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Cantor's argument and the Potential Infinite.
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Zuhair
Nov 16, 2012, 2:49:51 AM11/16/12
to
I'll here present my version of potential infinity, which intends to
capture that concept, and prove that Cantor's diagonal argument is
applicable to that context also. So it doesn't necessitate a completed
actual infinity interpretation. However I'll also show another kind of
potential infinity scenario, which I call the strict potential
infinity, under grounds of which Cantor's diagonal argument cease
working, and I'll discuss why that strict form of potential infinity
is defective.
Generally speaking the argument of potential infinity says that NO
infinite set exists in the sense of a complete actual infinite set, so
the set N of all naturals is never completed, it is in a continual
state of becoming, and all completed sets of naturals are finite. So
if we denote {x| x is a natural} to be an object that stands for the
"potential" of infinitude of naturals, then we'll have
For all x. x is a completed set of naturals -> x is a finite proper
subset of {x| x is a natural}.
Notice here that {x| x is a natural} do not mean an actual completed
set of all naturals, it is just an object that uniquely stands for the
predicate "natural". It is neither finite nor infinite since those
would be terms defined only for completed sets, and {x| x is a
natural} is not a completed set, it is viewed to be in continual
becoming; Lets call such objects Potential sets.
Under those grounds it is said that Cantor's argument of
uncountability of the reals vanishes.
But this is NOT true.
We still can characterize Cardinality in this setting.
Two potential sets are said to have equal cardinality iff there is a
potential injection from one to the other at each direction.
Example: the potential sets N and E
There is a potential injective F map from N to E that is {(n,x)| x=2n
& n is a natural & x is even}
Also in the other direction you have a potential injective map G that
is
{(x,n)| n=x & x is even & n is natural}
The idea is that one cannot demonstrate any element of N that is not
in the potential domain of F. Since that domain is clearly N itself.
But can we have a similar potential bi-injective mapping between N and
R?
The answer is NO. Cantor's diagonal argument is also applicable
here!!!
Say there can be a Potential injection from R to N, lets call it I
Lets take the converse of I, denoted it as I^-1, which will be an
injection from the range of I to R. Now define a diagonal in a
potential manner by changing the i_th member of the digit sequence
representing the real in R that the i_th natural in the domain of I^-1
is coupled to, where the ordering is the ordinary natural order which
of course can be potentially defined. Now take the Potential
collection of all changed elements, and we'll have a potential
diagonal that is not in the potential range of I^-1, i.e. not in R. A
contradiction.
So Cantor's diagonal is applicable to potential infinity context!
The next scenario that I'll discuss is the STRICT potential infinity
scenario:
Here in this scenario, there is NO representation of any object that
can stand uniquely for a predicate that is potentially infinite, so
the predicate "natural number" is of course a potentially infinite
predicate since every finite set of naturals is not a completed set of
all naturals but yet this scenario simply stipulates that there is no
object that can stand uniquely for such predicate. So sets (which are
objects) only stand uniquely for finite predicates, there is no actual
infinite set, and there is also no potential infinite set like that
described in first scenario. There are only "PREDICATES" that
qualifies to be potentially infinite, however those are further
stipulated to be only described by formulas which are parameter free,
which of course known to be countable in number. So at the end we
clearly have no grounds for any proof of uncountability.
The problem with this scenario is that it is too restrictive, a super-
task for example cannot be represented by it, it is actually not
faithful to the concept of potential infinity itself, since informally
a potentially infinite predicate yields a potentially infinite
collection of objects that stands for that predicate, which serve as a
potential extension of that predicate. Now to go and shun that object
from existence like that makes one wonder about the potential those
predicates are all about, it is a potential in vain, from one aspect
those predicates range over objects in a continual manner, and from
the other aspect we don't see that continual extension, it simply
vanished, just like that? its like continually blowing into
nowhere???
Actually to me a more faithful argument would be to call the above
scenario "finitism", this would suit it better, which is though
restrictive in the above manner, but yet it is faithful to its
original stance, albeit not fully so to speak.
The real faithful scenario is actually ultra-finistim which simply
says that there are no infinite extensions, nor there is anything in
continual being. Everything is finite and ends up by some large
finite, and that's it. So this doesn't only shun potential infinite
collections, it also shuns MOST of finite numbers from existence, and
only accept the few handy ones that we can experience with and can
communicate, those that our machines and us can reach with the
strongest abbreviation notions we can have (which is of course also
finite).
Of course under that scenario which is claiming to be a reality
scenario, I say under this scenario just mentioning the matter of
infinite whether potentially or complete is deemed as a fantasy, and
any thought about it relates to speech about fantasies whether that
argument was consistent in form or not, it is not significant since it
is not about the real world we are living in, that's how matters are
seen from this perspective.
However the subject of whether ultra-finitism is true or not, is
actually another subject that is not about potential infinity. What I
wanted to say is that concepts like the Actual infinite or even the
potential infinite that I've presented at the head post are more
faithful concepts to their informal background, than the argument of
strict potential infinity that from one angle attracts those who wish
to speak about the infinite in a potential manner, but yet on the
other hand stipulate a restriction that is not faithful to what it
began with in the first place.
