On 6/18/2013 10:05 PM, Julio Di Egidio wrote: > "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. >
Correct. That is why it is the omega-th entry.
The least ordinal in which omega appears as an object is omega+1. When you use the name 'omega' you are already in omega+1 or beyond.
>> 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 did that. He said nothing concerning the partitioning of the third side. So, the question is not well-construed. The fact that a set can be well-ordered does not mean that some particular construction is definitive enough to specify a particular well-order. All that can be said is that the initial segment and the terminal segment appear to be organized according to the succession of homogeneous symbols. Further, the construction suggests a countably infinite cardinality. So, absent any partitioning criteria, one has alpha+n where alpha is any countable limit ordinal and n is a finite ordinal.
1) countably infinite 2) initial segment given as a succession of homogeneous symbols 3) terminal segment given as a succession of homogeneous symbols
The geometry of construction has nothing to do with
1 1 1 1 ... 1 1 1 1
if that sequence is put in correspondence with an ordinal.
> 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? >
I do not know if they exist in some recognized mathematical context.
They do correspond with one interpretation of WM's theory of monotone inclusive marks.
Suppose a number is a choice function on its initial segment:
lambda(<1, ..., lambda>)=lambda
This is how I make sense out of a statement such as that made by Albrecht:
"Numbers count themselves"
But, I formulated the construction specifically when considering what WM had been posting.