# Carl Mummert

Marshall University

science.marshall.edu/mummertc

Age: 37

I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.
 1d comment Show that $\Sigma_n^0 \not= \Pi_n^0$ holds for every $n \in \mathbb{N}$ (arithmetical hierarchy)You should search for "Post's theorem" which is the standard name for the theorem that establishes the arithmetical hierarchy and relates it with computability theory. The result is in many textbooks. 2d awarded Good Answer Feb 10 revised Binary to Gray code using XOR boolean expressionsedited tags Feb 9 revised Existence of two disjoint closed sets with zero infimal distanceadded 3 characters in body Feb 9 comment Requests for Reopen & Undeletion Votes, etc. (volume 01/2015 - ) [current version]I think the reason we need to require context before questions are answered is because, after someone gets an answer, there is much less incentive to edit the question. In this case, the OP has been inactive since 2013, so asking for them to edit the question seems futile. We can and should look for more quality in questions than "here is a seemingly arbitrary list of requirements, can you cook up a function that meets them". Feb 9 comment How to find a differentiable function with bounded derivative satisfying some boundary conditions?!blindman: why are you trying to find an example of that form? That information would substantially improve the question by giving it some motivation. Feb 8 comment Is there a difference between induction in Peano Arithmetic and Presburger ArithmeticPerhaps one could say the same thing in a different way: the only difference between induction in PA and induction in Presberger arithmetic is that induction in PA applies to all formulas with addition and/or multiplication, while in Presberger arithmetic induction only applies to formulas that do not have multiplication, because all formulas in Presberger arithmetic are required to be in terms of addition only. Feb 8 revised Logic problem: "John's safe's passcode'" question from earlier, with more detailedited tags Feb 7 revised Proving the existence of a proof without actually giving a proofadded 528 characters in body Feb 6 awarded Nice Answer Feb 5 answered Proving the existence of a proof without actually giving a proof Feb 4 comment Law of Excluded Middle ControversyTo be fair, I should mention that the axiom of choice on its own does not imply nonconstructive principles - for example Martin-Lof type theory and higher-order Heyting Arithmetic both include the axiom of choice, and it is accepted in Bishop's constructive analysis. The combination of the axiom of choice with certain extensionality principles does imply LEM, as Diaconescu's theorem shows. But this could equally well be blamed on extensionality - constructive mathematics is often more attuned to setoids than sets. Feb 3 comment Deducing the compactness theorem from the completeness theorem (in first order logic)It will be provable from the formal definition of a derivation/formal proof. For example, one book might say that a derivation is a finite sequence of formulas such that each formula is an axiom, an assumption, or derivable form previous formulas by modus ponens. Another book might define a derivation to be a finite tree of a particular kind. There are logics other than first-order logic where a proof may involve infinitely many formulas, but in those logics compactness does not generally hold. @see Feb 2 comment Full classes in Kelley's bookConsider the example where $C = \{B\}$, $B = \{A\}$, $A = \emptyset$, and $X = \{A, B, C\}$. Then $X$ is transitive, because each element of $X$ is also a subset of $X$. But $A \not \in C$, so $X$ does not satisfy the second condition above. So it does seem that neither of the conditions above implies the other. Feb 1 awarded Autobiographer Feb 1 revised Can a diagonal be longer than the list being diagonalized?edited body Feb 1 comment Definite Integral of a infinitesimalI mentioned "some" sources, and in particular I was thinking of the way that Keisler's text handles derivatives. If I recall correctly, the only method he gives for computing them is only valid for standard inputs. I have been told that one viewpoint some have about nonstandard analysis is that within calculus we are primarily interested in "real" objects, and (accordingly) the values of the derivative or integral at nonstandard inputs are of secondary interest. That viewpoint could make it hard to see why the original question here is interesting from a purely geometrical point of view. Feb 1 revised Can a diagonal be longer than the list being diagonalized?added 898 characters in body Feb 1 comment Definite Integral of a infinitesimalAs I wrote. that is one definition of an integral. The integral is also "by definition" the limit of Riemann sums, and this particular integral should also be the area of a particular rectangle. There is nothing confusing about the fact that some presentations of nonstandard analysis use alternative definitions that no longer match the geometry of the nonstandard field that is being used. Feb 1 answered Can a diagonal be longer than the list being diagonalized?