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Objections against Cantor
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Zuhair
Nov 19, 2012, 5:47:24 AM11/19/12
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Cantor was the first to show the existence of sets that has
uncountably many elements. He showed that the set of all reals was
uncountable.
Also along lines of his diagonal proof it can be easily shown that the
set of all infinite binary sequences is uncountable. Various
objections have emerged to falsify this claim, however all of those
(possibly except one) are actually unsubstantiated.
Those are:
(1) Uncountability leads to undefinable sets, i.e. sets for which
there is no parameter free formula the dictates membership in them,
and since we cannot speak of such sets, then this leads us astray.
The answer to this is that the definition of a real, or of an infinite
binary sequence do not mention them to be definable, an infinite
binary sequence is nothing but a function from the domain N of all
naturals to the codomain {0,1}, that's all, nothing in that definition
per se mentions that this sequence must be definable. So this
objection against Cantor fails since it is about something else. That
some elements of the set of all reals cannot be described by a
parameter free formula and thus rendering them untouched by our
knowledge machinery
doesn't mean that we cannot make inferences about the whole set
itself, we can speak about general laws of the whole universe but we
know very well that there are areas of the universe that might never
be reached by human discovering endeavor. We can speak about the set
of reals, compare its size to other sets, define functions on it,
etc.., all of that is reachable! Not only that we can even prove
uncountability in the constructive universe of Godel in which all sets
are definable from prior stages in the hierarchy, which also
undermines this argument.
(2) Argument of Potential infinity: That only finite set exists and
some potential processes of infinite trend that do not at any stage
define an infinite complete set of elements. And under such picture of
course arguments of Cantor clearly fail from the outset since it is
speaking about matters that do not exist.
The problem with this objection is that it is not faithful to its own
motives since it clearly veers away from defining in an explicit
manner those potentials for the infinite, and if they do, then
Cantor's argument can be easily reproduced under those definitions,
and accordingly there is no reason to suppose that such a defective
account from the outset would be reality revealing.
(3) Argument of Finitism: Only finite sets and finite processes exist,
nothing else. So Cantor's argument is flying high up in imaginative
thinking far away from the grounds of reality.
The problem with that is that there is no clear justification of why
should the notion of "finiteness" be given so much credit over
"infiniteness" if we say that everything in our world is finite and
deem infinity as being at best a logically consistent imaginative
ideal, then the same can be exactly said about finitism also, since it
accepts large finite sets that we may not happen to even touch in any
finite way, like numbers that we cannot describe using all of our
abbreviation capacity, and so MOST of the finite world is also too
ideal to be real or even near real, so why accept those large finites?
Actually the infinite seems much simpler and easily touched by human
imagination than most of large finites.
(4) Ultrafinistism: Those restrict human mathematical reasoning to
only feasible length descriptions, so it is more consistent than
finitism, but yet it is too restrictive that most mathematicians see
no clear justification for it to be true.
The mere justification of what is available around us, and the finite
nature of our abilities, is not a clear evidence of why should the
universe abide by such inabilities. That's besides the fact that
actual infinity through set construction is intelligible, so why
commit ourselves to such a restriction based on some inability that
the universe and reality around us might not necessarily copy and yet
at the same time this non copying can still be touched by some of our
descriptive apparatus though not in full as with ultra-finite matters.
All the above 4 objections where actually at a level that is prior to
the argument of Cantor's.
The following are intra-argument objects, i.e., objections that try to
show some flaw in the logical frame of the argument itself.
(1) The argument is impredicative, and since paradoxes occur with
impredicative arguments, then it is false.
This objection is not correct, since the argument is produced in
predicative systems. And even if potentially impredicative this still
doesn't mean it is paradoxical, truly all paradoxes stem from
impredicative reasoning but the converse is not always true.
(2) The argument begins with a contradiction of assuming a set of all
reals that is shown to miss a real.
This is not a valid objection, since the argument can be reproduced in
another logical way other than "argument by negation". And even the
argument by negation method though non constructive yet it is valid in
classical logic, and there is no reason to consider it as not truth
revealing.
(3) The diagonal can be viewed as merely reflecting the potential of
having more and more reals, which is just to say that the reals are
infinite, it doesn't manage to prove anything a part from that which
is already known.
This objection is False, since the argument clearly prove that EVERY
injection from N to the reals (or to the set of all infinite binary
sequences) is always missing a real form its range and thus not
bijective, and thus it PROVES that the existence of a bijection from N
to R is impossible, and this what uncountability means.
