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Topic: Answer to Dik T. Winter
Replies: 441   Last Post: Feb 5, 2013 6:25 AM

 Messages: [ Previous | Next ]
 Virgil Posts: 311 Registered: 5/26/09
Re: Answer to Dik T. Winter
Posted: Jun 3, 2009 4:01 PM

In article
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 3 Jun., 04:25, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > In article
> > <mueck...@rz.fh-augsburg.de> writes:

> > > On 29 Mai, 03:58, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote:
> > ...
> > > > > > > > > The problem boils down to the following:
> > > > > > > > >
> > > > > > > > > En Am: m =< n <==> Am En m =< n [*]
> > > > > > > > > En Am: m =< n ==> Am En m =< n [**]

> > ...
> > > > > > > I said: For complete linear sets [*] is true.
> > > > > >
> > > > > > Not in the article to which I responded.

> > > > >
> > > > > But frequently I made use of what you call quatifier exchange and
> > > > > what
> > > > > is allowed in case of complete linear sets.

> > > >
> > > > You think so, but you have to prove that it is valid for infinite
> > > > complete
> > > > linear sets. Note that "classical logic is obtained from finite
> > > > sets".
> > > > Nowhere in that quote the word linear is mentioned.

> > >
> > > Nowhere in that quote the word union in mentioned. Nevertheless the
> > > logical rules of unions are obtained from unions of finite sets.

> >
> > What is the relevance of this?

>
> The word union and the word linear are not mentioned in the quote.
> Nevertheless the due logica rules were obtained from unions of finite
> sets and linear finite sets.

While they may be suggested by and be compatible with unions of finite
sets, definitions of union of a set of sets in any set theory
(excluding WM's) do not restricts sets, or their unions, to being finite
sets.
>
> >
> > > The
> > > logical rules of linear sets are obtained from finite linear sets.

> >
> > Aha, here we get to the heart of the matter. You do not believe in
> > infinite
> > sets (or infinite unions or infinite sets of finite linear sets). That is
> > possible, of course, but does not rule out theories in which those things
> > do exist.

>
> But it does rule out theories which are contradicted by the
> fundamental logical rules.

WM's notion of what contradicts "fundamental logical rules" conflicts
with logic's notion of what contradicts logical rules.

> And one of these rules is that a complete
> linear set has a last element.

A finite ordered set need not have a last element, so WM os wrong again.

> No. I use the fact that for complete linear sets always both
> implications are true :

But unless "completeness" requires "finiteness" it is false, and
nothing outside of WM's perverted vision of set theories requires
finiteness.
>
> If you disagree, then you should come up with a finite linear set for
> which only one implication is true.

(En in S) (Am in S) : m =< n <==> Am En m =< n
> >
> > > > > > I said that for the case involved you have to
> > > > > > *prove* that it is true, because it is not generally true.

> > > > >
> > > > > It is generally true for complete linear sets. You have to prove
> > > > > that
> > > > > it is not.

> > > >
> > > > It is not true for the infinite set of naturals.

> > >
> > > That is your claim. It is justified for potential infinity. It is
> > > wrong for complete sets.

> >
> > Now you are using words that are again completely incomprehensible. You
> > have
> > still failed to give a definition of "potential infinity" that is valid
> > within
> > ZF. Moreover, you have not proven (within ZF) that the statement is wrong.

>
> ZF uses potential infinity whenever the validity of
> En Am: m =< n <== Am En m =< n [***]
> for linear sets is denied.
>
> ZF claims that this denial is correct for complete linear infinite
> sets, but this is a wrong claim, as we can obtain from logic.

> >
> > > > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > > > {1, ..., n}.
> > > > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial,
> > > > take
> > > > n = m + 1.
> > > > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > > > false, take m = n + 1.
> > > > Which part of this proof is wrong?

> > >
> > > The proof is correct for potential infinity. The proof is incorrect
> > > for actual infinity.

> >
> > Can you provide me with definitions within ZF that shows the difference?
> >

> > > In that latter case you need not take an n that
> > > is surpassed by m. Why don't you start with an n that has no greater
> > > m?

> >
> > Because there is no such n. Remember: the set of natural numbers has no
> > largest element in ZF.

>
> That is the logic of potential infinity, i.e., of incomplete sets.

There is no logic of potentially infinite sets, as they are self

> In Cantor¹s diagonal argument you can use the same logic : There is no
> last line, therefore there is always a line beyond the checked lines.
> But there you don¹t.

Then WM does not understand that argument, since Cantor does not require
any constraints on a list other than that its members be binary
sequences, including whether or not there is a last "line".

> > Oh. I think that the term "classical logic" has changed a bit since the
> > last time you looked at it. And, if you refer to Weyl's quote, he stated
> > that classical logic was *derived* from the logic on finite sets, not that
> > it was *identical* to logic on finite sets.

>
> When it is derived from logic of finite sets, then it is not the
> reverse of the logic of finite sets. But that is claimed in ZF.

