Re: Galileo's Paradox

Robert Maas, see http://tinyurl.com/uh3t wrote:
From: Tony Orlow <t...@xxxxxxxxxxxxx>
I'm the one who said I had no problem with uncountably long
strings, and I don't.

"uncountable" and "string" are mutually inconsistent concepts.

In the standard world, that's apparently true, but I have special breathing apparatus.

You need to rephrase whatever you intended to say there.
Also "uncountably long" makes no sense at all.
"uncountable" is a quantity, not a length.

I consider that incorrect. Length is an integral quantity, when it comes to strings, since there are no partial symbols. It makes perfect sense. We can say what the successor and predecessor of ...3333 are, ...33332 and ...33334. But there are an infinite number of intermediate numbers between those and, say, ...2222.


On the other hand, starting from an enumeration (a computable
sequence), there are sub-sequences which are not computable, hence
which can't be enumerated in fact even if theoretically they
*could* be enumerated with the help of a supernatural oracle to
provide red/green answers for membership in that sub-sequence.
But the *length* of such a sub-sequence is the same as the length
of the full sequence. It's just that the density is less than the
full sequence, but that density is not computable.


The "density"? I think you're off-track. I'm talking about simultaneously addressing infinite, finite, and infinitesimal values combined, wth accuracy on all levels. But, I guess, maybe, we'll get to that...

or you can use the H-riffics and have two successors to each
point on the real line, such that the reals form and [sic] infinite tree.

I.e. that system doesn't have total ordering. IMO that sucks!
In particular, such a system can't be used to model classical
geometry.


It doesn't exactly represent a complete linear order. One can use infinitesimal units for that. The H-riffics, though, demonstrate uncountably separated countable neighborhoods of digital numbers. That's good... :)

Where two points can be distinguished, there is another between them.
Where they cannot be, there need not be.

I like that, just perfect for quote mining!


I hope you enjoy it for many moons to come.

Doing so, we see the piture [sic] of points with space in between.
Again, who other than you (Eckard Blumschein) would see a picture
of a line like that? Have you been looking at your dot-matrix
computer-display so long you mistakenly believe the row of discrete
dots on the screen *is* an actual geometrical line?
Why don't you go ask Leibniz and Newton what they saw?

I'm confident that both of those famous mathematicians accepted the
basic axioms of classical geometry, which include a new (third)
point between any two distinct points, in fact a whole segment
between any two distinct points, no empty space there at all.
If you know otherwise, please cite a reference.


Go Google "infinitesimal calculus" and see if you still think infinitesimal differences which cannot distinguish two standard values have never played a role in mathematics.

If by "DA2" you mean Cantor's second diagonal argument, ...
(I assume from below that you're affirming my guess about meaning of DA2:)
It's based on the false premise that the list is a square,

That makes no sense to me. What list? What square?
By "list", do you mean the supposed enumeration of *all* real
numbers, which is impossible?

Yes, I mean the one within which the anti-diagonal supposedly defines a string not on the list. That assumes the list ends no further down than the diagonal, which assumes the list is square, which it's not, using any base above unary.

By "square" do you mean the *quadrant* which runs down the supposed
enumeration along one axis and the decimal digits of members of
that enumeration along the other axis? It's rather sloppy to
conflate a square (which is bounded in all directions) and a
quadrant (which is unbounded in two directions).
If that's what you mean, nobody ever claimed a quadrant and a
sequence (enumeration) are the same thing, so you're talking
nonsense.


I'm not talking more nonsense than you, if you're talking about this diagonal line through some "quandrant". What, in your least sloppy language, is the meaning of the anti-diagonal generated by the random list?

It's a proof about combinatorics and language more than anything.

No, it's a proof that a particular kind of enumeration is impossible.


It means that the power set, and sets generated in like fashion, are necessarily larger.

It doesn't work in unary.

Who cares? That's totally irrelevant!

If it were a property of the set of numbers itself, the it would be independent of representation. Rather, it's a property of the representation used, and of the discrete logical gradation.

It doesn't exactly work in
binary either. You need at least base 3 to make it work directly.

Sure it works in binary. You can always generate an unused bit string.

I.e. you have to do the proof correctly to make it work. The fact
that doing it incorrectly doesn't work doens't prove anything.
Almost nothing works if you do it wrong.


Exhibit A.

It reflects, like the power set result of 2^n, that a language
with alphabet of size S and string length of L has S^L elements.

That's pretty much irrelevant if S and L are finite numbers.

And pretty much relevant if one or the other is not. :)

Tony
.

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