(4) at each step the diagonal produced when put on top of the original
list it would produce still a "countable" list, thus repeating this
process, will also, cause a countable list at the end.
This is wrong since the proof doesn't depend on such concept of
countable addition of diagonals to prior lists at each stage. In a
similar way if we prove that every FINITE subset of some set X would
be missing an element of X, then This is a proof that X is infinite?
Nobody objects to that, but yet according to this argument we can
still say if we add that element to prior finite subset the result is
a FINITE set, i.e. there is no change in finite-hood status and thus X
is FINITE? This is clearly false! We know of course here that the
additions are going infinitely, and we know that any finite number of
such additions would produce a FINITE subset of X, but still that
finite subset is of course not X itself.
In a similar manner Cantor's argument is saying that we cannot
countably many times repeat the diagonal on top of prior list process
to reach the set of all reals. We need to do the repetition process of
adding diagonals to prior lists "uncountably" many number of times in
order to recover the set of all reals!
(5) Alleged proofs of bijections between N and R.
Answer: all are proofs proved to be inconsistent and FALSE.
(6) The first argument of Cantor uses extended setting (i.e. setting
requiring an infinite countable domain having an omega_th entry) and
applies it to a situation where that setting is clearly absent, so the
argument is not addressing the matter coherently, and the result of
finding the missing real just reflects the result of running extended
setting on a background that lacks it, so it is a false result, it is
a deception brought about a perplexed approach to the issue at hand.
Answer: The above argument is just an argument of prejudice, the
pretense that extended setting must not be used for non-extended ones
and considering this issue as reality determinant is all just an
unbaked assertion. Since the argument is about Countability of the
reals then we are free to move and maneuver about different settings
as far as those are countable and related to the heart of the subject,
thus the alleged confusion is not really there, nor is its link to the
reality of the issue.
(7) The diagonal argument of Cantor uses higher setting; the diagonal
is a higher kind of set than the original list, and thus the argument
is springing from a confusion of lower and higher setting, thus
yielding the illusion of having a missed real, what is missing is a
real that belongs to a higher setting than the original list, but that
doesn't mean that there is always a missing real.
Answer: This is the same argument of (6) but in different terms, and
the same response goes to it, as far as we are maneuvering within
countable setting, then it doesn't matter what is the particular sub-
setting of it, the main setting is countability, and giving such
concepts a reality revealing status is just an unbaked pretense,
noting more.
The only important objection is the one emanating from Skolem paradox.
Skolem proved that every first order theory if consistent then it
would have a "countable" model. Thus ZFC which proves uncountability
of the reals would itself has a model that is countable? so this
uncountability in that model would be due to internal deficiency of
the model in having the needed bijection between N and R in that
model. And since countable models have less Ontology than higher
models (if they exist), then obviously we are to be committed to the
less ontology model that do the same job, a rational following
generally Ockham's
razor.
Answer, the argument is a reductionist argument, "if we can do with
less then what is more do not exist", and this reductionism is not
necessarily truth revealing, it is practical yes, but that doesn't
mean it has the final say on the reality of the matter. When we hold
that are certain theory is true, then this comes from our examination
of the very particulars of that theory, i.e. its axioms, logic behind
it, etc..., and not from a mere general feature of the logic
underlying it like that of having always a countable domain, so if I
say that ZFC is true, then this comes after examining its contents,
especially the axioms, and if there was a justification to believe in
its truth, then this justifies saying that the "intended model" of ZFC
does really exist, and this would be a model that copies to the most
degree its reality, and this would not be countable of course, Now to
believe that ZFC is true and yet not having its intended model is a
strange kind of an idea. Since uncountability of reals is proved in
pretty much very weak fragments of ZFC, actually of second order
arithmetic (formulated
in first order), and since those are generally thought to be true
depending on what their content is speaking of, then it follows
naturally to hold that their intended models are uncountable!
Cantor's argument per se is an argument that comes from the
particulars of the question at hand, while the above argument is
coming from the general feature of first order logic, that is besides
the ascending Skolem theorem tells us that there is no control over
the size of the universe of theories in first order logic, so we are
using a piece of knowledge that doesn't have much say on size concept
and we are giving it a truth value against an argument that springs
directly from the particulars of the issue in question and that
directly answers to size of matters.
Not only that the whole argument gives both first order logic and
Reductionist views (whether through Ockham or not), a reality
revealing status without clear justification, and actually it gives it
the final say on a matter that they are actually lacking any control
over or are biased to (to the less in reductionism). So it is also an
unjustified claim.