So in nutshell even under potential infinity background, still
Cantor's diagonals can be constructed and works to show that the
potential set R of reals is still having potentially more elements
than the potential set N of naturals.
Zuhair
capture that concept, and prove that Cantor's diagonal argument is
applicable to that context also. So it doesn't necessitate a completed
actual infinity interpretation. However I'll also show another kind of
potential infinity scenario, which I call the strict potential
infinity, under grounds of which Cantor's diagonal argument cease
working, and I'll discuss why that strict form of potential infinity
is defective.
Generally speaking the argument of potential infinity says that NO
infinite set exists in the sense of a complete actual infinite set, so
the set N of all naturals is never completed, it is in a continual
state of becoming, and all completed sets of naturals are finite. So
if we denote {x| x is a natural} to be an object that stands for the
"potential" of infinitude of naturals, then we'll have
For all x. x is a completed set of naturals -> x is a finite proper
subset of {x| x is a natural}.
Notice here that {x| x is a natural} do not mean an actual completed
set of all naturals, it is just an object that uniquely stands for the
predicate "natural". It is neither finite nor infinite since those
would be terms defined only for completed sets, and {x| x is a
natural} is not a completed set, it is viewed to be in continual
becoming; Lets call such objects Potential sets.
Under those grounds it is said that Cantor's argument of
uncountability of the reals vanishes.
But this is NOT true.
We still can characterize Cardinality in this setting.
Two potential sets are said to have equal cardinality iff there is a
potential injection from one to the other at each direction.
Example: the potential sets N and E
There is a potential injective F map from N to E that is {(n,x)| x=2n
& n is a natural & x is even}
Also in the other direction you have a potential injective map G that
is
{(x,n)| n=x & x is even & n is natural}
The idea is that one cannot demonstrate any element of N that is not
in the potential domain of F. Since that domain is clearly N itself.
But can we have a similar potential bi-injective mapping between N and
R?
The answer is NO. Cantor's diagonal argument is also applicable
here!!!
Say there can be a Potential injection from R to N, lets call it I
Lets take the converse of I, denoted it as I^-1, which will be an
injection from the range of I to R. Now define a diagonal in a
potential manner by changing the i_th member of the digit sequence
representing the real in R that the i_th natural in the domain of I^-1
is coupled to, where the ordering is the ordinary natural order which
of course can be potentially defined. Now take the Potential
collection of all changed elements, and we'll have a potential
diagonal that is not in the potential range of I^-1, i.e. not in R. A
contradiction.
So Cantor's diagonal is applicable to potential infinity context!
The next scenario that I'll discuss is the STRICT potential infinity
scenario:
Here in this scenario, there is NO representation of any object that
can stand uniquely for a predicate that is potentially infinite, so
the predicate "natural number" is of course a potentially infinite
predicate since every finite set of naturals is not a completed set of
all naturals but yet this scenario simply stipulates that there is no
object that can stand uniquely for such predicate. So sets (which are
objects) only stand uniquely for finite predicates, there is no actual
infinite set, and there is also no potential infinite set like that
described in first scenario. There are only "PREDICATES" that
qualifies to be potentially infinite, however those are further
stipulated to be only described by formulas which are parameter free,
which of course known to be countable in number. So at the end we
clearly have no grounds for any proof of uncountability.
The problem with this scenario is that it is too restrictive, a super-
task for example cannot be represented by it, it is actually not
faithful to the concept of potential infinity itself, since informally
a potentially infinite predicate yields a potentially infinite
collection of objects that stands for that predicate, which serve as a
potential extension of that predicate. Now to go and shun that object
from existence like that makes one wonder about the potential those
predicates are all about, it is a potential in vain, from one aspect
those predicates range over objects in a continual manner, and from
the other aspect we don't see that continual extension, it simply
vanished, just like that? its like continually blowing into
nowhere???
Actually to me a more faithful argument would be to call the above
scenario "finitism", this would suit it better, which is though
restrictive in the above manner, but yet it is faithful to its
original stance, albeit not fully so to speak.
The real faithful scenario is actually ultra-finistim which simply
says that there are no infinite extensions, nor there is anything in
continual being. Everything is finite and ends up by some large
finite, and that's it. So this doesn't only shun potential infinite
collections, it also shuns MOST of finite numbers from existence, and
only accept the few handy ones that we can experience with and can
communicate, those that our machines and us can reach with the
strongest abbreviation notions we can have (which is of course also
finite).
Of course under that scenario which is claiming to be a reality
scenario, I say under this scenario just mentioning the matter of
infinite whether potentially or complete is deemed as a fantasy, and
any thought about it relates to speech about fantasies whether that
argument was consistent in form or not, it is not significant since it
is not about the real world we are living in, that's how matters are
seen from this perspective.
However the subject of whether ultra-finitism is true or not, is
actually another subject that is not about potential infinity. What I
wanted to say is that concepts like the Actual infinite or even the
potential infinite that I've presented at the head post are more
faithful concepts to their informal background, than the argument of
strict potential infinity that from one angle attracts those who wish
to speak about the infinite in a potential manner, but yet on the
other hand stipulate a restriction that is not faithful to what it
began with in the first place.
