I've seen that there was a single-sorted definition of a category. In some ways, it seems more understandable than the original definition. I don't know much about category theory. But I would like to know how each definition is useful.

Can you provide any failed attempts to prove that ZF or ZFC to be inconsistent? References to articles in the literature if there are any will be much appreciated. Thanks!

I need to compute the following integral $$ I_{n,m} := \int_0^1 P_n(x) P_m(x) \; \mathrm{d}x $$ where $P_n$ is the Legendre polynomial. For an even sum $n+m=2l$ it is easy to show that $$ I_{n,m} = \frac{1}{2} \int_{-1}^1 P_n(x) P_m(x) \; \mathrm{d}x = \delta_{n,m} \frac{1}{2n+1} \,. $$ A lengt...

The homotopy groups of the spheres $S^n$ (see Wikipedia) vanish for the circle $S^1$ as, naively speaking, there are not higher order holes to be grasped by higher order homotopy groups. This intuitions already breaks down for the two sphere $S^2$, e.g. $\pi_3(S^2)$ is non-trivial because of th...