However this argument can be paraphrased against Cantor in somehow a
successful manner, like the following:
What has been asked is a PROOF of whether the uncountable exists.
All what we have PROVED is the existence of countable domains of
theories in first order logic.
All theories proved consistent like fragments of second order logic
(formulated in first order) are proved so by constructivist methods
that are linked to "countable ordinals" by ordinal analysis and the
alike, which are all within the countable arena of thought, although
internally some prove the existence of the uncountable, but yet
proving their consistency only came by defining countable models of
them, so the believe in the existence of their intended models needs
to be proved.
So all of what we have is a proof of consistency of those theories, we
didn't prove them TRUE, so that we hold their intended models to be
true in the real world, and even if we prove them true, it is still
the case that it can be argued that such a truth only occurs within a
countable mantle, and thus manifest itself by and only by a countable
model, still there is no proof so to say that the intended model
should exist if it was uncountable. And even if we go FULL second
order logic then we go to a system that doesn't support a proof system
and so can hardly be named as logic in order for us to take its
inferences as valid ones about the truth of the matter.
So Cantor's argument is not proved to be TRUE. It is conjectured to be
the most likely case, but NOT proved.
The answer to this argument is that proving the existence of a
countable model for every first order theories also depended on the
concept of uncountability, so still it was indispensable to reach into
such result in the first place, so all alleged truth status attached
to countability despite what internally those theories say even if
proved true, all of that will be only an empty assertion since the
original
assertion depended on the concept of the uncountable. I got this
response form a well-known set theorist, I'm myself not aware of its
particulars, and I'm taking his word for it. However also he remarked
that to there are programs to get rid of the uncountable altogether,
but he said they didn't succeed so far. However this only strengthen
the point in favor of uncountability.
The ultimate answer is that Cantor's argument is a direct argument at
the heart of the issue, the other argumentation are all involving
concepts that are either frankly erroneous or are not related to the
very issue to be solved.
So the final say is for Cantor on this issue.
There are uncountable sets!
Zuhair
uncountably many elements. He showed that the set of all reals was
uncountable.
Also along lines of his diagonal proof it can be easily shown that the
set of all infinite binary sequences is uncountable. Various
objections have emerged to falsify this claim, however all of those
(possibly except one) are actually unsubstantiated.
Those are:
(1) Uncountability leads to undefinable sets, i.e. sets for which
there is no parameter free formula the dictates membership in them,
and since we cannot speak of such sets, then this leads us astray.
The answer to this is that the definition of a real, or of an infinite
binary sequence do not mention them to be definable, an infinite
binary sequence is nothing but a function from the domain N of all
naturals to the codomain {0,1}, that's all, nothing in that definition
per se mentions that this sequence must be definable. So this
objection against Cantor fails since it is about something else. That
some elements of the set of all reals cannot be described by a
parameter free formula and thus rendering them untouched by our
knowledge machinery
doesn't mean that we cannot make inferences about the whole set
itself, we can speak about general laws of the whole universe but we
know very well that there are areas of the universe that might never
be reached by human discovering endeavor. We can speak about the set
of reals, compare its size to other sets, define functions on it,
etc.., all of that is reachable! Not only that we can even prove
uncountability in the constructive universe of Godel in which all sets
are definable from prior stages in the hierarchy, which also
undermines this argument.
(2) Argument of Potential infinity: That only finite set exists and
some potential processes of infinite trend that do not at any stage
define an infinite complete set of elements. And under such picture of
course arguments of Cantor clearly fail from the outset since it is
speaking about matters that do not exist.
The problem with this objection is that it is not faithful to its own
motives since it clearly veers away from defining in an explicit
manner those potentials for the infinite, and if they do, then
Cantor's argument can be easily reproduced under those definitions,
and accordingly there is no reason to suppose that such a defective
account from the outset would be reality revealing.
(3) Argument of Finitism: Only finite sets and finite processes exist,
nothing else. So Cantor's argument is flying high up in imaginative
thinking far away from the grounds of reality.
The problem with that is that there is no clear justification of why
should the notion of "finiteness" be given so much credit over
"infiniteness" if we say that everything in our world is finite and
deem infinity as being at best a logically consistent imaginative
ideal, then the same can be exactly said about finitism also, since it
accepts large finite sets that we may not happen to even touch in any
finite way, like numbers that we cannot describe using all of our
abbreviation capacity, and so MOST of the finite world is also too
ideal to be real or even near real, so why accept those large finites?
Actually the infinite seems much simpler and easily touched by human
imagination than most of large finites.