That may be how WM misinterpretes ZF, but no one else can be forced to
accept WM's misinterpretation, and no one else does.
> >
> > > > > I did never claim that quantifier exchange is allowed in case of
> > > > > non-
> > > > > linear sets, like cyclic sets as, for instance, your dice. That
> > > > > would
> > > > > be nonsense. A simple example: Every country has a country that
> > > > > lies
> > > > > west of it. But there is no country that lies west of all
> > > > > countries.

> > > >
> > > > But as Weyl did not include "linear" in his words, how can that quote
> > > > support your claim?

> > >
> > > There are many finite sets with many special properties that follow
> > > from classical logic. One of them is that a complete linear set has a
> > > lst element.

> >
> > Not "a complete linear set". But "a complete finite linear set". Why do
> > you drop the word "finite" in the second sentence? To obfuscate?

>
> The logic is derived from finite complete linear sets.

"Derived from" does not mean "identical to", and the set theory of ZF,
among others, is compatible with those set theories of finite sets which
do not impose finiteness.

> If logic is to
> be applied to infinite complete linear sets, then we cannot change it
> to the opposite. Either those sets obey that logic or they do not
> exist in a science that is subject to the application of logic.
>
> In fact thoses sets contradict their own existence under the
> government of logic.

That "government" does not rule anywhere except in WM's weird world of
MathuUnrealism.

If WM's arguments could be proved, he would long since have gathered
supporters, but he has not.
> >
> > > You drop the completeness condition in certain cases but you assume it
> > > in case of Cantor's proof. That is cheating.

> >
> > You again misunderstand the proof completely. There is an assumption that
> > a complete list is provided and that is proven false.

>
> Small wonder. There cannot be a complete list, because the existence
> of a complete infinite linear set like N contradicts logic.

Actually, the Cantor argument does not assume or require a complete
list, it allows ANY list, all of whose members are binary sequences and
then shows that that list must be incomplete in the sense of omitting at
least one potential member.

So WM does not even understand the very argument he is vainly trying to
oppose.

> >
> > What in the world is a "potentially infinite element"? And, in ZF all sets
> > are static, there is no place for non-static sets.

> "
> That is a blatant lie. Every static linear set has a last element.

Not necessarily in ZF. From what axiom of combination of axioms does WM
claim to derive his "Every static linear set has a last element"?

Unless he can do so, he lies.

> If you say you cannot choose the last elemement because there is none,
> then you apply potentially infinite sets.

Nonsense! There is no provision for potentially infinite sets in ZF.
If WM disputes this, he must show which definitions and/or axioms
produce this provision.

> Then for every element that
> you choose there is a larger one. If it has been there all time, why

For which element does WM argue there is no larger one?
Unless WM can tell us how to find "this one", we will continue to hold
that his his "this one", with no larger one, does not exist.

ZF says that for EVERY natural there is a successor natural.
> >
> > > > (1) define FISON(n) be the set of naturals from 1 to n, that is:
> > > > {1, ..., n}.
> > > > (2) A{m in N} E(n in N} such that FISON(m) subset FISON(n), trivial,
> > > > take
> > > > n = m + 1.
> > > > (3) E{n in N} A{m in N} such that FISON(m) subset FISON(n), trivially
> > > > false, take m = n + 1.

> >
> > > > Strange, I give above a proof that it does not hold. I did not use
> > > > "actual infinity" nor "potentially infinity"

> > >
> > > That is the point! You use the absence of element m when you choose n
> > > = m - 1.

> >
> > Where in the proof do I use an element m when I chose n = m - 1? In the
> > proof
> > I chose only elements that *follow* given elements, as is assured by the
> > axiom of infinity.

>
> That is assured by the axiom of potential infinity.

What does "the axiom of potentail infinity"

> say? If all elemeents
> were there, why then do you have to select one that is diminishingly
> small compared to most ?

Because every one of them is "diminishingly small compared to most".

WM keeps asking us to to select things which do not exist while
ignoring things which do.

> Show me a finite linear set that does not obey [*]. Or try to explain
> why in your opinion this law must be reversed for infinite sets which
> are actually existing, i.e., every elemenmt of them can be chosen for
> comparison with others.

No one can show anything to anyone as invincibly ignorant as WM.

--
Virgil

Date Subject Author
5/27/09 mueckenh@rz.fh-augsburg.de
5/27/09 Dik T. Winter
5/27/09 mueckenh@rz.fh-augsburg.de
5/27/09 mueckenh@rz.fh-augsburg.de
5/27/09 Virgil
5/27/09 mueckenh@rz.fh-augsburg.de
5/27/09 Virgil
5/27/09 Virgil
5/28/09 mueckenh@rz.fh-augsburg.de
5/28/09 Virgil
5/28/09 Dik T. Winter
5/28/09 G. Frege
5/28/09 Jesse F. Hughes
5/29/09 Dik T. Winter
5/29/09 G. Frege
5/29/09 Dik T. Winter
5/30/09 G. Frege
5/30/09 Jesse F. Hughes
6/2/09 Dik T. Winter
5/30/09 Jesse F. Hughes
6/1/09 mueckenh@rz.fh-augsburg.de
6/1/09 Jesse F. Hughes
6/1/09 Virgil
6/2/09 george
6/2/09 Denis Feldmann
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6/11/09 mueckenh@rz.fh-augsburg.de
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