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From: Transfer Principle on 2 Jul 2010 03:36
On Jun 29, 11:20 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
> "Transfer Principle" <lwal...(a)lausd.net> wrote
> > I still believe that it's possible to find a workable schema that
> > describe Herc's intuitions, but it won't be easy.
> phi( <[1] 2 3 4...> ) & An ((phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> ))
> ->
> phi( <[1 2 3 4...]> )

An interesting schema. But of course, this raises the question
as to what exactly the <> and [] symbols stand for.

For example, some posters use <> to denote an n-tuple, so that
<x y> would denote the (Kuratowski) ordered pair. But that
would make most of the "n-tuples" infinite -- which would turn
them into sequences. So under this interpretation:

<[1 2 ... n] n+1 n+2 ...>

is the sequence whose 0th entry is [1 2 ... n], whose 1st entry
is n+1, whose 2nd entry is n+2, and whose mth entry is n+m.

Reasonable enough. But now we must turn to the [1 2 ... n]
notation which uses [] brackets instead of <>. A few earlier
posters, including zuhair and tommy1729, used [] to denote
objects under the flattened mereology. Hopefully, this isn't
how Herc is using them, since this would make the theory a bit
more complicated.

What would help is to see more instances of this schema. We
already know that Herc intends to apply it to lists such as:

0.100000000...
0.110000000...
0.111000000...
0.111100000...

and conclude that the digit 1 appears in every position, or
something like that. Let me think about this for a while.
From: |-|ercules on 2 Jul 2010 04:10
"Transfer Principle" <lwalke3(a)lausd.net> wrote
> On Jun 29, 11:20 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote:
>> "Transfer Principle" <lwal...(a)lausd.net> wrote
>> > I still believe that it's possible to find a workable schema that
>> > describe Herc's intuitions, but it won't be easy.
>> phi( <[1] 2 3 4...> ) & An ((phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> ))
>> ->
>> phi( <[1 2 3 4...]> )
>
> An interesting schema. But of course, this raises the question
> as to what exactly the <> and [] symbols stand for.
>
> For example, some posters use <> to denote an n-tuple, so that
> <x y> would denote the (Kuratowski) ordered pair. But that
> would make most of the "n-tuples" infinite -- which would turn
> them into sequences. So under this interpretation:
>
> <[1 2 ... n] n+1 n+2 ...>
>
> is the sequence whose 0th entry is [1 2 ... n], whose 1st entry
> is n+1, whose 2nd entry is n+2, and whose mth entry is n+m.


Maybe <1* 2* 3* 4 5 6 ...> would be clearer and closer to standard induction,

I thought it was clear

< [1 2 3] 4 5 6...>

is the 3 digit prefix in the sequence of N.




>
> Reasonable enough. But now we must turn to the [1 2 ... n]
> notation which uses [] brackets instead of <>. A few earlier
> posters, including zuhair and tommy1729, used [] to denote
> objects under the flattened mereology. Hopefully, this isn't
> how Herc is using them, since this would make the theory a bit
> more complicated.
>
> What would help is to see more instances of this schema. We
> already know that Herc intends to apply it to lists such as:
>
> 0.100000000...
> 0.110000000...
> 0.111000000...
> 0.111100000...
>
> and conclude that the digit 1 appears in every position, or
> something like that. Let me think about this for a while.


What would be better is if the schema did NOT work on the above example.

The lack of constraint on the suffix, the fact that there is no distinction where the finite prefix ends
and the suffix starts blah blah..