So in nutshell even under potential infinity background, still
Cantor's diagonals can be constructed and works to show that the
potential set R of reals is still having potentially more elements
than the potential set N of naturals.
Zuhair
LudovicoVan
Nov 16, 2012, 3:36:21 AM11/16/12
to
"Zuhair" <zalj...@gmail.com> wrote in message
news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
> We still can characterize Cardinality in this setting.
And you keep missing the point, as the various objections of course involve
that the standard definition of cardinality for infinite sets is wrong!
> So Cantor's diagonal is applicable to potential infinity context!
Cantor's arguments are *only* applied to potentially infinite sets, in fact
in standard set theory there is no such thing as actual infinity at all.
Please get your head out of your ass and read and try to understand what you
are rebutting before you actually get to do it.
-LV
news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
> We still can characterize Cardinality in this setting.
that the standard definition of cardinality for infinite sets is wrong!
> So Cantor's diagonal is applicable to potential infinity context!
in standard set theory there is no such thing as actual infinity at all.
Please get your head out of your ass and read and try to understand what you
are rebutting before you actually get to do it.
-LV
Uirgil
Nov 16, 2012, 4:05:52 AM11/16/12
to
In article <k84tuf$t03$1...@dont-email.me>,
equally likely that the various objections are the things that are
wrong.
only allowed but actually required to exist, and no one yet has been
able to show that ZFC is not a perfectly sound set theory.
than those he is criticizing.
"LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zalj...@gmail.com> wrote in message
> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>
> > We still can characterize Cardinality in this setting.
>
> And you keep missing the point, as the various objections of course involve
> that the standard definition of cardinality for infinite sets is wrong!
But as far as any valid arguments are concerned, it appears AT LEAST
> "Zuhair" <zalj...@gmail.com> wrote in message
> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>
> > We still can characterize Cardinality in this setting.
>
> And you keep missing the point, as the various objections of course involve
> that the standard definition of cardinality for infinite sets is wrong!
equally likely that the various objections are the things that are
wrong.
>
> > So Cantor's diagonal is applicable to potential infinity context!
>
> Cantor's arguments are *only* applied to potentially infinite sets, in fact
> in standard set theory there is no such thing as actual infinity at all.
ZFC offers a standard set theory in which actually infinite sets are not
> > So Cantor's diagonal is applicable to potential infinity context!
>
> Cantor's arguments are *only* applied to potentially infinite sets, in fact
> in standard set theory there is no such thing as actual infinity at all.
only allowed but actually required to exist, and no one yet has been
able to show that ZFC is not a perfectly sound set theory.
>
> Please get your head out of your ass and read and try to understand what you
> are rebutting before you actually get to do it.
AS far as head-in-ass-itis, LV appears to have a far worse case of it
> Please get your head out of your ass and read and try to understand what you
> are rebutting before you actually get to do it.
than those he is criticizing.
LudovicoVan
Nov 16, 2012, 4:20:43 AM11/16/12
to
"Uirgil" <uir...@uirgil.ur> wrote in message
news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
rest is at best OT.
>> > So Cantor's diagonal is applicable to potential infinity context!
>>
>> Cantor's arguments are *only* applied to potentially infinite sets, in
>> fact
>> in standard set theory there is no such thing as actual infinity at all.
>
> ZFC offers a standard set theory in which actually infinite sets are not
> only allowed but actually required to exist, and no one yet has been
> able to show that ZFC is not a perfectly sound set theory.
That is only because you are so incoherent as to insist to call N an actual
infinity.
>> Please get your head out of your ass and read and try to understand what
>> you
>> are rebutting before you actually get to do it.
>
> AS far as head-in-ass-itis, LV appears to have a far worse case of it
> than those he is criticizing.
Sure, keep spamming and all that.
-LV
news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> In article <k84tuf$t03$1...@dont-email.me>,
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Zuhair" <zalj...@gmail.com> wrote in message
>> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>>
>> > We still can characterize Cardinality in this setting.
>>
>> And you keep missing the point, as the various objections of course
>> involve
>> that the standard definition of cardinality for infinite sets is wrong!
>
> But as far as any valid arguments are concerned, it appears AT LEAST
> equally likely that the various objections are the things that are
> wrong.
If an argument is wrong, you should show that it is so or just pass, the
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Zuhair" <zalj...@gmail.com> wrote in message
>> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>>
>> > We still can characterize Cardinality in this setting.
>>
>> And you keep missing the point, as the various objections of course
>> involve
>> that the standard definition of cardinality for infinite sets is wrong!
>
> But as far as any valid arguments are concerned, it appears AT LEAST
> equally likely that the various objections are the things that are
> wrong.
rest is at best OT.
>> > So Cantor's diagonal is applicable to potential infinity context!
>>
>> Cantor's arguments are *only* applied to potentially infinite sets, in
>> fact
>> in standard set theory there is no such thing as actual infinity at all.
