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bio website science.marshall.edu/mummertc
location Marshall University
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I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.

May
19
awarded  Nice Answer
May
16
comment Question 7F From S. Willard, *General Topology*
I have voted to "migrate" this question to our partner site mathematics.stackexchange.com. This site is for research-oriented questions, while that site is for all math questions, and this question is not a research-oriented question. To make the question more acceptable to the other site, you should edit it to indicate what you have attempted and where you ran into difficulty.
May
10
comment Is second-order ZFC categorical with regard to its proper class models
In particular, for people who have some favored set theoretic background (such as a "platonisitic $V$" consisting of "all sets"), the argument goes through in their background like any other. So rather than being more restrictive, the viewpoint that second-order logic is only defined relative to a set-theoretic background is more general. The source of categoricity is not only the use of second-order semantics; it is also in the choice of a particular set-theoretic background in which to interpret those semantics. @Andy
May
10
comment Is second-order ZFC categorical with regard to its proper class models
The real challenge is to decide how you want to formalize the question. As Joel David Hamkins' answer explains, if we choose an arbitrary model of $V$ of ZFC as out metatheory to give a definition of full second-order semantics, then that model $V$ will think that $V$ is itself the only class model of full second-order ZFC. Does this really show what you wanted to show?
Apr
25
comment A question on complexity notation
Because every $\Pi^1_n$ formula is $\Delta^2_0$, the comprehension schemes are ordinarily nested. I suppose you could view the latter as "improper" instances of the former scheme. But, to try to make the comprehension schemes disjoint, how would you separate the "properly" $\Pi^2_1$ instances of comprehension from the "improper" ones (making up words). That seems to involve more than just syntactic analysis, while the ordinary definition of the schemes is syntactic. For example, if $\phi$ is $\Pi^1_n$ and $\psi(X^2)$ is logically valid we might consider $(\exists X^2)[\psi(X) \land \phi]$.
Apr
18
awarded  Yearling
Apr
3
awarded  Nice Answer
Mar
19
comment This modal logic semantics is not S5, but is it something else well-known?
I think this may be related to your question: arxiv.org/abs/1401.0648 . It is not quite the same because we consider models that do not have all possible interpretations, but there seems to be some similarity on a quick reading of your question.
Feb
28
comment Completeness of a set of propositional formulas
I have voted to put this on-hold because it is not about research level mathematics - this question would be a better fit on mathematics.stackexchange.com. In any event: when we have a finite set of variables and a finite set of formulas, it is trivially decidable whether the set is complete, by using truth tables. When the set of formulas may be infinite, it is not decidable whether a given set is complete, by a simple diagonalization argument.
Feb
22
comment Uncomputability of the identity relation on computable real numbers
Thanks. The linked paper is "Computability and analysis: the legacy of Alan Turing" by Avigad and Brattka, arxiv.org/pdf/1206.3431v2.pdf . The citation they give is: H. Gordon Rice. Recursive real numbers. Proceedings of the American Mathematical Society, 5:784–791, 1954.
Jan
21
revised Total formulae in a theory equivalent to $\Delta_0$ formulae in the theory?
edited tags; edited title
Jan
4
comment Axiomatic ZFC Set Theory
Of course, it also matters exactly how replacement is stated. If it is stated in the form sometimes called "collection", it no longer implies the comprehension axioms (this form of replacement only says that the image of the function is a subset of some set). But if we already include the comprehension axioms, then it makes no difference which form of replacement is included. To see why this is particularly relevant: the axiom of replacement stated in Kunen's standard book is the "collection" form which does not imply the comprehension scheme.
Nov
27
awarded  Nice Answer
Oct
11
comment Forcing is intuitionistic
Some modern accounts also incorporate classical logic into the definition of forcing by beginning with a limited set of connectives (e.g. including $\lnot$, $\land$, and $\forall$, but not $\exists$ and not $\lor$), and then assuming the other connectives are given by their classical definitions, which are not intuitionistically correct.
Oct
11
answered Forcing is intuitionistic
Sep
30
awarded  Explainer
Sep
25
comment An interpretation of not-Con(PA)
Of course, some of the steps may also be nonstandard instances of axiom schemes, such as nonstandard instances of the induction scheme or nonstandard tautologies.
Sep
14
comment Deduction theorem
+1. The solution of Negri and Hakli is elegant - they redefine the meaning of $\vdash$ so that what they have is not quite the standard Hilbert system (because they have changed the N rule to a weaker rule), but which does have a deduction theorem. Of course the original Hilbert system, with the full N rule, does not satisfy the deduction theorem, in that it admits $A \vdash \Box A$ as a derived rule but not $\vdash A \to \Box A$. Presumably one could apply a similar method to certain systems for first order logic as well.
Aug
24
awarded  Nice Answer
Aug
24
comment Deduction theorem
@bellpeace: the original system is sound for a particular class of models, namely those in which $A \to B$ holds. Whenever we add new rules of inference, we have to restrict the set of models to those for which the new rules are sound. Of course the system I described, with the additional inference rule, is not complete, and no example that answers the question can be complete in that sense.