"fom" <fomJUNK@nyms.net> wrote in message news:C6KdnZCccIjltVzMnZ2dnUVZ_t6dnZ2d@giganews.com... > On 6/18/2013 10:05 PM, Julio Di Egidio wrote: <snipped>
>> 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.
I do not agree, to me the question is rather underspecified. Anyway, I'd propose we do not really bother too much about apoorv's "original intention": I had misinterpreted initially, I now see that he presented a finite triangle, but it's the infinite case that is interesting, the idea that the "triangle" is "equilateral" in the finite is obviously correct.
> 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.
I think there is a sense in which the geometry of construction is essential to the mathematical analysis: I will elaborate on this in a separate post. Here I just note that apoorv has never written "111...111", i.e. no "..." in the mathematical sense, his triangle was in fact finite. Then, I simply assume that we are given a sequence of finite triangles and we ask ourselves what is the (simplest) limit triangle, i.e. the limit of the sequence, and whether it remains true or not that all sides have the same order type (a fortiori, cardinality), i.e. whether the infinite "triangle" remains "equilateral", as our geometric intuition seems to suggest.
>> 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? > > chuckle... > > I do not know if they exist in some recognized mathematical > context.
It would be great if anyone could at list give a glimpse at what would be impacted, if anything, by using the second approach. In fact, despite the two intervals look of a different kind, in practice I am only seeing a rearrangement of symbols involved, i.e. nothing really substantial.
> "Numbers count themselves"
I will elaborate, in fact rewrite the whole thing, in a separate post. (Most probably, this will be later tonight.)
> But, I formulated the construction specifically when considering > what WM had been posting. > > I do not think that is what you have in mind.
Honestly, I concur with WM et al. that there is something *wrong* with the standard view, of logic as of math as of many more things: another story is that I just occasionally see anything useful or even just coherent in what he presents.