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Topic: Can L(<) be the language of the naturals?
Replies: 35   Last Post: Sep 10, 2013 2:12 AM

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 LudovicoVan Posts: 4,165 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
> Am Freitag, 6. September 2013 21:35:57 UTC+2 schrieb Julio Di Egidio:
>> "Albrecht" <albstorz> wrote in message
>>

>> > 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 "pre-formal" 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 pre-formal counting numbers
and the formal structured labelling then called
natural/ordinal/cardinal/what-not 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 non-well-founded
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

Date Subject Author
9/1/13 Jim Burns
9/1/13 Jim Burns
9/1/13 David Hartley
9/1/13 Peter Percival
9/1/13 Virgil
9/1/13 Peter Percival
9/1/13 Virgil
9/2/13 albrecht
9/6/13 albrecht
9/6/13 Robin Chapman
9/6/13 Tucsondrew@me.com
9/6/13 LudovicoVan
9/6/13 Tucsondrew@me.com
9/7/13 albrecht
9/6/13 Michael F. Stemper
9/7/13 albrecht
9/6/13 FredJeffries@gmail.com
9/7/13 albrecht
9/7/13 FredJeffries@gmail.com
9/8/13 albrecht
9/6/13 Robin Chapman
9/6/13 Brian Q. Hutchings
9/7/13 albrecht
9/6/13 LudovicoVan
9/7/13 albrecht
9/7/13 LudovicoVan
9/8/13 albrecht
9/8/13 LudovicoVan
9/8/13 albrecht
9/9/13 LudovicoVan
9/10/13 albrecht
9/1/13 Jim Burns
9/2/13 Shmuel (Seymour J.) Metz
9/2/13 Shmuel (Seymour J.) Metz
9/2/13 Shmuel (Seymour J.) Metz
9/2/13 Peter Percival