Re: Epistemology 201: The Science of Science

From: aeo6 (aeo6_at_cornell.edu)
Date: 02/14/05 

Date: Mon, 14 Feb 2005 15:32:20 -0500

Wolf Kirchmeir said:
> Tony Orlow (aeo6) wrote:
> [...]
> >
> > I don't have a better classification for infinities ready at this point,
> > but I am thinking about it. I think it may have been Albert that talked
> > about seeing infinities as the same size, but with different densities.
> > Perhaps the right measure would be, say, the inverse of the rate at
> > which the set approaches infinity. In any case, I think the key to finer
> > definition of infinities has to do with taking into account the values
> > of the elements in the set.
>
> Interesting notion: but it looks to me that you are talking about
> infinite series here. I'm not sure that a series is the same as a set in
> this context.
>
> > Question: Aren't the reals equivalent to the set of all digitial, say
> > decimal, numbers with unlimitied digits to either side of the decimal
> > point? For instance, couldn't all real numbers between zero and one be
> > represented with a decimal fraction 0.x, where x is an infinite number
> > of decimal digits? If so, can't all reals be represented by a matrix of
> > integers and such decimal fractions, which map to the integers, much
> > like rational numbers are represented by a matrix of integer numerators
> > and denominators? And if that's the case, then isn't it possible to map
> > the reals onto the integers like one does with the rationals? What am I
> > missing here?
>
> Cool, you're just a step away from Cantor's proof that you can't map
> reals 1-1 and onto integers.
>
> H'm should I leave for you to work out, or shall I show you how Cantor
> did it?
>
> Oh wotthhell. Here goes.
>
> Suppose such a matrix were possible, ie, that for each integer you have
> listed a real. Have yopu left reals out of the list? Cantor showed
> you've left out at least one, as follows: Take the first digit after the
> decimal point from the first real in your list, the second digit from
> the second real, the third digit from the thris real, and so on. You
> have just constructed a real that nowhere occurs in your list, since
> every one of its digits differs from another real at that position of
> the digit. But you started with the assumption that you did list every
> real. That assumption is contradicted. In fact you can construct as many
> reals as you like, add them to the list, and construct even more reals
> using the same method. Hence the reals are not countable (enumerable.)
>
> I tell you, when I understood what Cantor had done, the hair on the back
> of my neck stood up. Just like it did when I first hear the fourth
> movement of Beethoven's Ninth Symphony. Whooohah!
>
I'll have to take a hard look at that, but off the top of my head,
couldn't one do some similar thing with the digits of an integer? I
don't see how, if your set of reals includes all digits with all values
for each digit to the right of the deimal, how you could construct
another that doesn't already exist in the set. What was his initial
mapping? Maybe that was the problem.

It seems to me that the sets of combinations of digits to the right, and
to the left, of the decimal, are no bigger sets than the sets of
combinations of digits that make up integers, and as such, are no bigger
as a matrix than the two sets of integers that are used to construct
rationals. it seems to me one could construct an equivalent matrix and
traverse it diagonally as is done with the rationals.

I'll take a look again - it's been years. 

-- 
Smiles,
Tony

Relevant Pages

  • Re: Galileos Paradox and the Project of the Reals
    ... The way you get reals by a neverending process of generating digits ... and which thereby constructs a sequence of nested intervals whose ... It isn't *already* in the rationals you started with. ...
    (sci.math)
  • Re: Cantor
    ... |I think I'm getting sidetracked by giving poor examples. ... |reals prior to initiating the diagonalization method as the list is ... |differs in the first position from 0, ... what all of the other digits are doing, because of the nature of real ...
    (sci.math)
  • Re: Distinct linear orderings on Z
    ... rationals and reals seem more similar. ... >> Can't those digits be translated into a rational number? ... Remember, we're dealing with infinite sets, so you can't say we can never use ...
    (sci.math)
  • Re: Cantor and the binary tree
    ... >>>List all the infinite binary sequences with a bijection to the integers. ... to be referring to the "proof" of uncountability of the reals. ... By traversing the list diagonally ... the sense that there are as many digits in each number as numbers in the list. ...
    (sci.math)
  • Re: Math errors in python
    ... > They don't even share three digits beyond the decimal point. ... Uh, "constructive reals", such as those you can find at ... constructively interesting" subset that is _countably_ infinite. ... coefficients fit comfortably in memory (with space left over for some ...
    (comp.lang.python)
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