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From: Virgil on 15 Dec 2006 17:17
In article <aa902$458259a8$82a1e228$15301(a)news1.tudelft.nl>,
Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote:

> Virgil wrote:
>
> > In article <d1914$45811514$82a1e228$2825(a)news1.tudelft.nl>,
> > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote:
> >
> >>William Hughes wrote:
> >>
> >>>Therefore, as there are at least as many digit positions in 0.111...
> >>>as there are primes, there are an infinite number
> >>>of digit positions in 0.111...
> >>
> >>Binary or decimal?
> >
> > Irrelevant!
>
> So?
>
> Han de Bruijn

So!
From: Virgil on 15 Dec 2006 17:21
In article <1166184731.253435.242720(a)l12g2000cwl.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Virgil schrieb:
>
> > In article <1166092755.336596.309060(a)l12g2000cwl.googlegroups.com>,
> > mueckenh(a)rz.fh-augsburg.de wrote:
> >
> > > Virgil schrieb:
> > >
> > > > WM deceives himself over the number of lines versus the number of
> > > > elements in any one line. The number of lines is not finite but the
> > > > number of elements in any one line is finite.
> > >
> > > Each line differs by 1 element from the preceding line. If the number
> > > of lines is actually infinite then the number of differences must be
> > > actually infinite too.
> >
> > So far so good.
> >
> Their sum is an infinite number. If all elements are there, then also
> all sums are there.

That is an unjustifiable assumption claimimg that ALL sequences
converge to finite values.



> If not all sums are there, not all elements are
> there.

There are infinite sequences that do not converge.

And 1 + 2 + 3 + ... is one of them.
From: Virgil on 15 Dec 2006 17:34
In article <1166185042.110714.41980(a)f1g2000cwa.googlegroups.com>,
mueckenh(a)rz.fh-augsburg.de wrote:

> Virgil schrieb:
>
>
> > But the set of all sequences of edges is NOT countable, so what
> > restrictions on sequences does WM have to limit then to only a countable
> > subset of all such seqeunces?
>
> The set of all edges is countable. The set of all sequences could be
> uncountable, if all combinations of edges were possible.

It is sufficient that for each term in a sequence there be more than one
option in order for the set of all such sequences to be uncountable.




> But that is
> not the case in the binary tree. No path can separate from another one
> without an edge for each of them.

Which makes two choices at each term of every sequence and makes
uncountably many sequences. This uncountable "number" is essentially the
"number" of binary strings.

> > One is not separating individual paths but sets of paths with each set
> > uncountable as the set of all paths.
>
> And that is the very real reason of your confused mathematics: There
> are no seprated paths at all. But you claim them uncountable.

The set of all paths through any particular edge is separated from the
set of all paths not through that edge, and each edge produces two
uncountable sets in this way.

In a similar fashion, each node exxcept the root ode does likewise.

> > The union of a family of sets need not be a subset of any
> > of them.
>
> Not in general, but if the elements are linear ordered sets.
>
>
> > But unless we assume that there can only be finitely many we cannot
> > conclude that there is a last or largest conctaining all the others.
>
> It is enough to know that each one is finite.

That does not assure that the union of infinitely many such finite sets
is finite.

And, in fact, a union of infinitely many distinct finite sets cannot be
finite.
From: Lester Zick on 15 Dec 2006 17:34
On 15 Dec 2006 11:19:52 -0800, "Jonathan Hoyle" <jonhoyle(a)mac.com>
wrote:

>> And too bad Aristotle's approach to syllogistic inference doesn't
>> produce the truth he hoped it would yield. The best we can say for it
>> is that it produces a series of truisms which in turn produce the
>> empirical approaches to math and science which belabor the present.
>> That mathematics and science rely on it even today after two millenia
>> stands in mute testament not to its universal validity but to the fact
>> there is no other better formal system of logical inference available.
>
>I'm not following. What "truth" is not being produced, and what is
>missing in logic?