>
> ZFC offers a standard set theory in which actually infinite sets are not
> only allowed but actually required to exist, and no one yet has been
> able to show that ZFC is not a perfectly sound set theory.
infinity.
>> Please get your head out of your ass and read and try to understand what
>> you
>> are rebutting before you actually get to do it.
>
> AS far as head-in-ass-itis, LV appears to have a far worse case of it
> than those he is criticizing.
-LV
Uirgil
Nov 16, 2012, 4:31:02 AM11/16/12
to
In article <k850hm$a03$2...@dont-email.me>,
you do not have a convincing counter-proof but your attempts to
disprove Cantor have so far all fallen flat.
ZFC is internally inconsistent, which no one has yet done, we have
actual infinities in ZFC.
> > than those you are criticizing.
Cantor issue.
"LudovicoVan" <ju...@diegidio.name> wrote:
> "Uirgil" <uir...@uirgil.ur> wrote in message
> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> > In article <k84tuf$t03$1...@dont-email.me>,
> > "LudovicoVan" <ju...@diegidio.name> wrote:
> >> "Zuhair" <zalj...@gmail.com> wrote in message
> >> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
> >>
> >> > We still can characterize Cardinality in this setting.
> >>
> >> And you keep missing the point, as the various objections of course
> >> involve
> >> that the standard definition of cardinality for infinite sets is wrong!
> >
> > But as far as any valid arguments are concerned, it appears AT LEAST
> > equally likely that the various objections are the things that are
> > wrong.
>
> If an argument is wrong, you should show that it is so or just pass, the
> rest is at best OT.
You are the one claiming that Cantor is wrong, but he has a proof and
> "Uirgil" <uir...@uirgil.ur> wrote in message
> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> > In article <k84tuf$t03$1...@dont-email.me>,
> > "LudovicoVan" <ju...@diegidio.name> wrote:
> >> "Zuhair" <zalj...@gmail.com> wrote in message
> >> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
> >>
> >> > We still can characterize Cardinality in this setting.
> >>
> >> And you keep missing the point, as the various objections of course
> >> involve
> >> that the standard definition of cardinality for infinite sets is wrong!
> >
> > But as far as any valid arguments are concerned, it appears AT LEAST
> > equally likely that the various objections are the things that are
> > wrong.
>
> If an argument is wrong, you should show that it is so or just pass, the
> rest is at best OT.
you do not have a convincing counter-proof but your attempts to
disprove Cantor have so far all fallen flat.
>
> >> > So Cantor's diagonal is applicable to potential infinity context!
> >>
> >> Cantor's arguments are *only* applied to potentially infinite sets, in
> >> fact
> >> in standard set theory there is no such thing as actual infinity at all.
> >
> > ZFC offers a standard set theory in which actually infinite sets are not
> > only allowed but actually required to exist, and no one yet has been
> > able to show that ZFC is not a perfectly sound set theory.
>
> That is only because you are so incoherent as to insist to call N an actual
> infinity.
In ZFC, the N is an actually infinite set. So until you can show that
> >> > So Cantor's diagonal is applicable to potential infinity context!
> >>
> >> Cantor's arguments are *only* applied to potentially infinite sets, in
> >> fact
> >> in standard set theory there is no such thing as actual infinity at all.
> >
> > ZFC offers a standard set theory in which actually infinite sets are not
> > only allowed but actually required to exist, and no one yet has been
> > able to show that ZFC is not a perfectly sound set theory.
>
> That is only because you are so incoherent as to insist to call N an actual
> infinity.
ZFC is internally inconsistent, which no one has yet done, we have
actual infinities in ZFC.
>
> >> Please get your head out of your ass and read and try to understand what
> >> you
> >> are rebutting before you actually get to do it.
> >
> > AS far as head-in-ass-itis, LV appears you have a far worse case of it
> >> Please get your head out of your ass and read and try to understand what
> >> you
> >> are rebutting before you actually get to do it.
> >
> > than those you are criticizing.
>
> Sure, keep spamming and all that.
I notice in your own spamming a lack of any arguments relevant to the
> Sure, keep spamming and all that.
Cantor issue.
LudovicoVan
Nov 16, 2012, 4:40:45 AM11/16/12
to
"Uirgil" <uir...@uirgil.ur> wrote in message
news:uirgil-8D50A0....@BIGNEWS.USENETMONSTER.COM...
> In article <k850hm$a03$2...@dont-email.me>,
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Uirgil" <uir...@uirgil.ur> wrote in message
>> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
<snipped>
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Uirgil" <uir...@uirgil.ur> wrote in message
>> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
>> > ZFC offers a standard set theory in which actually infinite sets are
>> > not
>> > only allowed but actually required to exist, and no one yet has been
>> > able to show that ZFC is not a perfectly sound set theory.
>>
>> That is only because you are so incoherent as to insist to call N an
>> actual
>> infinity.
>
> In ZFC, the N is an actually infinite set. So until you can show that
> ZFC is internally inconsistent, which no one has yet done, we have
> actual infinities in ZFC.
although informal as it needs be, N is an "actual infinity" in ZFC?