Herc

From: |-|ercules on 2 Jul 2010 04:18
"|-|ercules" <radgray123(a)yahoo.com> wrote...
> "Transfer Principle" <lwalke3(a)lausd.net> wrote
>> On Jun 28, 9:58 pm, herbzet <herb...(a)gmail.com> wrote:
>>> Tim Little wrote:
>>> > In short, just another crank.
>>> Yeh, well, I actually defended him from the beginning when he
>>> showed up in sci.logic with his project of providing formal
>>> systems in which the assertions of various cranks can be
>>> demonstrated. I think that is a valid intellectual exercise,
>>> at least, and could provoke some actually interesting
>>> discussions about fundamental assumptions we routinely make
>>> in logic/math.
>>
>> I actually attempted to do this a few times in this thread, but
>> when Herc stated that he was trying to use some form of
>> induction, all I could muster was a schema of the form:
>>
>> (phi(.1) & An (phi(n 1's) -> phi(n+1 1's))) -> phi(.111...)
>>
>> Of course, such schemata are invalid in standard theory, and I
>> even attempted to warn Herc that the majority of posters in this
>> thread are likely to reject such schemata.
>>
>> I still believe that it's possible to find a workable schema that
>> describe Herc's intuitions, but it won't be easy.
>
> phi( <[1] 2 3 4...> ) & An (phi ( <[1 2 ... n] n+1 n+2 ...>) -> phi( <[1 2 ... n n+1] n+2 n+3 ...> ))
> ->
> phi( <[1 2 3 4...]> )
>
> Herc


I was going to check yesterday WHAT TIME I made this 'discovery'.

Just realized it was 4:20! The most famous time of day, on par with 3:14!

I asked my Lord if that was a good thing, and scrolled to this comment on RicksBlog.com

<<is clear to those that can see that they have the tools and the products to win the war.>>

I find that amazing as Vertical Horizon's song "HE SAYS ALL THE RIGHT THINGS AT EXACTLY THE RIGHT TIME"
is about me. The world runs to clockwork.

I hereby rename Prefix Induction Schema to Stoners Formula! There is only 1 infinity!!!!!!

Herc

From: Jesse F. Hughes on 2 Jul 2010 07:47
Transfer Principle <lwalke3(a)lausd.net> writes:

> On Jun 30, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> 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.

Pardon me?

Who denies him any such freedom?

You failed to understand my point entirely. When Herc says ZFC is wrong
or bad or some theorem in ZFC is wrong, then an argument ensues. If
Herc were to say, here is a mathematical theory and here are some of its
consequences (and his claims were sensible and correct), why should
anyone argue?

In any case, Herc has not been denied any freedoms -- at least, not by
sci.math and not for his claims regarding Cantor. But when he says
something that is apparently false or groundless, then he is corrected
(and often insulted). Nonetheless, he's perfectly free to persist in
his mathematical delusions.

--
"I'd step through arguments in such detail that it was like I was
teaching basic arithmetic and some poster would come back and act like
I hadn't said anything that made sense. For a while I almost started
to doubt myself." -- James S. Harris, so close and yet....
From: MoeBlee on 2 Jul 2010 17:40
On Jul 2, 12:12 am, Transfer Principle <lwal...(a)lausd.net> wrote:

> at what point
> beyond which a theory differs from ZFC such that we should no
> longer call the objects which satisfy them sets?

I don't know. But we can adopt certain definitions, such as:

x is a set <-> (x=0 or Eyz y in x in z))

That will work as long as the theory defines '0' appropriately.

We may consider at least four predicates

class (has a member or is 0)
set (as defined above)
urelement (has no member but is not 0)
proper class (class but not itself a member of anything)

Then certain theories may prove which of those exist or do not exist.

> 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.

ZF does not prove there exists an infinite yet Dedekind infinite set,
right? Rather, it is undecidable in ZF whether there exists such a
thing. Same with sets without a choice function, right?

> So I wonder, where is the line that when crossed we can
> no longer call them sets, but, to use MoeBlee's name, "zets"?

Just to be clear, this has NOTHING to do with what I meant about using
the term 'zet'.

> 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.

What engenders such remarks is not merely discussing alternative
theories, but certain OTHER behavior. This has been pointed out to you
hundreds and hundreds of times (literally); I don't know why you don't
get it.

MoeBlee

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