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From: Transfer Principle on 30 Jun 2010 02:30

On Jun 28, 8:57 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> "Sylvia Else" <syl...(a)not.here.invalid> wrote
> > I have questioned whether what you're doing is induction. P(n) -> P(n+1)
> > is the result of an inductive proof, not the proof itself.
> > Sylvia.
> You and I start a landscaping business Herc And Syl's Landscaping Ad Infinitium
> I handle all the mowing, and you do the fencing.
> We get a call from Mr Fenceme and Mrs Mowme Blockheads.
> We drive to the property which appears to be divided into 2 blocks, both infinite
> rectangular lawns.

Interesting analogy.

Of course, one might wonder what exactly an infinite rectangular
region is
suppsed to be. For example, we consider the coordinate plane. Is a
single
quadrant (say the first quadrant) an infinite rectangle, or is a half-
plane
(say the union of the first two quadrants) an infinite rectangle?

> On one block, you start doing the fencing for Mr Fenceme, completing the perimeters of larger and larger concentric
> rectangular paddocks.

But then we notice that neither a quadrant nor a half-plane is the
union of
any number of concentric rectangular regions. Indeed, the union of
infinitely many concentric rectangular regions is either a (finite)
rectangular
region (if the dimensions are increasing but bounded), a strip (if the
lengths
are increasing to infinity but not the widths), or the entire plane.

Since a finite rectangle can hardly be described as "infinite," and
since the
entire plane leaves no room for Mrs. Mowme's lawn, I conclude that an
"infinite rectangle" is actually an infinite strip.

> I get to the mowing for Mrs Mowme, completing larger and larger rectangular mown lawn areas, each building
> upon the earlier smaller rectangular lawn area.
> Being a man, I'm well on my way to mowing the whole lawn.

OK. I assume that on the first step, Herc mows a small finite segment
of the strip (whose width is the same as that of the entire strip),
then
on subsequent steps, mows rectangles on either side of the starting
rectangle so that after each step, the center of the mowed rectangle
remains constant.

If each step takes, say, half as long as the previous step, then the
entire lawn can be mowed in finite time.

> Because you're a woman

[snip sexism]

From: Jesse F. Hughes on 30 Jun 2010 08:54

Transfer Principle <lwalke3(a)lausd.net> writes:

> I refuse to believe that the only way to discuss alternate theories
> is to incite emotional reaction and be a "troll," just as I refuse to
> believe that the only posters who oppose ZFC are those who
> don't understand ZFC.

Let's take these one at a time.

(1) The only way to discuss alternate theories is to incite emotional
reaction and be a "troll".

No one believes this is so.

(2) The only posters who oppose ZFC are those who don't understand ZFC.

This is a matter of fact, not necessity. I imagine there are perfectly
intelligent philosophers of mathematics who understand ZFC yet believe
that it is a "bad" theory for various philosophical reasons. Those
people aren't posting here on the group. It is obvious that the people
who post about the evils of ZFC on this group are, indeed, a confused
lot that don't understand mathematics.

If, on the contrary, they *did* understand mathematical theories and ZFC
in particular, why must they wait for a hero like you to make their
so-called theories rigorous? Surely, they would see the need and do the
work on their own.

> It's possible to be an _expert_ of the proof of Cantor's Theorem in
> ZFC and _still_ prefer to work in a theory in which its negation is a
> theorem, and it's possible to discuss such a theory without an
> emotional "trolling" reaction.

Of course. But the cranks here do not simply claim to "prefer" to work
in an alternate theory. First, they do not have an alternate theory
that they claim to prefer. Second, their criticisms do not come off as
mere preference.

To give you an example: for a while, I did a little research in ZFA,
anti-well-founded set theory. I liked the theory. I suppose that I
found it preferable to ZFC at the time. I did not argue that ZFC was
wrong, or give tired, silly arguments that the axiom of regularity is
false because it's false in ZFA. The fact that ZFC was regarded as a
foundation of mathematics while ZFA was not did not bother me. I worked
in ZFA because it provided a nice setting for the things I was doing.

That's rather different than the behaviors we see here.
--
"Another factor one has got to look at is the amount of liquidity in
the system. In other words, is there enough liquidity to enable
markets to be able to correct? And I am told there is enough liquidity
in the system to enable markets to correct." -- Guess who.

From: |-|ercules on 30 Jun 2010 21:25

"Transfer Principle" <lwalke3(a)lausd.net> wrote
>
> Of course, one might wonder what exactly an infinite rectangular
> region is
> suppsed to be. For example, we consider the coordinate plane. Is a
> single
> quadrant (say the first quadrant) an infinite rectangle,

Yes that's what I had in mind.

http://i721.photobucket.com/albums/ww214/ozdude7/fencingVSmowing.png

Here's the story again that tells of the difference to Sylvia's finite sequence induction.

