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herbzet
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 Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: Why "meta diagonals" are irrelevant |
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Horand.Gassmann@googlemail.com wrote:
[quote]
On Oct 9, 4:07 pm, Virgil <Vir...@gmale.com> wrote:
In article <98e9a$48ee1394$82a1e228$13...@news1.tudelft.nl>,
Han de Bruijn <Han.deBru...@DTO.TUDelft.NL> wrote:
Daryl McCullough wrote:
If you really think that ZF can show that the same set is
countable and uncountable, then that means that you think
that you can prove a contradiction using only the axioms of ZF.
Do you believe that? Then why can>t you do it? After years and
years of claiming that ZF is inconsistent, why haven>t you been
able to write down a proof of a contradictionn using the axioms of ZF?
Why should someone who likes to play socker join _your_ team of football
players? Why should socker be an inferior sports compared with football?
WM playes the game of mathematics with different rules. So what? Why not
look at it from _his_ perspective? Or at last give it a _honest_ try and
by that I mean: not just _suggesting_ you>re giving it a honest try.
Han de Bruijn
Do you mean what we call "soccer"?
WE do not play by WM>s rules because WM>s rules do not provide a level
playing field.
WM>s rules amount to "WM wins, because WM says so!"
Well, that>s the one rule he uses consistently. He has other rules
that are made up (and broken) as he goes along. Since nobody can
figure out the rules (and he seems unable himself), it is no wonder
that no one wants to play his game.
[/quote]
It seems that everyone and his brother wants to play his game.
--
hz |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: A consideration concerning the diagonal argument of G. C |
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On 11 Oct 2008 08:25:04 +0300 Aatu Koskensilta wrote:
[quote]Herbert Newman <nomail@invalid> writes:
On the other hand, some of the "intuitions" of cranks have already
been formulated in a more or less satisfactory manner. Finitism and
ultrafinitism come to mind.
What more or less satisfactory formulation of ultrafinitism do you
have in mind?
Finitism: more[/quote]
Ultrafinitism: less
"Ultrafinitism is a form of constructivism, but even constructivists
generally view the philosophy as unworkably extreme. The logical foundation
of ultrafinitism is unclear; in his comprehensive survey Constructivism in
Mathematics (1988), the constructive logician A. S. Troelstra dismissed it
as "no satisfactory development exists at present". This was not so much a
philosophical objection as it was an admission that, in a rigorous work of
mathematical logic, there was simply nothing precise enough to include.
Serious work on ultrafinitism has been led, since 1959, by Alexander
Esenin-Volpin."
http://en.wikipedia.org/wiki/Ultrafinitism
Herb |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: A consideration concerning the diagonal argument of G. C |
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On Fri, 10 Oct 2008 18:34:24 -0700 (PDT) georgie wrote:
[quote]
So [mathematical] cranks are people who are incoherent?
Some (if not many or even all) of them are.[/quote]
Herb |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: A consideration concerning the diagonal argument of G. C |
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On Fri, 10 Oct 2008 18:35:26 -0700 (PDT) MoeBlee wrote:
[quote]
a real/true crank will never try to make himself familiar with what
has been done/achieved in this field already
The literature of the subject is to a crank what citric acid is to a
termite, what garlic is to a vampire, what a press interview is to
Sarah Palin.
:-)[/quote]
Oh, oh, you mean the "pitbul with lipstick"? :-)
Herb |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: A consideration concerning the diagonal argument of G. C |
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On Fri, 10 Oct 2008 18:50:43 -0700 (PDT) georgie wrote:
[quote]
a real/true crank will never try to make himself familiar with what
has been done/achieved in this field already
The literature of the subject is to a crank what citric acid is to a
termite, what garlic is to a vampire, what a press interview is to
Sarah Palin.
I would have thought that the Cantor fanatics were also Palin
fanatics. Your statement seems to indicate your religous
fanaticism is strictly Cantorian.
Actually there are no "Cantor fanatics", there are only "Anti-Cantor[/quote]
fanatics" - usually mathematical cranks.
Herb |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: Question regard implication and being a theorem of FOL. |
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On Fri, 10 Oct 2008 19:55:16 -0700 (PDT) george wrote:
[quote]
The left arguments of |= MUST be of type INTERPRETATION!
They must be of type MODEL.
Depends. My |= here is indeed "overloaded".[/quote]
L |= phi iff for any Model M: if M |= L, then M |= phi,
where L is a set of wffs and phi is a wff.
Actually, this is a rather common (dual) usage of |=.
See http://plato.stanford.edu/entries/logic-classical/
So one might write
{F & ~F} |= G.
