Re: Well Ordering the Reals

Ross A. Finlayson said:
> Virgil wrote:
> >
> > TO has given no satisfactory reason to doubt the standard resolution.
>
> Well, others have put forward that notion.
>
> I think Tony is refreshing, besides reasonable and voluble. He's not
> always right, that's a far cry from others not accepting where he is
> right. To be honest I'm impressed with Tony.
Gee, thanks, Ross! :) Of course I AM always right! Just kidding. I have had to
reconsider plenty of points and refine my thinking for sure. It's not the
easiest subject, but that's what makes it fun, and important, maybe.
>
> Virgil basically attacks Tony, and as above tries to segregate him.
> There are a variety of theories, where standardness is arbitrary, there
> are others who agree with Tony on a variety of these issues. So,
> Virgil commits a fallacy, he tells the reader that according to him
> Tony is wrong, where according to others, Tony is not wrong, that's
> omission of relevant truths. His invocation of the
> community-at-large's "we" is unwarranted, and he certainly doesn't
> speak for me. It's carefully phrased to reflect on Tony, but that's
> dishonest, in the context of the larger discussion.

Oh, Virgil is like the attack terrier, always yapping at me, every time I walk
through the garden. I am used to it. I don't think it ultimately reflects badly
on me. I think Virgil has a very vested interest in the standard model. He does
seem to be extraordinarily personally invested in simply discrediting any
"crackpot" ideas in this area. Maybe it's just for fun, maybe for ego
gratification, or maybe it has political purpose in what has become a bit of an
ongoing debate within mathematics. I suppose it's to be expected, and not given
too much weight. I used to get more upset about dishonest clips on his part,
but I think he quit that tactic. I don't know. I am not paying that much
attention to that any more.

>
> In well-ordering the reals, basically I'm at the notion of the initial
> segment, and yes there is still the notion of the set of nested
> intervals, in terms of any well-ordering of the reals, the set of
> intervals over the well-ordering. Having adjacent points _would_ remove
> that consideration of the spot or point discontinuity.

Yes, I agree that the use of infinitesimals is valid, though maybe not in the
standard model. It also seems like it would necessarily produce an
"uuncountably" infinite set, without the kind of discontinuities we find in the
"ordinals" or "cardinals", and lead to "infinite descending chains". I am of
the opinion now that what is required for any acceptable well ordering of the
reals is some kind of predecessor discontinuity, such that we can lump all the
"countable" reals, with finite numbers of bits, into a countable set, and then
declare some first infinite-bit real, and some countable succession of such
infinite reals. However, the distasteful kludginess and tomfoolery of this
approach notwithstanding, we still will still not have our well ordering, since
we will still have an uncountable infinity after removing the countable
infinity. I suppose the way this is handled in the standard model is to have a
countable set of uncountable values, like the alephs? It's hard to even know
what this theory expects of itself, which I guess is half the point.
>
> About nonstandard models of the natural integers, there is some
> consideration that the one-point compactification or point at infinity
> is implicit. That's a notion that the naturals "contain" themselves,
> just like the universe would "contain" itself. Now, there is no
> universe in ZF, so where there must be a universe, it is not
> unreasonable to presume that there is, and, there are reasonable,
> sound, rational, logical, and rigorous reasons to consider why there
> is.

Ah! I think I see where this goes. Certainly, there should be a universe,
within which everything resides. This makes sense to me. I see a system that
starts with a concept of space, and the expanse, within which are defined
smaller sets of points, some infinite and some finite, which can be measured
according to the properties of the space. Perhaps this is the kind of universe
you are envisioning, an all-encompassing set of all sets. To me, the continuous
real number line IS the union of all quantitative sets, and the substrate withi
which all quantitative sets are embedded, including the unit discrete infinite
set, the naturals, with one point per unit of space. Does that sound at all
like it might satisfy the need for the universal set?
>
> Ross
>
>

--
Smiles,

Tony
http://www.people.cornell.edu/pages/aeo6/WellOrder/
.

Relevant Pages

  • Re: An uncountable countable set
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  • Re: infinity
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  • Re: Well Ordering the Reals
    ... > Tony Orlow wrote: ... >> Virgil may say that I need infinite values to have infinite sets, ... so your argument is a piece of nonsense. ...
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  • Re: Relative Cardinality
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  • Re: Well Ordering the Reals
    ... >> Ross A. Finlayson said: ... >>> You have all this vitriol for Tony and none for me? ... sets dense in the reals who complement ... there is no infinite descending chain. ...
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