LudovicoVan
Posts:
4,025
From:
London
Registered:
2/8/08


Re: Can L(<) be the language of the naturals?
Posted:
Sep 7, 2013 9:28 AM


"Albrecht" <albstorz@gmx.de> wrote in message news:079ffa06a43140e6b85c90066726b234@googlegroups.com... > Am Freitag, 6. September 2013 21:35:57 UTC+2 schrieb Julio Di Egidio: >> "Albrecht" <albstorz> wrote in message >> news:de62b240ff1c4ed3afc80de4b3f89f8a@googlegroups.com... >> >> > The natural numbers of normal people starts with an object or entity or >> > sign and increases in succesive adding further objects or entities or >> > signs step by step. >> >> > E.g.: >> >> > I >> > II >> > III >> > IIII >> > IIIII >> > IIIIII >> > ... >> >> What is wrong with this? >> >> 0 = {} = 0 >> 1 = {0} = 1 >> 2 = {0, 1} = 2 >> 3 = {0, 1, 2} = 3 >> ... >> n = {i < n} = n >> ... >> w = {i < w} = w (i.e. aleph_0) >> >> An objection, as I have got it, is that there must be "w+1" lines in that >> sequence (i.e. from the first line with 0 up to and including the line >> with >> w), so w is the "(w+1)th" number... But, IMO, there is no incongruence >> really, rather a relabeling: it is the "(w+1)th" starting from "1". (To >> make it explicit, I have quoted the "preformal" counting.) > > First, this notation obscures the fact, that any numbering sarts with one > (one > object, in this case the {}).
Primitive counting, as supported by anthropology, starts with none/some and develops from there. Zero has just gone missing in mathematics for a bunch of centuries. In fact, counting from zero is absolutely natural, it is the initial value of an accumulator, it is the starting point on a ruler, etc. etc.
> And second, this notation makes so much people believing that there are > more > natural numbers as there are natural numbers. A completely idiotic idea.
Some people may miss the distinction between the preformal counting numbers and the formal structured labelling then called natural/ordinal/cardinal/whatnot numbers. That the number of natural numbers is greater than any natural number is obvious in von Neumann's construction, and I think it remains true even in a nonwellfounded construction, such as:
0 = {0} = 1 1 = {0, 1} = 2 2 = {0, 1, 2} = 3 ... n = {i <= n} = n+1 w = {i <= w} = w+1
Here, the cardinal numbers may be off by one re the ordinal numbers, but the "count of lines" is still the same. IOW, numbers do count themselves, the rest is "labelling tricks".
Julio