And writing
F & ~F |= G
is just a (again rather common) simplification for that.
Herb |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: Question regard implication and being a theorem of FOL. |
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On Fri, 10 Oct 2008 19:55:16 -0700 (PDT) george wrote:
[quote]
The left arguments of |= MUST be of type INTERPRETATION!
They must be of type MODEL.
Depends. My |= here is indeed "overloaded".[/quote]
L |= phi iff for all Models M: if M |= L, then M |= phi,
where L is a set of wffs and phi is a wff.
Actually, this is a rather common (dual) usage of |=.
See http://plato.stanford.edu/entries/logic-classical/
So one might write
{F & ~F} |= G.
And writing
F & ~F |= G
is just a (again rather common) simplification for that.
Herb |
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herbzet
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: Question regard implication and being a theorem of FOL. |
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Scott wrote:
[quote]
I still don>t see how you are explaining the problem away. Perhaps a
change in notation would be helpful. Let F and G be any wff.
I think we all agree we have:
|- (F&~F) -> G
and for all intepretations the above is true so we have:
|= (F&~F) -> G
Suppose we have (F&~F) then we get:
{ F&~F, (F&~F)->G } |- G
[/quote]
(F&~F)->G is superfluous. {F&~F} |- G.
[quote]then for some interpretations M1 and M2 we can have:
M1: { F&~F, (F&~F)->G } |= G
M2: { F&~F, (F&~F)->G } |/= G
[/quote]
No. M2 does not exist: there is no interpretation that satisfies
F&~F and fails to satisfy G, because there is no interpretation
that satisfies F&~F.
There>s no point in discussing the rest of your post
until you understand these facts.
--
hz |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: A consideration concerning the diagonal argument of G. C |
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On Sat, 11 Oct 2008 02:22:40 -0400 herbzet wrote:
[quote]georgie wrote:
You just don>t get it, do you?
I get it. You>re a troll.
Most probably.[/quote]
Herb |
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Herbert Newman
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: Question regard implication and being a theorem of FOL. |
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On Sat, 11 Oct 2008 04:14:58 -0400 herbzet wrote:
[quote]Scott wrote:
then for some interpretations M1 and M2 we can have:
M1: F & ~F |= G
M2: F & ~F |/= G
Actually, his notation doesn>t make any sense. (See Moe>s reply.)[/quote]
I guess he meant
M1 |= F & ~F and M1 |= G,
but
M2 |= F & ~F and M2 |/= G.
[quote]
No. M2 does not exist: there is no interpretation that satisfies
F & ~F and fails to satisfy G, because there is no interpretation
that satisfies F & ~F.
Right.[/quote]
[quote]
There>s no point in discussing the rest of your post
until you understand these facts.
Right.[/quote]
Herb |
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Aatu Koskensilta
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| Posted: Sat Oct 11, 2008 8:05 am Post subject: Re: A consideration concerning the diagonal argument of G. C |
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Herbert Newman <nomail@invalid> writes:
[quote]On 11 Oct 2008 08:25:04 +0300 Aatu Koskensilta wrote:
What more or less satisfactory formulation of ultrafinitism do you
have in mind?
Finitism: more
Ultrafinitism: less
[/quote]
There are certainly coherent and satisfactory formulations of
finitism. Of the proposed formulations of ultra-intuitionism -- do
note that Esenin-Volpin is an ultra-intuitionist -- pretty much no-one
seems to be able to make any coherent sense. As to ultra-finitism, the
formalisations and formulations, such as Sazanov>s, are usually
presented in the framework of classical first-order logic, making use
of ordinary mathematical results. This is obviously not a particularly
satisfactory situation from a philosophical and conceptual point of
view, even if the formulations turn out to be of some technical
interest.
--
Aatu Koskensilta (aatu.koskensilta@uta.fi)
"Wovon man nicht sprechen kann, darĂ¼ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus |
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WM
Guest
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| Posted: Sat Oct 11, 2008 10:16 am Post subject: Re: Why "meta diagonals" are irrelevant |
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On 10 Okt., 15:48, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
[quote]In article <f0c369b2-873e-4ecf-a323-39c76bcff...@p31g2000prf.googlegroups..com> WM <mueck...@rz.fh-augsburg.de> writes:
...
> But all constructive sets belong to your mathematics and can be
> treated by your mathematics? Or are there exceptions?
The question now becomes whether the diagonal of the list of lists actually
*can* be constructed. But now we require a proper definition of "constructed"
before we can go any further.
[/quote]
The question is, whether it can be in a constructed list. It cannot,
because it cannot be constructed. That supports my claim that there
are countable sets that cannot be sequenced.
