Re: Galileo's Paradox and the Project of the Reals

Six Letters wrote:
On Sat, 30 Dec 2006 17:23:51 -0800, rem642b@xxxxxxxxx (Robert Maas, see
http://tinyurl.com/uh3t) wrote:

I'm surprised you take so many pains against a troll. I can only
assume it was not for my benefit. But I should thank you, howsoever
fortuitous my profit.
It was a wonderful post, full of interesting things and joyous
scorn. It had a slightly deranged quality which suggested you had been at
the bottle, but that's OK. It was that time of year, and parties can be
such a drag,

So true! Schmolidays!

Robert is a very standard thinker, from what I have been able to tell. He seeks to clear up confusion about the facts, leading to me to believe he's done a lot of teaching. I've also found that he misses a lot of the nonstandard point, because of that, and can be somewhat insulting, but I think he means well. Or maybe he just wants to feel smart by making others feel dumb. I hope not...

I particularly liked that bit where you made the even numbers
appear to be twice as frequent as the odd numbers. If you could do the same
thing for the squares and non-squares, only this time make the non-squares
disappear altogether, then I would be ready to believe that |N| = |S|. And
if you could try just a little bit harder, I'm sure you could make a rabbit
appear out of a hat. It reminded me of the Enigma-machine-like crankings
one has to do to demonstrate a bijection between two infinite sets, for
example the knit-one perl-one method of showing there are as many naturals
as rationals.
And you may be right, despite the fact that your post spectacularly
misses the point and is nine tenths irrelevant.


Some bijections are very straightforward, and can lead us to simple rules, which perhaps can be compounded into a system. I've gotten a git devoted to that cause over the last couple years, finally.

So let us assume this Peano ordering. For my purposes, I am not
concerned whether arithmetic has been axiomatised by Peano, Poirot, or
Zorro. Since such a project will have to try and reflect or implement the
ordinary, familiar objects the natural numbers. To be quite careful, I am
not sure if assuming the Peano ordering commits me to some nonsense about
infinity. Let us talk instead of a Beano ordering. The Beano ordering is
exactly like the Peano ordering except it does not commit to views about
infinity in which I do not believe and the overturning of which was the
whole point of my post.

Good one! I read that as "Bay-anno" at first. Very subtle. :)

Peano defined his axioms. One can reject one or another, and apply a name to what they create. One the one hand his set defines order in terms of successor, but that boils down to inequality, since:

Succ(x)={y| y>x ^ (~E z| z>x ^ y>z) }

Then ">" is used to define the spaces between "successors" to create rationals and reals. But ">" defines successor to begin with, so when we talk of reals, let's include rationals and reals, and include naturals in the rationals, and include polynomials and transcendentals in the irrationals, and define it all in terms of inequality, along the real line, to the "right" or to the "left". Set membership follows from logical or natural inequality and equality. I advocate the quantitative approach. :)

So let us assume the Beano ordering on the naturals, with all its
wonderful 'underlying' structure. By the way, is any ordering required to
demonstrate a bijection between, say, the squares and the naturals? I think
so. Unless one takes an infinitley big bag of random naturals and an
infinitely big bag of random squares, and counts them out to infinity to
see if there are any naturals left. But of course bijection is very
"simple", "pre-systematic". Extending bijection to infinite sets is
"nothing especially new". "We simply include not just the integers....but
also the entire set of *all* the integers (whose size we call Aleph-null)".
No worries there then. Unfortunately there is an equally simple, pre-systematic and
persuasive intuition that there are more naturals than squares. Are we
'exploiting' the Beano substructure? No, we're just taking it for granted
in the same way we do with bijection intuitions. Actually we are using a
very general intuition which applies to sets, collections, series, you name
it. If you take something away from a collection, set etc., then you reduce
it. I would have liked to have referred to the intuiton that |N| > |S| as
the argument from composition (the set of naturals is composed of the
squares and non-squares) but then you would get some idiot complaining that
you have confused the whole/part relation with the set/element relation.

I agree. A part can be viewed as an element, among other parts. A singleton is nothing more than an element, when it comes right down to it. A member of a member is a member in some sense.

Taking something away leaves less then there was before, and that's very basic. It has to do with measure. Measure is the part that set theory tries to pretend is irrelevant, when it's the whole point. Truth of a statement itself is a measure in [0,1].

Just because there is all this structure associated with the
natural numbers does not mean that it is all being employed in the Galieleo
Paradox. You should also be aware of the non-remarkable fact that the two
arms of a paradox will employ different intuitions which converge upon the
same object or issue. (It would indeed be difficult to see how a paradox
could arise which exploited the same intuitions about a given issue.) There is a natural way to resolve this paradox which you haven't
yet really considered. When you are ready to consider it, or at least
address the issue properly, then I am ready to listen.

I have recently come down to the conclusion that infinite "sets" don't exist. Only infinite "sequences", "structures" and "processes", all of which rely on a second primitive operation, ">", as opposed to "e". So, set theory is extremely limited in what it has to say about infinity. In fact, it says essentially nothing.


There are a few comments in the text.


From: Six Letters
<snippy doodles>

Have a nice day.

Tony
.

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