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Jun
10
comment Is there any formal definition or reasonably good heuristic for mathematical 'interestingness?'
I suggest you take a look at the area called "automated theory formation" (belonging to AI).
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/…
Oct
28
comment Why does undecidability of arithmetic not follow from that of first-order logic?
Indeed, it is also well known that these two sets (arithmetic truths versus firs-order validities) have different Turing (undecidability) degrees. In particular, there is no computable reduction reducing "arithemtic truths" to "first-order validities". If you want to take a deeper look at this I suggest you start looking at the artihmetical hierarchy (for instance, at wikipedia en.wikipedia.org/wiki/Arithmetical_hierarchy )
Aug
27
comment Does ZFC pin down precisely which theorems PA can and cannot prove?
This comment develops (just a bit) Carl's comment. It is easy to prove from the assumption that "for all sentences $\phi$ in the language of PA, it holds that either $ZFC \vdash (PA \vdash \phi)$ or $ZFC \vdash (PA \not \vdash \phi)$" (together with assuming that $ZFC$ is consistent) that there is an algorithm to compute "provability in PA". The algorithm is the trivial one you expect (use provability in ZFC until you find the proof that shows ...). Using the well-known fact that PA is non-computable one concludes that the answer to your question is NO.
Jul
22
comment Illustrative examples of a phenomenon in the logic of mathematical induction
@Doug: Humans in general do not write the external parenthesis (although in a syntactic way they are there). So please do not be so formal: do you seriously think that mathematicians should never write $3+2$ and start writing $(3+2)$?