Re: Zenkin's paper on Cantor

From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 10/01/04 

Date: Thu, 30 Sep 2004 19:48:31 -0700

Virgil wrote:

> In article <3c6b9c1e.0409291047.28ee7f87@posting.google.com>,
> raf@tiki-lounge.com (Ross A. Finlayson) wrote:
>
> > Use your strong analytical capabilities to either a) refute, or b)
> > validate that EF is a bijection from N to R[0,1]. Show your work, and
> > mark any derivations.
> >
> > Here's one way to consider EF, n/d, start with small values of d. For
> > d = 2, the range includes zero, one half, and one. For d = 3, the
> > range includes 0, 1/3, 2/3, and 1. For d = 4, the range includes 0,
> > 1/4, 1/2, 3/4, and 1.
> >
> > You might claim that for any finite d that each of those elements of
> > the range is rational. The variable d is unbounded. The range
> > includes zero and one for any positive value of d.
> >
> > You might claim that you can generate an antidiagonal, ie, use the
> > "diagonal argument", that is an element of R[0,1] and different in its
> > representation than each element of the range.
>
> You may be able to
> > generate a single one,
>
> For any list (function from the naturals to the reals), I can generate
> as least as many reals not in that list as are in the list to start with.
>
>
> > The single one you generated, were it not already on the list in a
> > different representation, then used to generate another, leads to the
> > inability to generate another, because it goes in its place in the
> > sorted order. There is not an antidiagonal that doesn't exist in the
> > range.
>
> WRONG!
>
> Given any list, one can generate a non-member by the Cantor method.
> Prepend this to the list and apply the Cantor method again to give a new
> non-member different fron the first non-member. Iterations of this
> prepend and repeat generation will contintue to generate new non-members
> to the original list ad infinitum.
>
> And there are infinitely many modifications to the Cantor method which
> will also generate their own non-members.
>
> > So, if you care about mappings of the natural numbers to the real
> > numbers, take a while and examine the mathematical basis of these
> > claims. A bijection between the naturals and reals is not accepted by
> > many.
>
> That last word should not contian an "m".

Forget about the antidiagonal argument, just prepend a zero to each element.

That's facile, Virgil, the list of elements in this case is specifically the
range of the function. I make it clear that the function is monotonically
increasing thus that the elements of the range are in an ascending, sorted
order as are the elements of the domain. That feature is also used in
extending Cantor's first proof about the inability of most functions to
biject between the naturals and reals.

I proofread my posts before I send them.

So anyways, I'm open-minded. I describe the range of the function which is
a sorted list, and you say here's an antidiagonal. As the range is the unit
interval of real numbers, under a different representation it already exists
on the list, but I place it on the list anyways. It goes in its exact
specified lexicographically ordered place, which happens to be the next
location after the one with which it shares value, right off of the range.
I think about a requirement thusly: no duplicates on the list. Where I
claim the number is already on the list, the list is unchanged by addition
of your number to the list, and your process always returns the same value,
approximately one.

My perception _was_ that you could toss out antidiagonals all day and after
one there were no more to generate. Yes, that's my perception still: the
generated antidiagonal already exists on the list and your process always
returns the same value, and never an element not on the list. The list is
sorted and you may only put the list item in its place. You are not allowed
to prepend.

It's a one-to-one mapping of the naturals to the reals, not a one-to-many
mapping. Via composition, all kinds of funny things are possible, although
I might have to look farther than the open<->closed compositions, about
mappings between open and closed sets of different measure. Part of the
deduction of the characteristics of the real numbers as a point set is that
each is dependent on the previous and next. In extension of Cantor's
arguments, the mapping must be that way.

Do not the natural numbers fullfill the requirements of being a compact
set? (In a sense they do.)

Here's an idea on actually a different tack, about the monotone
characteristic of the function. If you accept iota being the least positive
real, then you might consider mapping even values of N to 0+ n/2* iota and
odd values to 0- (n+1)/2*iota. That function is nowhere monotone, and its
image is [-1/2, 1/2], EF2. Alternatively a form mapping to [-1/2, 1/2]
could be piecewise-monotone with one non-monotone point, with the sign bit.
Generalize to the complex plane. So, you don't have a problem with iota
anymore?

It's key for your argument as defending your status quo that you not allow
the argument to hinge upon the powerset mapping result, because that would
demand systematic change of the foundations of mathematics in its defense.

It does.

What's your opinion of the past few months of F.O.M. postings? Do you
surmise progress?

Virgil is actually quite aware of several of my lines of argument and could
readily use several of them.

To my American readers, please vote for regime change here, in America.
Demand that your vote be counted.

Warm regards,

Ross F.

Relevant Pages

  • Re: Zenkins paper on Cantor
    ... > as least as many reals not in that list as are in the list to start with. ... A bijection between the naturals and reals is not accepted by ... My perception _was_ that you could toss out antidiagonals all day and after ... the mapping must be that way. ...
    (sci.math)
  • Re: Calculus XOR Probability
    ... But this mapping denumerates only some of the reals (0,1] and omits ... What set of naturals is TO using as his domain for this function. ... and the number of unit intervals on the real line is Big'un, ...
    (sci.math)
  • Re: Uncountable sets in CZF?
    ... naturals bijectively to the reals. ... relatively modern response to the vagaries of infinitesimals of Newton ... from the naturals to the reals was the definition of the Natural/Unit ... properties similar to EF in monotonically mapping. ...
    (sci.math)
  • Re: induction vs Cantor
    ... > that creates a new mapping for each natural number. ... > between the naturals and the reals. ... It's a pity that you don't understand what a Proof By Contradiction means. ...
    (sci.math)
  • Re: induction vs Cantor
    ... > that creates a new mapping for each natural number. ... > between the naturals and the reals. ... It's a pity that you don't understand what a Proof By Contradiction means. ...
    (sci.logic)