3,228 reputation
1833
bio website jakitoimgeisterhaus.blogspot.…
location Munich, Germany
age 38
visits member for 3 years, 2 months
seen 1 hour ago

My past research interests included differential algebraic equations, nonlinear analysis and relations between symmetries and structural properties.

I recently investigated hierarchical structures, starting from group cohomology, continuing with semi-group theory and ending with lattices and universal algebra.


May
19
comment Difference between bound and free variable
philosophy.stackexchange.com/questions/7827/…
May
19
comment Difference between bound and free variable
math.andrej.com/2012/12/25/…
May
19
comment Difference between bound and free variable
@MauroALLEGRANZA I tried to incorporate your suggestiongs into my anser. I agree that $P(x)$ has no fixed meaning. There are many reasonable ways to assign meaning to it. It makes most sense to strive for functoriality of truth values for these meanings. The drawback of the definition I gave is that it hides the fact that even $\forall x (P(x)\implies Q(x))$ can take truth values in a Boolean algebra. But I hope that it might be relatively easy to understand for somebody with a "bivalued" classical mindset.
May
19
revised Difference between bound and free variable
tried to incorporate comments by Mauro ALLEGRANZE
May
19
answered Difference between bound and free variable
May
17
comment Not enough memory for GMRES
While computing the vector product, you might also consider using your existing Gauss-Seidel as a preconditioner, because the memory consumption of GMRES is proportional to the number of GMRES iterations (once you avoid the "obvious" mistakes).
May
17
comment Not enough memory for GMRES
You can just replace all instances of "A*v" in that code by a callback to "your code", which computes this matrix vector product (and uses a more appropriate representation of "A" internally). Many of the common numerical C++ frameworks (like PETSc or Trilinos) will also have implementations of GMRES. However, if you are interested in this sort of answer, please ask on Computational Science instead. People "closer associated" to these frameworks are active there, and they would probably correct misleading answers (which probably won't happen here, even if you get answers).
May
17
accepted Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
I think I can appreciate why Gödel needs a $\beta$-function to bootstrap everything. But the encoding of the sequence is not the only place where a $\beta$ function might implicitly be involved. The statement "such that $s$ has some arithmetically definable property relative to $n$" might also rely on "Satz VII" from Gödel's 1931 paper, and hence implicitly on a $\beta$-function. So a (non-computable?) axiom scheme expressing Goodstein's theorem could have instances for all $\Phi(n,s)$ which "provably" express Goodstein's theorem in the "intended" way.
May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
I'm not sure about the "usual definition of things" regarding $\beta$-functions and Gödel's $\beta$-function. The wikipedia article refers to Mendelson 1997:186. In Gödel's 1931 article, in Hilfssatz 1 of Satz VII the same function as in the wikipedia article appears, but it is not named $\beta$. I don't have access to Mendelson, but lemma 10.6.11 in Ebbinghaus defines the two conditions that a $\beta$-function satisfies and states that such a function exists indeed. A footnote claims that the notation $\beta$ for such a function and the two conditions are due to Gödel.
May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
I tried to reflect on why I expected an axiom scheme, and my comment just tried to illustrate this with $\beta$-functions (motivated by lemma 10.6.11 in Ebbinghaus and how it presents these). There are more obvious encodings of zero sequences of natural numbers than specific simple $\beta$-functions. For example, a zero sequence $(a_n)_{n>0}$, which is zero for $n>k$, can be encoded as the number $z=p_1^{a_1}\cdot\ldots\cdots p_k^{a_k}$ (where $p_i$ is the $i$-th prime number). Because this encoding is computable, it can also be expressed by a first-order formula in PA.
May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
How an axiom scheme might really help here is by generating instances of $(\forall n)(\exists s)\Phi_\beta(n,s)$ for all (first-order PA) definable $\beta$-functions $\beta(x_1,x_2,x_3)$ (which can be proved in PA to be $\beta$-functions). This axiom scheme might be a stronger statement than an arbitrary individual instance of this scheme (also I don't know whether it is really stronger).
May
17
comment A consistent first-order theory whose impredicative second-order variant is inconsistent
@AsafKaragila Thanks, this really helped. I see now that the main obstacle to the second part of my question is the word "simple". If I drop it, I can just fix a Gödel numbering, and add $\lnot\mathsf{Con(PA)}$ as a single axiom to $\mathsf{PA}$. My best guess for an answer is then probably that either a "simple" independent statement for PA like Goodstein's theorem can be expressed as a single formula (so that I can add the negation of that formula as an axiom), or to see whether Robinson arithmetic $\mathsf{Q}$ allows for a simple formula (provably) independent of $\mathsf{Q}$.
May
17
comment Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
Great, I know $\beta$-functions from lemma 10.6.11 in Ebbinghaus et. al. I will read it again now. An axiom scheme might replace $(\forall n)(\exists s)\Phi(n,s)$ with each specific instance of $(\exists s)\Phi(\mathbf{n},s)$ for $\mathbf{n}=0,S(0),S(S(0)), \ldots$. This helps if $\Phi(n,s)$ cannot be defined in PA, but $\Phi(\mathbf{n},s)$ can.
May
17
asked Can Goodstein's theorem be expressed as an axiom or axiom scheme in PA?
May
17
awarded  Talkative
May
17
comment Non-i.i.d Empirical Risk Minimization
Maybe you can learn something from the "final" resolution of my question How to sample numerically from an arbitrary smooth distribution? However, I have the impression that you are stuck much earlier, and wouldn't actually care too much about the accuracy issued that bothered me there. Still, it may be interesting as an example where the composition method was the most appropriate solution.
May
17
revised Non-i.i.d Empirical Risk Minimization
Mentioning only rejection sampling, but not also the composition method is not good...
May
15
awarded  Nice Question
May
13
reviewed Approve suggested edit on Continuous probability distribution with no first moment but the characteristic function is differentiable