"fom" <fomJUNK@nyms.net> wrote in message news:l9CdnVddyNhtl1zMnZ2dnUVZ_jKdnZ2d@giganews.com... > On 6/18/2013 1:38 PM, Julio Di Egidio wrote: >> "Julio Di Egidio" <email@example.com> wrote in message >> news:firstname.lastname@example.org... <snipped>
>>> Consider this: >>> >>> 1-> 1 >>> 2-> 12 >>> 3-> 123 >>> ... >>> n-> 123...n >>> ... >>> ___ >>> w-> 123...n...___w >>> >>> The order type of the w-th entry is again w+1. >>> >>> So, the "triangle" is "equilateral", at every step as in the limit. >>> >>> There just seems to be a sort of dissymmetry, so that the n-th entry >>> has order type n but the w-th entry has order type w+1. > > The omega-th entry is the union of its predecessors > as is typical of its definition as a limit ordinal. > > The successor of the omega-th entry takes omega as an > element and has order type omega+1. > > Your ">>___" is the omega-th entry.
Nope, the "___" indicates that we get to the w-th entry via a limit, it's not itself the w-th entry. The w-th entry is the limit entry, a specific ordinal.
> You should consider the possibility that apoorv's > question is not well-construed. <snip>
I cannot tell for sure what apoorv had in mind, but he mentioned ordinals and I have tried to present what seems to me a pretty reasonable, simple reading and approach. We are given that a diagonal side has order type w+1, we also assume (not stated) that we shall use comparable "machinery" when determining the base side.
And I am unclear if you'd at least agree concede (beside the ruminations that have followed) that the "triangle" I describe is "equilateral", i.e. its side and base have (can have) the same order type.
> This is why your omega-th entry is in a set with order > type omega+1.
As to my ruminations on dissymmetries, you describe the usual construction a la von Neumann, where lambda = [0, lambda), but my questions where about a kind of dissymmetry, then about constructions where lambda = [1, lambda]. Do these exist? Does the question even make sense?
> Anyway, I expect to see the same response you > gave to ZG. Sorry about further regurgitation.