Recent MathOverflow Questions

Fundamental group of a topological group

Math Overflow Recent Questions - Mon, 03/25/2019 - 12:28

It is well known that the fundamental group of a path-connected topological group is abelian. Suppose that $G$ is a connected topological group and let $Ab(G)$ the abelianization of the topological group $G$. Is there a relation between $\pi_{1}G$ and $\pi_{1}(Ab(G))$ ?

Partial sums of primes

Math Overflow Recent Questions - Mon, 03/25/2019 - 09:56

$2+3+5+7+11+13...$ is clearly the sum of the primes.

Now I consider partial sums such:

$2+3+5+7+11=28$ which is divisible by $7$

My question is:

are there infinitely many partial sums such that:

$p_1+p_2+p_3+...+p_{k}+p_{k+1}=m*p_{k}?$ with $m$ some positive integer? With Pari/gp apparently up to 10^10 there are only two examples $7$=$p_k$ and $8263=p_k$. Heuristically do you think that infinitely many such partial sums should exist? Note: 7 and 8263 are both primes belonging to primes on the left side of the triangle formed by listing successively the prime numbers in a triangular grid. See https://oeis.org/A078721 Note in both cases $2+3+5+7=17$ is prime and $2+3+5+...+p_{1036}=3974497$ is prime.

Can we get rid of the primitive symbol $V$ in Ackermann's set theory this way?

Math Overflow Recent Questions - Mon, 03/25/2019 - 00:46

I want to get rid of the primitive $V$ in Ackmerann set theory, without changing the axioms so much.

I have the following try in my mind, but I'm not sure if it works.

So we instead work in the pure language of set theory, keep Extensionality, replace axiom schema of comprehension in Ackermann by axiom schema of separation of Zermelo, also generalize foundation over all sets, then modify Reflection schema to the following:

for $m=3,4,5,\dotsc$; $n=1,2,3,\dotsc$; if $\{\varphi_1,..,\varphi_n\}$ is the set of all special formulas of length not exceeding $m$ characters, in which symbols "$X$, $x$" do not occur, in which all and only symbols "$y$, $x_1$, $x_2$" occur free; then: \begin{align*} \exists X \bigl(&\text{$X$ is supertransitive} \wedge X \neq \ \emptyset \ \wedge \\ &\quad\forall x_1,x_2 \in X [ \forall y (\varphi_1 \to y \in X) \to \exists x \in X \forall y (y \in x \leftrightarrow \varphi_1)] \wedge \\ &\quad\forall x_1,x_2 \in X [ \forall y (\varphi_2 \to y \in X) \to \exists x \in X \forall y (y \in x \leftrightarrow \varphi_2)] \wedge \\ &\quad\vdots \\ &\quad\forall x_1,x_2 \in X [ \forall y (\varphi_n \to y \in X) \to \exists x \in X \forall y (y \in x \leftrightarrow \varphi_n)]\bigr) \end{align*} is an axiom. (See definition of supertransitive.)

where a special formula is a formula restricted in a particular manner as to allow only finite number of formulas per each specific length, like for example requiring that quantification order dictates the symbol given to the quantified variable, like in saying that $\varphi_j$ is a special formula if and only if the $i^{\text{th}}$ quantified variable in $\varphi_j$ must be symbolized as $``x_{2+i}"$.

Question: can the proof of Reinhardt's of Ackermann set theory being equal to ZF, be applied to that situation? for example by proving that for each formula $\varphi$ of length $m$ there is a set $X$ that takes the role of $V$ in the proof. It needs to be noticed that this theory clearly interprets Zermelo set theory, we just need to take a sufficiently long formula length $m$; the whole problem is in proving that each set constructed by Replacement after formula $\varphi$ in ZF, can have its parallel in this theory, as it is the case with Ackermann's set theory.

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