-LV
Zuhair
Nov 16, 2012, 5:34:16 AM11/16/12
to
On Nov 16, 11:36 am, "LudovicoVan" <ju...@diegidio.name> wrote:
> "Zuhair" <zaljo...@gmail.com> wrote in message
Good advice for you actually, since you don't know what you are
speaking about. So just try to read what is written here, and if you
don't understand what is written, or you have some problem with it,
then just try to ask politely about it, so that I or others who are
more informed that you can explain matters to you. Anyhow standard set
theory "ZFC" is of course not limiting itself to the potential
scenario, not even to the one I've presented here, that's why it
accepts Impredicative definitions, as well as non well founded
versions of it, the reason is that it doesn't have a problem with
considering the possibility that all sets in the universe of discourse
are GIVEN beforehand, and Godel's have stated that there is nothing
wrong with this assumption, so there is no problem with considering
that the set N is already Given, i.e. it is there beforehand with all
its elements, i.e. N is a completed actual infinite set, in standard
set theory understanding of N is not limited to the potential of
becoming that I've presented here. However here I showed that even if
we assume potential infinity in the sense I've presented, which is as
I showed here the most faithful to that concept itself, then still
Cantor's diagonal argument applies to it. All of what I'm saying here
is that standard set theory as customarily understood doesn't not
restrict itself to a potential infinity context, but even if so then
if we faithfully represent that concept of potentiality then Cantor's
argument can be still carried on.
Zuhair
> "Zuhair" <zaljo...@gmail.com> wrote in message
speaking about. So just try to read what is written here, and if you
don't understand what is written, or you have some problem with it,
then just try to ask politely about it, so that I or others who are
more informed that you can explain matters to you. Anyhow standard set
theory "ZFC" is of course not limiting itself to the potential
scenario, not even to the one I've presented here, that's why it
accepts Impredicative definitions, as well as non well founded
versions of it, the reason is that it doesn't have a problem with
considering the possibility that all sets in the universe of discourse
are GIVEN beforehand, and Godel's have stated that there is nothing
wrong with this assumption, so there is no problem with considering
that the set N is already Given, i.e. it is there beforehand with all
its elements, i.e. N is a completed actual infinite set, in standard
set theory understanding of N is not limited to the potential of
becoming that I've presented here. However here I showed that even if
we assume potential infinity in the sense I've presented, which is as
I showed here the most faithful to that concept itself, then still
Cantor's diagonal argument applies to it. All of what I'm saying here
is that standard set theory as customarily understood doesn't not
restrict itself to a potential infinity context, but even if so then
if we faithfully represent that concept of potentiality then Cantor's
argument can be still carried on.
Zuhair
Zuhair
Nov 16, 2012, 5:40:12 AM11/16/12
to
On Nov 16, 12:31 pm, Uirgil <uir...@uirgil.ur> wrote:
> In article <k850hm$a0...@dont-email.me>,
> > > "LudovicoVan" <ju...@diegidio.name> wrote:
> > >> "Zuhair" <zaljo...@gmail.com> wrote in message
coherently what such a trial require so that he even make a reasonable
attempt to try. The man is just ignorant that highly shouts at others
to convince himself of being not.
Empty vessels make the most noise.
Zuhair
> In article <k850hm$a0...@dont-email.me>,
>
>
>
>
>
>
>
>
>
> "LudovicoVan" <ju...@diegidio.name> wrote:
> > "Uirgil" <uir...@uirgil.ur> wrote in message
> >news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> > > In article <k84tuf$t0...@dont-email.me>,
>
>
>
>
>
>
>
>
> "LudovicoVan" <ju...@diegidio.name> wrote:
> > "Uirgil" <uir...@uirgil.ur> wrote in message
> >news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> > > "LudovicoVan" <ju...@diegidio.name> wrote:
> > >> "Zuhair" <zaljo...@gmail.com> wrote in message
> > >>news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>
> > >> > We still can characterize Cardinality in this setting.
>
> > >> And you keep missing the point, as the various objections of course
> > >> involve
> > >> that the standard definition of cardinality for infinite sets is wrong!
>
> > > But as far as any valid arguments are concerned, it appears AT LEAST
> > > equally likely that the various objections are the things that are
> > > wrong.
>
> > If an argument is wrong, you should show that it is so or just pass, the
> > rest is at best OT.
>
> You are the one claiming that Cantor is wrong, but he has a proof and
> you do not have a convincing counter-proof but your attempts to
> disprove Cantor have so far all fallen flat.
>
>
LV tried to disprove Cantor? that's funny really, can he even state
>
> > >> > We still can characterize Cardinality in this setting.
>
> > >> And you keep missing the point, as the various objections of course
> > >> involve
> > >> that the standard definition of cardinality for infinite sets is wrong!
>
> > > But as far as any valid arguments are concerned, it appears AT LEAST
> > > equally likely that the various objections are the things that are
> > > wrong.
>
> > If an argument is wrong, you should show that it is so or just pass, the
> > rest is at best OT.
