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 Mar5 comment What technical books should I read to understand clearly Gödel's theorems and their implications for math? I also suggest the lecture notes "Gödel Without (Too Many) Tears" by the very Peter Smith. They are available from logicmatters.net/igt/godel-without-tears Feb18 comment Reference for Algebraic Categories Look at the book "Algebraic Theories: A Categorical Introduction to General Algebra" by Adamek et al. books.google.es/books?id=siNlAn8Bm30C The book is available from Adamek's web page iti.cs.tu-bs.de/~adamek/algebraic_theories.pdf Jan19 comment When do finite linear orders have the same theory in MSO? Hint: Combine the following two facts to answer your question 1) finite structures have the same first-order theory iff they are isomorphic, 2) the monadic second order language is more expressive than the first-order one. Jan12 comment Fuzzy Logic Question Help Hint: First, think about how solving it with classical logic. And then use the same idea. Oct8 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? Oct8 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? Sep7 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 Sep2 comment Colimits in the category of “sets with partial mappings” Thanks for the clarification. I guess you mean "colimits" (since limits are not preserved). May30 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 ) Apr11 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 $f$? Apr11 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. Apr11 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. Mar28 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/… ) Mar28 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) Mar28 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? Mar27 comment Mathematical intro to Turing machines One example is the following "classic" written by Martin Davis amazon.com/Computability-Unsolvability-Prof-Martin-Davis/dp/… Mar19 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) Mar18 comment Books (and supporting material) that are useful in deconstructing one's intuition? @Sabysachi: torus solution? Mar15 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? Mar13 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.