6,051 reputation
12039
bio website science.marshall.edu/mummertc
location Marshall University
age 35
visits member for 4 years, 3 months
seen 3 hours ago
I work in mathematical logic. My main areas of interest are arithmetic, reverse mathematics, computability, and proof theory.

Jul
12
awarded  Enlightened
Jul
12
awarded  Nice Answer
Jul
9
revised Infinite decreasing sequence by the Turing jump
added 826 characters in body
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
May
5
comment What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
@Joel: Yes, exactly. Some other models have a different, unchangeable tape symbol to mark the left edge of the tape.
May
5
comment What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
@Joel: By the way, there is a technical reason to start the machine on the first input symbol, which is to accommodate machines that have a one-sided (i.e. semi-infinite) input tape, which are common in the computational complexity literature. These need to know where the left end of the tape is, because in the usual framework they have no way to find it otherwise.
May
5
revised What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
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May
5
comment What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
@Joel: Silly Turing machines - everyone has their own idea how they work! In my mind they need to start on the first symbol of input. This is also (likely) why the author said that $h(x) = x-1$ takes $O(|x|)$ steps; it would also take only one step if the machine started on the last digit. I'll edit the answer, however.
May
5
comment What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
@Dan Turetsky: as I point out in my answer, the question is somewhat trivially unsolvable in that context.
May
5
answered What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
May
5
comment What new primitive recursive functions are needed to reconcile Turing time complexity with Gödel time complexity?
Why would $f(x) = x-1$ require time $2^{|x|}$ as a primitive recursive function? Evaluating it requires (regardless of $x$) exactly one call to the primitive recursion combinator to determine whether $x = 0$ or $x > 0$, and in either of those cases the result is then returned immediately (i.e. by a projection function). In other words the definition is $f(0) = 0$ and $f(x+1) = x = \pi^2_1(f(x),x)$. The natural primitive recursive definition of $f(x) = x-1$ doesn't even use the successor function.
Apr
29
answered Are all functions in Bishop's constructive mathematics continuous?