8EgC&pg=PA3 )
Apr
11
comment The idea of “generators” for arbitrary categories
I am not able to understand what you mean with "$\{ x \circ f \mid x \in X \}$"; can you tell me what is $f$?
Apr
11
comment The idea of “generators” for arbitrary categories
@espen180: Be careful because the upset generated is not the same thing than the filter generated. The notion of filter is isually used in the context of lattices (a particular kind of partial orders), and filters are always closed under finite meets, while upsets might not be closed under finite meets.
Apr
11
comment The idea of “generators” for arbitrary categories
@Mariano: I am afraid that this wikipedia notion does not coincide with the one I have written above for partial orders (although the same word "generator" is used), so it is not adequate for what I really want to generalize.
Mar
28
comment nonisomorphic graph drawings
You can try to use some software for drawing them, this is always helpful (see for instance the "sage" code in the question ask.sagemath.org/question/3473/… )
Mar
28
comment How to explain the power of PA to non-logicians
This essentially means that all $\Sigma_1$-setences that are true in the standard model are provable in PA. This result can be find in most of logic books dealing with PA (and Robinson's Q). In particular, you can take a look at the notes by Peter Smith logicmatters.net/igt/godel-without-tears (in the current version is Theorem 17 in Episode 5, page 40)
Mar
28
comment How to explain the power of PA to non-logicians
Do you consider the fact that "PA (indeed Robinson Q is enough) is $\Sigma_1$-complete" strong enough?
Mar
27
comment Mathematical intro to Turing machines
One example is the following "classic" written by Martin Davis amazon.com/Computability-Unsolvability-Prof-Martin-Davis/dp/…
Mar
19
comment Show that a recursively inseparable pair of recursively enumerable sets exists
Yes, there is a standard proof (for two sets whose definiton only involves Turing machines). One place where you can find this is in Theorem 3.3.5 of the wonderful notes "Syllabus Computability Theory" written by Sebastiaan A. Terwijn math.ru.nl/~terwijn/teaching.html After you know these two sets are inseparable you can easily prove the inseparability of sets with a "logic" flavour (like in Biderman answer)
Mar
18
comment Books (and supporting material) that are useful in deconstructing one's intuition?
@Sabysachi: torus solution?
Mar
15
comment Book on the first-order modal logic
@user132181: The standard translation only works for propositional modal logic (and it translates these formulas into first-order, non-modal, formulas). Thus, right now I am afraid that your question does not make sense: could you clarify your question?
Mar
13
comment When is a Decidable Set Decidable?
I suspect Russell is confusing two different uses of the term "undecidable". We have "undecidability in a formal theory" which applies to a sentence and refers to its unprovability. And then we also have "undecidability in a computability setting" which applies to a family (set) of finite objects and refers to the fact that this set cannot be computed by a Turing machine. For this second use there are some people which have advocated for using the word "uncomputable" instead of "undecidable", in order to avoid these misunderstandings.
Feb
15
comment Complexity of Recursively Inseparable Sets
@Easterly: What do you mean here with the word "Complexity"?
Jan
10
comment Why are mathematical proofs that rely on computers controversial?
@DumpsterDoofus (and the upvoters): What is the statement you talk about?
Jan
5
comment Determining if a theory in first-order logic is decidable
@Sid: Use the "informal proof" given by André. In the proof it is crucial that you know that your theory is complete. Indeed, for complete theories (but not in general) it is known that "decidable" coincides with "recursively axiomatizable".
Dec
22
comment Good textbook on geometries
And the recent "trilogy" by Borceux. I think up to now only the first two volumes have been published: amazon.com/An-Axiomatic-Approach-Geometry-Geometric/dp/… amazon.com/An-Algebraic-Approach-Geometry-Geometric/dp/…