Date: Jun 21, 2013 9:43 AM
Author: LudovicoVan
Subject: Re: Matheology § 288

"Sam Sung" <no@mail.invalid> wrote in message 
news:kq1k8g$u5d$1@dont-email.me...
> Julio Di Egidio write:
>> "Sam Sung" <no@mail.invalid> wrote in message
>> news:kq1ica$juk$1@dont-email.me...

>>> Julio Di Egidio wrote:
>>>

>>>> << Edward Nelson criticizes the classical conception of natural numbers
>>>> because of the circularity of its definition. In classical mathematics
>>>> the
>>>> natural numbers are defined as 0 and numbers obtained by the iterative
>>>> applications of the successor function to 0. But the concept of natural
>>>> number is already assumed for the iteration. >>
>>>>
>>>> <http://en.wikipedia.org/wiki/Ultrafinitism#Main_ideas>

>>>
>>> So what?

>>
>> You have snipped the context.

>
> You refer to:
>

>>> What does pi count?
>>> Isn't it a number?

>>
>> pi counts pi, of course...

>
> ? Sorry then.


The point in fact was "numbers count themselves, how else?"

>>> Take any (uniquely operated) iteration and count the
>>> steps by whichever method you like - the result is isomorphic
>>> to N in whichever 'theory'.

>>
>> No,

>
> Why not?
>

>> confront strict finitism.
>
> "Classical finitism vs. strict finitism
> In her book Philosophy of Set Theory, Mary Tiles characterized those who
> allow countably infinite objects as classical finitists, and those who do
> not allow countably infinite objects as strict finitists. Historically,
> the
> written history of mathematics was thus classically finitist until Cantor
> invented the hierarchy of transfinite cardinals in the end of the 19th
> century."
>
> They may 'not allow' whatever they want, but cannot 'forbid' appliyng
> endlessly an operation like a successor function to 0.


Indefinitely large yet (ever) finite: there just is no set N in the strict
finitist view...

Julio