>
> You are the one claiming that Cantor is wrong, but he has a proof and
> you do not have a convincing counter-proof but your attempts to
> disprove Cantor have so far all fallen flat.
>
>
coherently what such a trial require so that he even make a reasonable
attempt to try. The man is just ignorant that highly shouts at others
to convince himself of being not.
Empty vessels make the most noise.
Zuhair
LudovicoVan
Nov 16, 2012, 6:07:54 AM11/16/12
to
"Zuhair" <zalj...@gmail.com> wrote in message
news:1b9b0f3e-bb0e-4309...@r5g2000yqo.googlegroups.com...
-LV
LudovicoVan
Nov 16, 2012, 6:10:25 AM11/16/12
to
"Zuhair" <zalj...@gmail.com> wrote in message
news:d84ff6d8-34ac-405f...@3g2000yqn.googlegroups.com...
> On Nov 16, 11:36 am, "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Zuhair" <zaljo...@gmail.com> wrote in message
>> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>>
>> > We still can characterize Cardinality in this setting.
>>
>> And you keep missing the point, as the various objections of course
>> involve
>> that the standard definition of cardinality for infinite sets is wrong!
>>
>> > So Cantor's diagonal is applicable to potential infinity context!
>>
>> Cantor's arguments are *only* applied to potentially infinite sets, in
>> fact
>> in standard set theory there is no such thing as actual infinity at all.
>>
>> Please get your head out of your ass and read and try to understand what
>> you
>> are rebutting before you actually get to do it.
>
>> "Zuhair" <zaljo...@gmail.com> wrote in message
>> news:5e28971d-adb1-49ae...@o8g2000yqh.googlegroups.com...
>>
>> > We still can characterize Cardinality in this setting.
>>
>> And you keep missing the point, as the various objections of course
>> involve
>> that the standard definition of cardinality for infinite sets is wrong!
>>
>> > So Cantor's diagonal is applicable to potential infinity context!
>>
>> Cantor's arguments are *only* applied to potentially infinite sets, in
>> fact
>> in standard set theory there is no such thing as actual infinity at all.
>>
>> Please get your head out of your ass and read and try to understand what
>> you
>> are rebutting before you actually get to do it.
>
> Good advice for you actually, since you don't know what you are
> speaking about. So just try to read what is written here, and if you
> don't understand what is written, or you have some problem with it,
> then just try to ask politely about it, so that I or others who are
> more informed that you can explain matters to you. Anyhow standard set
> theory "ZFC" is of course not limiting itself to the potential
> scenario, not even to the one I've presented here, that's why it
> accepts Impredicative definitions, as well as non well founded
> versions of it, the reason is that it doesn't have a problem with
> considering the possibility that all sets in the universe of discourse
> are GIVEN beforehand, and Godel's have stated that there is nothing
> wrong with this assumption, so there is no problem with considering
> that the set N is already Given, i.e. it is there beforehand with all
> its elements, i.e. N is a completed actual infinite set, in standard
> set theory understanding of N is not limited to the potential of
> becoming that I've presented here. However here I showed that even if
> we assume potential infinity in the sense I've presented, which is as
> I showed here the most faithful to that concept itself, then still
> Cantor's diagonal argument applies to it. All of what I'm saying here
> is that standard set theory as customarily understood doesn't not
> restrict itself to a potential infinity context, but even if so then
> if we faithfully represent that concept of potentiality then Cantor's
> argument can be still carried on.
As usual, you are not even wrong.
> speaking about. So just try to read what is written here, and if you
> don't understand what is written, or you have some problem with it,
> then just try to ask politely about it, so that I or others who are
> more informed that you can explain matters to you. Anyhow standard set
> theory "ZFC" is of course not limiting itself to the potential
> scenario, not even to the one I've presented here, that's why it
> accepts Impredicative definitions, as well as non well founded
> versions of it, the reason is that it doesn't have a problem with
> considering the possibility that all sets in the universe of discourse
> are GIVEN beforehand, and Godel's have stated that there is nothing
> wrong with this assumption, so there is no problem with considering
> that the set N is already Given, i.e. it is there beforehand with all
> its elements, i.e. N is a completed actual infinite set, in standard
> set theory understanding of N is not limited to the potential of
> becoming that I've presented here. However here I showed that even if
> we assume potential infinity in the sense I've presented, which is as
> I showed here the most faithful to that concept itself, then still
> Cantor's diagonal argument applies to it. All of what I'm saying here
> is that standard set theory as customarily understood doesn't not
> restrict itself to a potential infinity context, but even if so then
> if we faithfully represent that concept of potentiality then Cantor's
> argument can be still carried on.
Keep spamming.
-LV
forbi...@gmail.com
Nov 16, 2012, 12:27:00 PM11/16/12
to
On Friday, November 16, 2012 1:20:55 AM UTC-8, LudovicoVan wrote:
> If an argument is wrong, you should show that it is so or just pass, the
> rest is at best OT.
This rests at the heart of the matter.
> If an argument is wrong, you should show that it is so or just pass, the
> rest is at best OT.
