I have recently become suspicious of the real numbers. For nearly 3 decades I accepted their axiomatic existence as a complete, ordered archimedian field. The Dedekind-cut, and Cauchy sequence, and "infinite decimal" constructions all made sense to me.
And then I started reading about models...
If τ is the co-finite topology on an infinite set X, does there exist an injection from τ to X? i'm having trouble wrapping my mind around this.
on the one hand, for A in τ, we have A = X - S, for some finite set S. so it seems that there is a 1-1 correspondence:
A <--> S, of τ with the...
Well, if it weren't the wee hours, and me being bored as sin, I'd likely not be doing this. But, you only live once, right? Unless the Buddhists are right...um, let's not go there!
So, here it is. I'm looking for a woman. Sort of.
Get your minds out of the gutter, it's not like that! Well...
I was asked recently to prove that if N is normal in G:
G solvable iff G/N and N solvable.
now, I was able to do so, but I wasn't very happy with my proof.
==> the way I approached this was to form the images of the Gk/G(k+1) under the canonical homomorphism G-->G/N to prove G/N...