I've covered this in detail previously. Below is a reply I posted a
couple of weeks back on the problem of truth and empiricism in the
context of syllogistic inference:

>Tonico, let me revisit my comments below to present the same analysis
>in slightly different terms.
>
>If we take some conclusion, C, and want to know if it's true there are
>two different ways to proceed.Aristotelian syllogistic inference bases
>the truth of C on the truth of its constituent premises such as B. In
>other words it says "if B then C" which is a truism because C would
>certainly be true if B were true.
>
>However then we're just faced with exactly the same problem with B. So
>we further regress analysis of the truth of C to the truth of B to
>find the truth of B relies on the truth of some constituent premise of
>B such as A with the result that we wind up with "if A then B then C".
>
>This is exactly how classical syllogistic inference has always worked
>in the context of science and mathematics. To support the truth of
>some conclusion such as C there is an indefinite regression of
>problematic premises and this regression is what I call empiricism. In
>ordinary science this regression stops at what would appear a logical
>boundary of sensory and perceptual experience whereas in mathematics
>it stops with axioms and axiomatic assumptions of truth.
>
>Now this doesn't mean that truisms like "if A then B then C" cannot be
>true only that their truth can never be known in exhaustive terms. The
>most we can hope for is to stumble on some syllogistic regression that
>turns out to be true and employ it to ground further speculations. In
>effect syllogistic regressions such as "if A then B then C" become a
>line of reasoning or in the parlance of modern math a "model" of truth
>because the truth can never be known absolutely with such a method.
>
>Now I analyze the same problem from exactly the opposite perspective.
>Instead of asserting the truth of C relies on the truth of constituent
>premises I maintain the truth of any conclusion such as C relies on
>the falsity of alternatives to C, in other words what is "not C'.
>
>Thus we form a tautological regression of "C, not C" instead of the
>syllogistic regression "if A then B then C" and find that C can and
>must be true only if "not C" must be false and "not C" must be false
>only if it is self contradictory.
>
>In any event I hope this clears up what I mean by empiricism and truth
>in the context of mathematics and science.

~v~~

>I agree we don't understand the term "empirical" the same way. Not my
>fault since I've discussed the subject at length over the past couple
>years here and elsewhere. Basically any tautologically undemonstrated
>judgment is empirical. Doesn't matter whether the judgment is sensory,
>perceptual, cognitive, or whatever. If you assume an axiom such as "a
>straight line is the shortest distance between points" the assumption
>is empirical until and unless demonstrated true analytically. The same
>applies to definitions.
>
>Most people completely misunderstand the meaning of an empirical
>judgment. Most think it means getting out the tape measure, scales,
>and so forth. The problem originated with Aristotle and his concept of
>syllogistic inference. Aristotle was history's first empiricist in
>formal terms. He found he could not establish the truth of any
>conclusion syllogistically except by regression to further premises
>whose truth he could not establish either except by further regression
>ad infinitum. Which meant he could establish no truth syllogistically
>at all without some kind of true basic premises which he set out to
>find in unreducible perceptual terms. Which left us epistemologically
>exactly where we are today in terms of all kinds of mathematical and
>scientific methodologies.
>
>In point of fact however empirical judgment is nothing more than input
>to a process of tautological regression whose ultimate goal is
>reduction to self contradictory alternatives. That's how the mind and
>brain work, tautological rather than syllogistic inference because it
>can produce reductions to truth in exhaustive mechanical terms.

In other words, Jonathan, Aristotelian syllogistic inference only
yields truisms not truth. It can't tell us what the truth of anything
actually is only whether or not it's true in relation to constituent
premises of equally problematic truth. The truth of conclusions is not
inherited from the truth of constituent premises as syllogistic
inference implies and Aristotle thought. The truth of any conclusion
must be demonstrated in its own right by demonstrating the falsity of
alternatives by finite tautological regression to self contradictory
alternatives.

~v~~
From: Lester Zick on 15 Dec 2006 17:38
On 15 Dec 2006 13:11:27 -0800, "Jonathan Hoyle" <jonhoyle(a)mac.com>
wrote:

>> Therefore I fault him. Except for lenses Aristotle had every contrivence
>> and technology that was available to Galileo. Greek shipwrights could
>> have made straight smooth wooden ramps to be used as inclined planes.
>> The Greeks had Egyption drip clocks at their disposal. So it wasn't lack
>> of tools or technology that prevented Aristotle from checking. It was
>> attitude.
>
>I see your point. I'm a big fan of Aristotle and the work he did, but
>his physics does appear to lack the attention to detail that his other
>works included. If he is deserving of the praise for his other works,
>he likewise deserves the criticisms of his failures.
>
>> We would be a millenium ahead or extinct. Aristotle's followers cost us
>> over a thousand years of progress.
>
>And the worst part to this was his "followers", for all those
>centuries, not *one* amongst the untold millions even attempted to
>coroborate his work on "Natural Philosophy" to see if it held true.
>This is more than simply "being sheep". This is not caring enough to
>even try. :-/

You know, Jonathan, you could help yourself out a lot by including
headers in your replies so we knew who said what to whom about what.

~v~~
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