bio | website | science.marshall.edu/mummertc |
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location | Marshall University | |
age | 36 | |
visits | member for | 4 years, 8 months |
seen | Dec 20 at 1:28 | |
stats | profile views | 2,032 |
I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.
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. |
Jul 12 |
awarded | Enlightened |
Jul 12 |
awarded | Nice Answer |
Jul 9 |
revised |
Infinite decreasing sequence by the Turing jump
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Jul 9 |
answered | Infinite decreasing sequence by the Turing jump |
Jun 5 |
comment |
Is the Invariant Subspace Problem arithmetic?
Here "equivalent" means "provably equivalent in a sufficiently weak system", which is more or less what the question is asking. Of course every true statement is uninerestingly equivalent to $0=0$. |
Jun 5 |
comment |
Is the Invariant Subspace Problem arithmetic?
Do we know that the statement in question is true in the setting of computable analysis? If it is not, then that immediately prevents it from being equivalent to any arithmetical formula, because every true arithmetical formula is satisfied by the model of second-order arithmetic with the standard natural numbers and only computable sets. Actually, every true $\Pi^1_1$ formula is true in that model. This is a standard method for showing that particular theorems are not expressible by excessively simple formulas. |
May 18 |
comment |
Necessity of omega-models in second order arithmetic
I am not aware of any reverse math research in which the base theory includes true first-order arithmetic. But there is another point: the induction schemes in reverse mathematics all include set parameters. This is the case even for schemes like B$\Sigma^0_2$. Formulas of true first order arithmetic don't include set parameters. |
May 18 |
answered | Necessity of omega-models in second order arithmetic |
May 18 |
revised |
Are there “non-constructive” sets in second-order arithmetic?
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May 17 |
answered | Are there “non-constructive” sets in second-order arithmetic? |
May 17 |
comment |
Proof complexity of two directions of equivalency?
This sort of question comes up very naturally, but it is often hard to find precisely the notion we want to capture. The length of a formal proof is not a very interesting metric, because we don't normally look at formal proofs anyway (so the length is meaningless for practice) and because the length depends as much on the proof system as on the theorem being proved. Reverse Mathematics can capture the set-existence axioms required for each direction. But trying to capture how "easy" or "natural" each direction is to prove is a challenge that has not been solved. |
May 15 |
comment |
Does formalizing math require search and creativity, or is it near-mechanical?
In his paper about the formalization of the prime number theorem, Avigad reported that once he was up to speed it took about a day to formalize a page of mathematics in Isabelle. I recommend that paper for its discussion of the formalization process. repository.cmu.edu/philosophy/31 |