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.
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?