Let $A$ be a recursive set. $A$ is recursively enumerable, so $A$ may be defined by a $\Sigma^0_1$ formula, i.e. by $\exists \overrightarrow{a} \phi (\overrightarrow{a}, n)$, where …

Many years ago, when I was still a high school student, I came up with a certain first-order axiomatization of PA over the signature (+, x, ≤). Out of nostalgia, I've decided t …

I have a unit cube, and operating in the continuum limit (i.e. not on a lattice), I sequentially place spheres of some radius $r$ inside the cube until a filled volume "jamming lim …

I've heard from others about the WO($\kappa$) as a counterpart of AC($\kappa$), but I cannot find a suitable way to express it in ZF since "every set of cardnality $\kappa$ can be …

I encountered this quantity in my calculations and tried to simplify it. Approximate numeric calculations suggested it could be zero (more precisely, it is certainly less than $10^ …

I am reading Beauville's chapter IX on Elliptic surfaces. Let $S$ be a minimal elliptic surface with $\kappa=1$ and $p:S\rightarrow C$ be the elliptic fibration. We know $K^2=0$ …

We say that $G$ is a homocyclic group, if it is direct product of isomorphic cyclic groups. Is there any classification of finite odd-order groups which all their Sylow subgroups a …

Let $\alpha$ be any positive irrational and $\beta$ be any positive real. We have the following results. H. Weyl (1909): The fractional part of the sequence $\alpha n$ is equidist …

The short version of my question goes: What is known to follow from the existence of Siegel zeros? A longer version to give an idea of what I have in mind: The "expectional zeros" …

What is a reference for the subject of "free resolutions for Lie algebras"? Does the term "standard resolutions" means "free resolutions"? What is a "bar resolution"? Is there o …

In Brian Conrad's notes here for the 2007 Arizona winter school, bottom of p18, he says that there is an affinoid rigid-analytic space and a sheaf of abelian groups on it equipped …

Let say I have a system of 3 polynomials, f1(x,y,z), f2(x,y,z), f3(x,y,z). How to find the resultant of these 3 polynomials. What I meant is, is there any special method to do this …

For me second-countability always felt like to be the more important and fundamental concept from general topology than separability. I wonder whether there are any points which ca …

The question is the title. Working in ZF, is it true that: for every nonempty set X, there exists a total order on X ? If it is false, do we have an example of a nonempty set t …

How can we prove that if $\int_U{f}=0$,then the homogeneous Neumann problem $\Delta u=f$on U,and $\frac{\partial u}{\partial n}=0$ on $\partial U$ has a weak solution in $H^1(U)$? …