Prev: JSH: Judging difficulty
Next: HOW DID GOVERNMENT BEGIN?
From: Transfer Principle on 21 Jun 2010 23:49
On Jun 20, 8:14 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> "Sylvia Else" <syl...(a)not.here.invalid> wrote
> > Well, it's true that I rather assumed, without proof, that any number
> > can be created in the diagonal. But I don't think anything relevant
> > turns on that.
> You admit your error to him but not me?

But then again, Else made the same error that Herc made, namely
assuming that any real can appear on the diagonal. Of course,
the conjecture that _almost_every_ real (i.e., every real
except in a set of Lebesgue measure zero) can appear on the
diagonal remains open.

> What are we married or something!

Married? What's next -- will Herc claim that Else is actually
Genesis Eve?

> Atleast you dismissed your error as irrelevant in true [...]

[snip sexism and homophobia]
From: Transfer Principle on 21 Jun 2010 23:56
On Jun 21, 8:49 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jun 20, 8:14 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> > "Sylvia Else" <syl...(a)not.here.invalid> wrote
> > > Well, it's true that I rather assumed, without proof, that any number
> > > can be created in the diagonal. But I don't think anything relevant
> > > turns on that.
> > You admit your error to him but not me?
> But then again, Else made the same error that Herc made

By which I mean the mistake that Herc made _years_ago_
when he came up with this idea of swapping to create a
diagonal in the first place.

Now, of course, he concedes that 0.222... can't appear
in the diagonal if 0.111... is in the list, but only
after either I (about a month ago) or some other poster
showed him the error.
From: Graham Cooper on 22 Jun 2010 04:06
On Jun 22, 1:56 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Jun 21, 8:49 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
>
> > On Jun 20, 8:14 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> > > "Sylvia Else" <syl...(a)not.here.invalid> wrote
> > > > Well, it's true that I rather assumed, without proof, that any number
> > > > can be created in the diagonal. But I don't think anything relevant
> > > > turns on that.
> > > You admit your error to him but not me?
> > But then again, Else made the same error that Herc made
>
> By which I mean the mistake that Herc made _years_ago_
> when he came up with this idea of swapping to create a
> diagonal in the first place.
>
> Now, of course, he concedes that 0.222... can't appear
> in the diagonal if 0.111... is in the list, but only
> after either I (about a month ago) or some other poster
> showed him the error.


You are posting pure fantasy. I was well aware of rigged
diagonals defeating my algorithm years ago that's why I
claimed random diagonals will fit.

You may have mentioned 0.11.. In some other argument
but if you would provide a cite every time you say
"herc was wrong" you would realize nobody has ever
corrected me. Not even my bad spelling!

Even the captcha is EURNOS. NO ERROR!
Herc
From: Mike Terry on 22 Jun 2010 13:55
"Transfer Principle" <lwalke3(a)lausd.net> wrote in message
news:11547e1f-04e5-4f87-8dcd-5b6208c23251(a)v29g2000prb.googlegroups.com...
> On Jun 21, 5:59 pm, "Mike Terry"
> <news.dead.person.sto...(a)darjeeling.plus.com> wrote:
> > "Tim Little" <t...(a)little-possums.net> wrote in message
> > > More generally still, it is not easy to define any reasonable meaning
> > > for an infinite composition of non-disjoint permutations.
> > Aaaaargh, it's all starting to come back! This is how I started off with
> > Herc a few years ago, and Herc believed that IF he could permute the
list in
> > order to achieve ANY diagonal of his choosing, then he could engineer it
so
> > that the corresponding antidiagonal was in the list, although by
Cantor's
> > argument it is not. Well, this much is correct, but obviously it won't
be
> > possible to permute the list in this way...
> > But then Herc went on to invent a succession of ever more complex
schemes
> > involving building up these "permutations" by successively
> > swapping/shuffling rows of a carefully constructed starting list.
Actually
> > the whole idea was quite good fun to think through by comparison with
> > today's Herc issues, and in the end the problem boiled down to that,
say,
> > row 1 would initially be swapped to row 20, and then in step 20 it would
be
> > swapped to say row 1000, and in step 1000...etc.
> > Well, in the end row 1 would end up being shuffled right out of the list
> > altogether!
>
> Ah yes, a bit similar to Hilbert's Hotel.
>
> I was wondering whether something like this could happen, but
> Little's comment about 0.111... = 1/9 (or 1/2 in the original
> ternary that Cooper is using) means that it is futile to
> put 1/2 on the diagonal before worrying about this Hilbert's
> Hotel paradox at all.
>
> But of course, I assume that Herc proposed some sort of
> _algorithm_ to get the real r on the diagonal. It most likely
> went something like this -- if the dth digit of the dth real
> doesn't match the dth digit of r, then we swap the dth real
> with the nth real, where n is the first natural greater than
> d such that the dth digit of the nth real is the correct dth
> digit of r. Since the list contains all the computable reals,
> and there are infinitely many computable reals with the
> correct digit in the dth place, and there have been only
> finitely many reals before reaching the dth real, there's
> guaranteed to be a computable real remaining in the list with
> the correct digit in the dth place.

Exactly. I believe he was working with an initial list that was crafted
specifically to make it obvious that there would be suitable "swap line"
later in the list when needed. So we end up with a countably infinite
succession of swap operations, which Herc was assuming would still be just a
permutation of the original list.

>
> Thus r is guaranteed to appear on the diagonal. But if r were
> say, any member of the Cantor set (so that it contains only
> zeros and twos in its ternary expansion), then 1/2 = 0.111...
> will be swapped, and swapped again, indeed producing the
> Hilbert's Hotel effect as described by Terry.
>
> Still, will this happen with _almost_every_ real that we try
> to put on the diagonal?

I don't know - there are uncountably many numbers not in the original list,
so as has been said, there will be a lot of flexibility in the diagonals
that can be produced this way. The point is really that it doesn't matter
when it comes to arguing about Cantor's proof. (As a question in its own
right, interesting maybe..)



First  |  Prev  | 
Pages: 1 2 3 4 5 6 7 8 9
Prev: JSH: Judging difficulty
Next: HOW DID GOVERNMENT BEGIN?