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Topic: § 488 Who said this?
Replies: 187   Last Post: May 7, 2014 10:06 AM

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 Virgil Posts: 8,833 Registered: 1/6/11
Re: � 488 Who said this?
Posted: May 3, 2014 2:55 PM

mueckenh@rz.fh-augsburg.de wrote:

> On Friday, 2 May 2014 22:55:00 UTC+2, Virgil wrote:
>
>

> > Only because they cannot INDIVIDUALLY constructed or defined, but they
> > can be collectively defined

>
> If you cannot individually address a number, then it does not exist in
> mathematics understood as dialogue.

That it may be allowed in WM's wild weird world of WMytheology has
nothing to do with whether it can exist in mathetamics, which,
fortunately for mathematics, is quite freet of WM's corruptions.

Then the collectively defined set exists,

> but it has no elements.

There is only one such set which has no elements. the empty set, and the
number of such sets is easily counted, and is one. But in ZF and ZFC and
any number of other set theories, for every set theereis a power set of
strictly greater cardinality.

> It does not obey the ZFC axioms according to which
> the elements must be ditinguishable.

Which of the other 8 ZFC axioms cited at
http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#8._Axiom
_of_power_set
does WM claim the power set axiom violates?

And how?

Mere wild claims, without specifics and without proofs is WM's standard
modus operandi, but is a method that convinces no one but himself.

The Axiom of Power Set of ZF and ZFC states that for any set x, there is
a set y that contains every subset of x.

And it is easy to prove that no set can surject onto its own power set.

> In short, it is not a set in ZFC. Or
> even shorter, it is not a set.

WM can only speak for what he alows inside his wild weird world of
WMytheology, since there are proofsmany of which have been posted here,
that what WM claims must hold in other sysems, like ZF and ZFC are
false.
>
>

> > one can define sets which contain
> > undefineable members

> No.

Yes! If |N is defineable in ZF/ZFC, snd itprovably is, then 2^|N is
also, and provably is, and 2^|N is provably uncountable in ZF?ZFC.
That WM does not like it or accept it is his problem, not ours.

> Further, Pair, subset, sum set: All these axioms cannot be
> satisfied.

Claimed, but not proved, and not provable outside of WM's wild weird
world of WMytheology.
> >
> > Consider D as the set of all finite definitions and 2^D as the set of
> > all subsets of D.
> >

> I do.
> >
> > Does WM cLaim that D MUST contains a unique different member for each
> > member of 2^D?

>
> Not in actual infinity.

> >
> >
> >
> > That would require a surjection f: D --> 2^D .

>
> Yes.

> >
> >
> >
> > But for ANY function f from D to 2^D , S ={d in D: d not in f(d)} is a
> > member of 2^D which is not of form f(d).

>
> So you have proved the idea wrong that infinite bijections have any meaning.

Not at all, I have merely proved that a certain form of mapping (from a
set to its power set) CAN be injective, but CANNOT be surjective.

> > Thus no f:D -> 2^D can be surjective and thus there are "more" members
> > in 2^D that in D.

>
> There are more definable members.

Not necessarily, but there are certainly always more members if they are
not required to be individually defineable.
>
> Such results in mathematics are called contradictions.

They re only called that well outside of mathematics in such dismal
places as WM's wild weird world of WMytheology,

> > Thus there cannot be any such surjection, and must alwasys be
> > undefineable objects in any system having a set of definitions that is in
> > any way limited.

>
> You have not shown an undefinable object.

But I have shown the necessity for them.

And in mathematics, one can prove the existence of something without
necessarily providing any concrete example of it or individual
definition of it. Like objects that must exist as members of
well-defined sets but can have no individual definitions.

> You have shown that the definable objects are uncountable.

Not at all. If there are only countably many definitions, then clearly
the number of INDIVIDUALLY defined objects must also be countable, but t
objects that only exist by being members of a well-defined set need not
all be all individually defined.

> I have shown that according to the standards of set
> theory, the definable objects are countable. You cannot reject that

I do not reject that claim if WM's "definable" means
individually-defineable, but WM ignores collective definitions which
defines only the set of which the individuals are members without
actually defining each of those individual members individually.

The definition of power sets is an example of defining a set without
providing any individual definitions for each individual member of the
resulting set.

For finite sets, one can, at least in principle, provide additional
definitions for each member of their power sets, but for actually
infinite sets, there are too many members of their power sets to do this
even in principle.
>
> Transfinite set theory has been proven self contradictory.

Not outside of WM's wild weird world of WMytheology, it hasn't.

As of today, there is no proof that ZF or ZFC or NBG or any of many
assumptions that those theories do not make.

And WM lies when he claims otherwise.
--

Date Subject Author
4/29/14 mueckenh@rz.fh-augsburg.de
4/29/14 Virgil
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4/29/14 Virgil
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4/29/14 Virgil
4/29/14 FredJeffries@gmail.com
4/29/14 Brian Q. Hutchings
4/29/14 Virgil
4/29/14 Brian Q. Hutchings
4/30/14 mueckenh@rz.fh-augsburg.de
4/30/14 Virgil
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4/30/14 mueckenh@rz.fh-augsburg.de
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4/30/14 Virgil
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