Regards, WM |
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Guest
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| Posted: Sat Oct 11, 2008 11:52 am Post subject: Re: solutions manual and ebook (text book) |
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Hi,
I am requesting for the solutions of Investments (6th edition) by
William Sharpe, Gordon Alexander and Jeffery Bailey
Regards
Leckjam |
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Nam Nguyen
Guest
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| Posted: Sat Oct 11, 2008 1:34 pm Post subject: Re: Godel cant tell us what makes a mathematical statement t |
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MoeBlee wrote:
[quote]On Oct 8, 10:37 pm, Nam Nguyen <namducngu...@shaw.ca> wrote:
MoeBlee wrote:
What is most frustrating is that in all these cases we never get an
answer as to whether they>re trolling or whether they really are
cracked, and if they really are cracked, how do we explain it.
All that childish utterance only sounds like the line of reasoning done
by Bernardo Gui of the Inquisition in the movie "The Name of the Rose".
They, the Inquisitors, apparently thought that to reason was to simply
*decree* what they believed as true, no matter what others might have
supplied as evidences!
The irony of projection. In this case, I dare to pun that it>s "the
Eco effect".
[/quote]
I presented to you what I think as good evidences of model subjectivity
through sets; and you flatly decreed "No amount of talk by you about
'set notation', etc. changes the above fact [whatever you>d think as 'fact']".
So I don>t see any irony of projection on my part.
[quote]
the following is *just a set*,
not an interpretation or model!
{ (A {0}) (a 0) (m 0) (< {(0 0)}) }
A model IS a certain kind of set. (In the Enderton treatment) a model
is a certain kind of function. The above is a function of the kind
that is a model.
[/quote]
"Kind of set" is very problematic!
*Being certain kind of set* isn>t the same as *being a set*! And that>s
where your not understanding the issue originates from. Basically, if
you use all that>s available to you from the language of set, say L(ZF),
then "a kind of set" isn>t definable, hence there>s no interpretation or
model. You have to ascend to a meta level where "a kind of set" is
conceivable; but in *meta level* things are apt to be *subjective*!
Take for example Kuratowski>s definition of an ordered pair (x,y),
in L(ZF), where x, y are *just sets*:
(x,y) df= {{x}, {x,y}}
So what do we mean by saying something like
"The 1st element of (x,y) is x" or
"The last element of (x,y) is y"?
The long and short of it is to the extend x is *just a set* in
the definiendum (x,y) it>s *not* "a kind of set" as "The 1st element"
or "The last element" is purported to be. Otoh, to the extend
"The 1st element" and "The last element" are (subjective) "kinds of set",
would they mean the same set x, or y? In fact, could they both mean the
same set?
It>s all subjective in what we mean by the 1st or last element of (x,y),
or in our saying the 1st or last element is a kind of set, isn>t it?
In summary, interpretation and model, like the notion of "the 1st|last element",
are purely subjective: you can>t express them in 1st order language, without
going to a meta level, which is subjective |
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Herbert Newman
Guest
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| Posted: Sat Oct 11, 2008 2:00 pm Post subject: Re: A consideration concerning the diagonal argument of G. C |
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On 11 Oct 2008 11:22:43 +0300 Aatu Koskensilta wrote:
[quote]
What more or less satisfactory formulation of ultrafinitism do you
have in mind?
Finitism: more
Ultrafinitism: less
There are certainly coherent and satisfactory formulations of
finitism. Of the proposed formulations of ultra-intuitionism -- do
note that Esenin-Volpin is an ultra-intuitionist -- pretty much no-one
seems to be able to make any coherent sense. As to ultra-finitism, the
formalisations and formulations, such as Sazanov>s, are usually
presented in the framework of classical first-order logic, making use
of ordinary mathematical results. This is obviously not a particularly
satisfactory situation from a philosophical and conceptual point of
view, even if the formulations turn out to be of some technical
interest.
I wouldn>t dare to doubt your point of view. It>s in agreement with the[/quote]
statement:
"The logical foundation of ultrafinitism is unclear; in his comprehensive
survey Constructivism in Mathematics (1988), the constructive logician A.
S. Troelstra dismissed it as 'no satisfactory development exists at
present'."
Source:
http://en.wikipedia.org/wiki/Ultrafinitism
Herb
P.S.
Personally, finitism seems to be a fine (philosophical) position to me. I
never could figure out reasonable arguments for an ultrafinitistic position
(concerning math). My personal mantra: Math is no empirical science, and a
mathematical theory is no physical theory, it does not deal with (our
physical) "reality"; hence physical restrictions are of no interest
concerning a mathematical theory. |
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