You and I start a landscaping business Herc And Syl's Landscaping Ad Infinitum

I handle all the mowing, and you do the fencing.

We get a call from Mr Fenceme and Mrs Mowme Blockheads.

We drive to the property which appears to be divided into 2 blocks, both infinite rectangular lawns.

On one block, you start doing the fencing for Mr Fenceme, completing the perimeters of larger and larger concentric
rectangular paddocks.

I get to the mowing for Mrs Mowme, completing larger and larger rectangular mown lawn areas, each building
upon the earlier smaller rectangular lawn area.

I'm well on my way to mowing the whole lawn.

You never ever come close to fencing the entire lawn! ;-)

The limit of mown lawn area as mowing time->oo is infinity.

However, infinitely many fence sizes all have finite perimeters.

Herc

From: Transfer Principle on 2 Jul 2010 03:12

On Jun 30, 7:54 pm, herbzet <herb...(a)gmail.com> wrote:
> Transfer Principle wrote:
> > But now what? I want to be able to discuss theories other than
> > ZFC, including those theories which might prove the negation of
> > Cantor's Theorem.
> It seems evident to me that such a theory would be talking about
> objects somewhat different from what is being discussed in more
> standard theories, even if such objects are called "sets" in
> both theories.
> Even in a not-too-distant from standard set theory like NF, it
> seems fairly evident to me that objects which could conceivably
> satisfy NF are somewhat different from objects that would satisfy,
> say, ZFC.

This raises another interesting question, namely at what point
beyond which a theory differs from ZFC such that we should no
longer call the objects which satisfy them sets? This question
has come up in other threads as well. By this line of argument,
one could even point out that there are objects satisfying _ZF_
that are different from those satisfying ZFC, namely those
without choice functions, are infinite yet Dedekind finite, and
so on. So I wonder, where is the line that when crossed we can
no longer call them sets, but, to use MoeBlee's name, "zets"?

> And so what? What's at stake?

If there's a theory, say NF, which satisfies Herc's intuitions,
then perhaps he might find it more reasonable than ZFC. If it's
true that Herc has criticized ZFC and exhibited behaviors that
cause him to be considered a "troll," then perhaps knowledge of
a theory such as NF will convince him to be less of a "troll."

> > Is it possible to do so _without_ inciting a
> > highly emotional reaction? If so, then I'd _love_ to do so, so that
> > I can discuss theories other than ZFC without fear that the
> > "troll" label will come up.
> Please clarify: do you want to discuss theories other than ZFC, or
> do you wish to discuss why people are so mean to the trolls, cranks,
> and loons who show up with their revolutionary breakthroughs and
> poo-flinging?
> If just the former, I don't think you'll have any problems.

I want to discuss theories other than ZFC -- and hope that I
can do so with neither the five-letter insults nor the
behavior that inspires those words appearing.

From: Transfer Principle on 2 Jul 2010 03:24

On Jun 30, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Transfer Principle <lwal...(a)lausd.net> writes:
> > I refuse to believe that the only way to discuss alternate theories
> > is to incite emotional reaction and be a "troll," just as I refuse to
> > believe that the only posters who oppose ZFC are those who
> > don't understand ZFC.
> (2) The only posters who oppose ZFC are those who don't understand ZFC.
> If, on the contrary, they *did* understand mathematical theories and ZFC
> in particular, why must they wait for a hero like you to make their
> so-called theories rigorous? Surely, they would see the need and do the
> work on their own.

Was ZFC the work of a single set theorist? Sure, Cantor laid
the foundation, but the fact that ZFC is named after _two_
set theorists, Zermelo and Frankel, speaks for itself.

Similarly, if we work together, we might be able to come up
with a rigorous set theory as well. Not only have I helped,
but WM's writings have inspired Herc as well.

As the saying goes, "two heads are better than one."

> To give you an example: for a while, I did a little research in ZFA,
> anti-well-founded set theory. I liked the theory. I suppose that I
> found it preferable to ZFC at the time. I did not argue that ZFC was
> wrong, or give tired, silly arguments that the axiom of regularity is
> false because it's false in ZFA. The fact that ZFC was regarded as a
> foundation of mathematics while ZFA was not did not bother me. I worked
> in ZFA because it provided a nice setting for the things I was doing.
> That's rather different than the behaviors we see here.

As Hughes is allowed to use a theory which proves the negation of
Regularity, likewise Herc should be allowed to use a theory which
proves the negation of Cantor's Theorem. I wonder what combination
of axioms and posting behavior will lead to Herc being granted
that freedom.