It appears not always taken to heart by those who claim it.
Maybe we are at the bounds of language where all strings of
characters are expressions in private language not interpretable
between individuals. It is the three blind men describing an
elephant. The bullying tactic of trading insults when concepts
fail shows a level of frustration when private language cannot
be communicated. It seems to me that failure of an individual
to understand leaves open the question as to the point of failure
but when most fail to comprehend then the point of failure is a
bit more conclusive. The same is true in the other direction,
that is where the majority agree there has been communication
but a few say otherwise.
Uirgil
Nov 16, 2012, 4:54:05 PM11/16/12
to
In article <k856vb$b40$2...@dont-email.me>,
The response of someone who can't find any actual errors but still
disagrees.
disagrees.
Uirgil
Nov 16, 2012, 5:03:56 PM11/16/12
to
In article <k851n8$fgs$1...@dont-email.me>,
{} is a member of N, and
If x is a member of N, so is x \/ {x}, and
N is a subset of every set S such that
{} is a member of S and
If x is a member of S, so is x \/ {x}
Such a set is provably not finite, as finiteness of a set would require
that it biject with some MEMBER of such an N, which N provably does not.
"LudovicoVan" <ju...@diegidio.name> wrote:
> "Uirgil" <uir...@uirgil.ur> wrote in message
> news:uirgil-8D50A0....@BIGNEWS.USENETMONSTER.COM...
> > In article <k850hm$a03$2...@dont-email.me>,
> > "LudovicoVan" <ju...@diegidio.name> wrote:
> >> "Uirgil" <uir...@uirgil.ur> wrote in message
> >> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> <snipped>
>
> >> > ZFC offers a standard set theory in which actually infinite sets are
> >> > not
> >> > only allowed but actually required to exist, and no one yet has been
> >> > able to show that ZFC is not a perfectly sound set theory.
> >>
> >> That is only because you are so incoherent as to insist to call N an
> >> actual
> >> infinity.
> >
> > In ZFC, the N is an actually infinite set. So until you can show that
> > ZFC is internally inconsistent, which no one has yet done, we have
> > actual infinities in ZFC.
>
> That's interesting: would you be so kind to show me how/why, technically
> although informal as it needs be, N is an "actual infinity" in ZFC?
>
> -LV
>
ZFC requires the existence of a set N such that
> "Uirgil" <uir...@uirgil.ur> wrote in message
> news:uirgil-8D50A0....@BIGNEWS.USENETMONSTER.COM...
> > In article <k850hm$a03$2...@dont-email.me>,
> > "LudovicoVan" <ju...@diegidio.name> wrote:
> >> "Uirgil" <uir...@uirgil.ur> wrote in message
> >> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
> <snipped>
>
> >> > ZFC offers a standard set theory in which actually infinite sets are
> >> > not
> >> > only allowed but actually required to exist, and no one yet has been
> >> > able to show that ZFC is not a perfectly sound set theory.
> >>
> >> That is only because you are so incoherent as to insist to call N an
> >> actual
> >> infinity.
> >
> > In ZFC, the N is an actually infinite set. So until you can show that
> > ZFC is internally inconsistent, which no one has yet done, we have
> > actual infinities in ZFC.
>
> That's interesting: would you be so kind to show me how/why, technically
> although informal as it needs be, N is an "actual infinity" in ZFC?
>
> -LV
>
{} is a member of N, and
If x is a member of N, so is x \/ {x}, and
N is a subset of every set S such that
{} is a member of S and
If x is a member of S, so is x \/ {x}
Such a set is provably not finite, as finiteness of a set would require
that it biject with some MEMBER of such an N, which N provably does not.
LudovicoVan
Nov 17, 2012, 2:24:37 AM11/17/12
to
"Uirgil" <uir...@uirgil.ur> wrote in message
news:uirgil-C5CD34....@BIGNEWS.USENETMONSTER.COM...
> In article <k851n8$fgs$1...@dont-email.me>,
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Uirgil" <uir...@uirgil.ur> wrote in message
>> news:uirgil-8D50A0....@BIGNEWS.USENETMONSTER.COM...
>> > In article <k850hm$a03$2...@dont-email.me>,
>> > "LudovicoVan" <ju...@diegidio.name> wrote:
>> >> "Uirgil" <uir...@uirgil.ur> wrote in message
>> >> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
>> <snipped>
>>
>> >> > ZFC offers a standard set theory in which actually infinite sets are
>> >> > not
>> >> > only allowed but actually required to exist, and no one yet has been
>> >> > able to show that ZFC is not a perfectly sound set theory.
>> >>
>> >> That is only because you are so incoherent as to insist to call N an
>> >> actual
>> >> infinity.
>> >
>> > In ZFC, the N is an actually infinite set. So until you can show that
>> > ZFC is internally inconsistent, which no one has yet done, we have
>> > actual infinities in ZFC.
>>
>> That's interesting: would you be so kind to show me how/why, technically
>> although informal as it needs be, N is an "actual infinity" in ZFC?
