Date: Nov 25, 2012 2:43 PM Author: ross.finlayson@gmail.com Subject: Re: Cantor's first proof in DETAILS On Nov 20, 8:08 am, "Ross A. Finlayson" <ross.finlay...@gmail.com>

wrote:

> On Sunday, November 18, 2012 9:15:22 PM UTC-8, Ben Bacarisse wrote:

> > "Ross A. Finlayson" <ross.finlay...@gmail.com> writes:

>

> > > On Sunday, November 18, 2012 4:32:40 PM UTC-8, Ben Bacarisse wrote:

>

> > >> "Ross A. Finlayson" <ross.finlay...@gmail.com> writes:

>

> > >> <snip>

>

> > >> > Basically this set is of an interval, that it is non-empty, and

>

> > >> > non-degenerate (not point width): it contains rationals.

>

> > >> We've both stated our positions and the argument is not advancing. You

>

> > >> wrote a lot after this, but I could not connect it to the document I was

>

> > >> commenting on. It purports to show that the rationals and the

>

> > >> irrationals have the same cardinality. If you posted it for a critique,

>

> > >> I gave it a shot, but if I did not find the error, where is it? And if

>

> > >> you are sure of your reasoning why is it not published?

>

> > <snip>

>

> > > This article has never been submitted to a journal for publishing,

>

> > But you don't say why. You seem convinced by it and it's dynamite.

>

> > > Now, the emphasized point is that for each irrational there are the

>

> > > rationals less than it. Then for those, as each q_h is less than q_i,

>

> > > here as it's defined, for the rationals less than the irrationals less

>

> > > than q_i, their union doesn't include all the irrationals less than

>

> > > q_i, because it only includes the rationals less than q_i. Then for

>

> > > that there's a p_h. Else: exists q_h > p_h or some p_h = p_i,

>

> > > contradiction.

>

> > Your subscripts are getting the way again. For an irrational p you

>

> > define Q(p) as the set of all rationals < p. You then remove from that

>

> > set all the rationals less than all the irrationals less than p:

>

> > L(p) = Q(p) \ Union{r < p} Q(r) for irrational r

>

> > You claim that L(p) = {} leads to a contradiction but I can't see how.

>

> > I simply can't see how your "else" clause follows. Perhaps it's the

>

> > subscripts. Maybe writing it like this will make it clearer when you

>

> > explain it.

>

> > On the other hand, it seems clear to me that having any rational w in

>

> > L(p) does lead to a contradiction. It would follow that w < p (because

>

> > it must come from Q(p)) but also that there was no Q(r) that included

>

> > it. I.e. that no irrational r > w but still less than p exists, and

>

> > that's not the case.

>

> > I think your point is that L(p) is not empty because for every r < p

>

> > (i.e. for every member of the union) there is always a rational w with

>

> > r < w < q, and therefore a member of Q(p) that is not removed. But

>

> > that's not true in the limit, at least so I believe. If this is wrong,

>

> > then it means, surely, that the union is undefined.

>

> > We are both convinced (though I am less than certain due the uncountable

>

> > nature of the union) which I why I don't think there can be progress

>

> > unless someone else chips in.

>

> > <snip>

>

> > --

>

> > Ben.

>

> Matter: particle and wave.

>

> The set of all subsets of the universe is itself.

>

> Measure theory uses countable, not uncountable, additivity to establish results.

>

> Infinite sets are infinite.

>

> The universe is infinite or reference frames would be finite.

>

> The closer science looks at the atom the smaller it is, the farther we look to space the bigger it is.

>

> The rationals and irrationals are disjoint, and between any of them, from the complete ordered field, there are more of them. Ditto the irrationals and rationals.

>

> Discovering applications of transfinite cardinals would be important news for many, and much work has gone into that research, without placement into the applied.

>

> These are basically reasonings in as to why to consider the region about the rational, free of rationals, still with irrationals, and vice versa, because they're not the same.

>

> Selecting each greatest element of that intersection and from that, in the limit, each next, derives an ordering of elements dense in the reals in their natural order.

>

> Defined? It exists. And so does the square root of negative one.

>

So, we know from modern particle physics that the particle, is both

particle, and wave. Now, where these are the closest things in

reality to modeling the continuum of space, mathematically, then where

is it discovered that the mathematical infinitesimal reflects and

defines our most fundamental substrate of reality?

One might point to string theory, with its general notions that the

strings are as many orders of magnitudes finer than the atom as the

atom is to our meter, here in allusion to the fluxions of each fluent

as Newton so described the infinitesimals in the reals, but why is it

ignored the obvious parallel to the atom, the post-Democritan atom, as

being both particle and wave?

The real numbers are continuous in their continuum, infinitesimals as

they are are in them and of them. And, there would be the infinity

and infinities, too.

Look farther, it's bigger, look closer, it's finer: our very real and

most modern experiments in physics don't just hint but advise

scientifically that the features of the most fundamental laws, most

easily described as generally applicable mathematically, define truths

of the real natural continuum beyond what our standard and modern

mathematics does. As well and simply logically, we find from our

regular theories as incomplete that there are true facts about the

objects of our theories, not decided by our standard theories, as is

generally attributed to Goedel.

Then, we do have the remarkable results in the infinities (though he

would aver to there being no infinitesimals as a pox on mathematics)

of Cantor then Russell and so formulated for its great applicability

generally in the finite and in infinite ordinals, well-ordered, of

Zermelo and Fraenkel (who advises that ZF isn't the end-all or be-

all). Compelling as they are, the conscientious mathematician also

finding the real natural continuum as compelling, mathematically,

works to discover how they are at once one and the other.

Then, here with a simple demurral to the dually-self-infraconsistent

dialetheic of the natural formulation of an axiomless system of

natural deduction, with conservation and symmetry of truth, the

technical philosopher satisfies the conscientious mathematician's

rigor, in the all, from the none, the all. Results follow in the

conciliation of our standard, and our emergent, discoveries of truth,

here in mathematics, logic, and technically, philosophy.

An entire new realm of application is simply waiting there to be

discovered. An entire new realm of application was simply waiting

there to be discovered.

Good luck with that, right then, warm regards,

Ross Finlayson