Date: Feb 2, 2013 3:57 AM
Author: fom
Subject: Re: Matheology § 203
On 2/1/2013 9:48 AM, WM wrote:

> On 1 Feb., 16:35, William Hughes <wpihug...@gmail.com> wrote:

>> Let P(n) be

>> 0.111... is not the nth line

>> of

>>

>> 0.1000...

>> 0.11000...

>> 0.111000...

>> ...

>>

>> Clearly for every natural number n

>> P(n) is true.

>>

>> This means there is no natural

>> number m for which P(m) is true.

>>

>> It is not simply that we cannot find m,

>> we know that m does not exist.

>

> More. We know that P(n) = 0.111... = 1/0 does not exist as an

> actually infinite sequence of 1's.

Hmm....

As I watch you make these arguments, it occurs to me...

What proof do you have that some sequence is not infinitely

long?

How do you prove this assertion?

To even have the discussion presupposes that you ascribe

meaning to the phrase "actually infinite sequence."

This is an old problem. Nevertheless, you are

maintaining your assertions relative to the fact

that one cannot ostensively prove the existence

of an infinite sequence.

But, one cannot prove that a sequence is not

infinite unless one can get to an end. So,

since your position entails a claim concerning

all sequences, the handful of finitary sequences

that you accept do not constitute a proof

of your own claim.

To assume some finite sequence of symbols

represents an exact real number presupposes an

infinite terminal sequence of constant 0's

in so far as that number is understood

only in relation to a system of names (to be

contrasted with understanding through the act

of manipulating strings by a long division).

So, you are arguing that mathematics is

only as exact as the limitations associated

with some given acts of measurement.

Every real world situation is understood

relative to some normative ideal. Jurisprudence

should be just, politics should be virtuous,

etc.

Why is a normative ideal with respect

to which acts of measurement are given

a science so objectionable?