>
> "LudovicoVan" <ju...@diegidio.name> wrote:
>> "Uirgil" <uir...@uirgil.ur> wrote in message
>> news:uirgil-8D50A0....@BIGNEWS.USENETMONSTER.COM...
>> > In article <k850hm$a03$2...@dont-email.me>,
>> > "LudovicoVan" <ju...@diegidio.name> wrote:
>> >> "Uirgil" <uir...@uirgil.ur> wrote in message
>> >> news:uirgil-981B6A....@BIGNEWS.USENETMONSTER.COM...
>> <snipped>
>>
>> >> > ZFC offers a standard set theory in which actually infinite sets are
>> >> > not
>> >> > only allowed but actually required to exist, and no one yet has been
>> >> > able to show that ZFC is not a perfectly sound set theory.
>> >>
>> >> That is only because you are so incoherent as to insist to call N an
>> >> actual
>> >> infinity.
>> >
>> > In ZFC, the N is an actually infinite set. So until you can show that
>> > ZFC is internally inconsistent, which no one has yet done, we have
>> > actual infinities in ZFC.
>>
>> That's interesting: would you be so kind to show me how/why, technically
>> although informal as it needs be, N is an "actual infinity" in ZFC?
>
> ZFC requires the existence of a set N such that
> {} is a member of N, and
> If x is a member of N, so is x \/ {x}, and
> N is a subset of every set S such that
> {} is a member of S and
> If x is a member of S, so is x \/ {x}
>
> Such a set is provably not finite, as finiteness of a set would require
> that it biject with some MEMBER of such an N, which N provably does not.
Sure, N is the minimal set with 0 and closed under the successor operation.
> {} is a member of N, and
> If x is a member of N, so is x \/ {x}, and
> N is a subset of every set S such that
> {} is a member of S and
> If x is a member of S, so is x \/ {x}
>
> Such a set is provably not finite, as finiteness of a set would require
> that it biject with some MEMBER of such an N, which N provably does not.
But that remains a characterization of a *potential infinity*.
Well (not) done.
-LV
Uirgil
Nov 17, 2012, 12:42:38 PM11/17/12
to
LudovicoVan
Nov 17, 2012, 2:32:20 PM11/17/12
to
"Uirgil" <uir...@uirgil.ur> wrote in message
news:uirgil-4EE436....@BIGNEWS.USENETMONSTER.COM...
-LV
Shmuel Metz
Nov 17, 2012, 8:19:11 PM11/17/12
to
In <k84tuf$t03$1...@dont-email.me>, on 11/16/2012
but it can't be wrong.
>Cantor's arguments are *only* applied to potentially infinite sets,
>in fact in standard set theory there is no such thing as actual
>infinity at all.
As usual, you've got it backwards; in standard set theory there is no
such thing as "potential infinity", but only sets[1], some of which
are not finite.
>Please get your head out of your ass
PKB.
[1] Or sets and classes, depending on which common set theory
you consider standard.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
at 08:36 AM, "LudovicoVan" <ju...@diegidio.name> said:
>And you keep missing the point, as the various objections of course
>involve that the standard definition of cardinality for infinite
>sets is wrong!
ROTF,LMAO! A definition can be ambiguous, unconventional or vacuous,
>And you keep missing the point, as the various objections of course
>involve that the standard definition of cardinality for infinite
>sets is wrong!
but it can't be wrong.
>Cantor's arguments are *only* applied to potentially infinite sets,
>in fact in standard set theory there is no such thing as actual
>infinity at all.
such thing as "potential infinity", but only sets[1], some of which
are not finite.
>Please get your head out of your ass
[1] Or sets and classes, depending on which common set theory
you consider standard.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Shmuel Metz
Nov 17, 2012, 8:39:00 PM11/17/12
to
In <k87e41$nr0$1...@dont-email.me>, on 11/17/2012
using WMzfc instead?
at 07:24 AM, "LudovicoVan" <ju...@diegidio.name> said:
>But that remains a characterization of a *potential infinity*.
There is no such thing as "potential infinity" in ZFC. Perhaps you're
>But that remains a characterization of a *potential infinity*.
using WMzfc instead?
Uirgil
Nov 17, 2012, 10:59:13 PM11/17/12
to
In article <k88oog$tno$1...@dont-email.me>,
there is an inductive set, and since in ZFC there is nothing else but
actual sets, that set is actual, not merely potential.
"LudovicoVan" <ju...@diegidio.name> wrote:
> >> Sure, N is the minimal set with 0 and closed under the successor
> >> operation.
> >>
> >> But that remains a characterization of a *potential infinity*.
> >>
> > In ZFC that particular set is actual.
>
> Yet you have shown no ground to call it actual, in fact the opposite.
It is well known by anyone who knows anything about ZFC that in ZFC
> >> Sure, N is the minimal set with 0 and closed under the successor
> >> operation.
> >>
> >> But that remains a characterization of a *potential infinity*.
> >>
> > In ZFC that particular set is actual.
>
> Yet you have shown no ground to call it actual, in fact the opposite.
there is an inductive set, and since in ZFC there is nothing else but
actual sets, that set is actual, not merely potential.
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