Re:Spinoza’s God: Einstein believed in it, but what was it?
by nanis on Friday October 15, 2021 @06:58AM
(#28876436)
> natural numbers are infinite but do not contain negative integers
While Z is not a subset of N, N maps 1-1, onto to Z (i.e., they have the same cardinality). That means every element of one set has a corresponding element in the other and vice versa.
Similarly, ismorphisms, homeomorphisms, diffeomorphisms are formalizations of the idea of "different but equivalent in the relevant structures".
I do not have a deeper point to make, but I think your intro argument is not as solid as you might imagine. Plus, things get weird when you go beyond discrete infinities and into the continuum and beyond.
While Z is not a subset of N, N maps 1-1, onto to Z (i.e., they have the same cardinality). That means every element of one set has a corresponding element in the other and vice versa.
Similarly, ismorphisms, homeomorphisms, diffeomorphisms are formalizations of the idea of "different but equivalent in the relevant structures".
I do not have a deeper point to make, but I think your intro argument is not as solid as you might imagine. Plus, things get weird when you go beyond discrete infinities and into the continuum and beyond.