(4) Ultrafinistism: Those restrict human mathematical reasoning to
only feasible length descriptions, so it is more consistent than
finitism, but yet it is too restrictive that most mathematicians see
no clear justification for it to be true.
The mere justification of what is available around us, and the finite
nature of our abilities, is not a clear evidence of why should the
universe abide by such inabilities. That's besides the fact that
actual infinity through set construction is intelligible, so why
commit ourselves to such a restriction based on some inability that
the universe and reality around us might not necessarily copy and yet
at the same time this non copying can still be touched by some of our
descriptive apparatus though not in full as with ultra-finite matters.
All the above 4 objections where actually at a level that is prior to
the argument of Cantor's.
The following are intra-argument objects, i.e., objections that try to
show some flaw in the logical frame of the argument itself.
(1) The argument is impredicative, and since paradoxes occur with
impredicative arguments, then it is false.
This objection is not correct, since the argument is produced in
predicative systems. And even if potentially impredicative this still
doesn't mean it is paradoxical, truly all paradoxes stem from
impredicative reasoning but the converse is not always true.
(2) The argument begins with a contradiction of assuming a set of all
reals that is shown to miss a real.
This is not a valid objection, since the argument can be reproduced in
another logical way other than "argument by negation". And even the
argument by negation method though non constructive yet it is valid in
classical logic, and there is no reason to consider it as not truth
revealing.
(3) The diagonal can be viewed as merely reflecting the potential of
having more and more reals, which is just to say that the reals are
infinite, it doesn't manage to prove anything a part from that which
is already known.
This objection is False, since the argument clearly prove that EVERY
injection from N to the reals (or to the set of all infinite binary
sequences) is always missing a real form its range and thus not
bijective, and thus it PROVES that the existence of a bijection from N
to R is impossible, and this what uncountability means.
(4) at each step the diagonal produced when put on top of the original
list it would produce still a "countable" list, thus repeating this
process, will also, cause a countable list at the end.
This is wrong since the proof doesn't depend on such concept of
countable addition of diagonals to prior lists at each stage. In a
similar way if we prove that every FINITE subset of some set X would
be missing an element of X, then This is a proof that X is infinite?
Nobody objects to that, but yet according to this argument we can
still say if we add that element to prior finite subset the result is
a FINITE set, i.e. there is no change in finite-hood status and thus X
is FINITE? This is clearly false! We know of course here that the
additions are going infinitely, and we know that any finite number of
such additions would produce a FINITE subset of X, but still that
finite subset is of course not X itself.
In a similar manner Cantor's argument is saying that we cannot
countably many times repeat the diagonal on top of prior list process
to reach the set of all reals. We need to do the repetition process of
adding diagonals to prior lists "uncountably" many number of times in
order to recover the set of all reals!
(5) Alleged proofs of bijections between N and R.
Answer: all are proofs proved to be inconsistent and FALSE.
(6) The first argument of Cantor uses extended setting (i.e. setting
requiring an infinite countable domain having an omega_th entry) and
applies it to a situation where that setting is clearly absent, so the
argument is not addressing the matter coherently, and the result of
finding the missing real just reflects the result of running extended
setting on a background that lacks it, so it is a false result, it is
a deception brought about a perplexed approach to the issue at hand.
Answer: The above argument is just an argument of prejudice, the
pretense that extended setting must not be used for non-extended ones
and considering this issue as reality determinant is all just an
unbaked assertion. Since the argument is about Countability of the
reals then we are free to move and maneuver about different settings
as far as those are countable and related to the heart of the subject,
thus the alleged confusion is not really there, nor is its link to the
reality of the issue.
(7) The diagonal argument of Cantor uses higher setting; the diagonal
is a higher kind of set than the original list, and thus the argument
is springing from a confusion of lower and higher setting, thus
yielding the illusion of having a missed real, what is missing is a
real that belongs to a higher setting than the original list, but that
doesn't mean that there is always a missing real.
Answer: This is the same argument of (6) but in different terms, and
the same response goes to it, as far as we are maneuvering within
countable setting, then it doesn't matter what is the particular sub-
setting of it, the main setting is countability, and giving such
concepts a reality revealing status is just an unbaked pretense,
noting more.
The only important objection is the one emanating from Skolem paradox.
Skolem proved that every first order theory if consistent then it
would have a "countable" model. Thus ZFC which proves uncountability
of the reals would itself has a model that is countable? so this
uncountability in that model would be due to internal deficiency of
the model in having the needed bijection between N and R in that
model. And since countable models have less Ontology than higher
models (if they exist), then obviously we are to be committed to the
less ontology model that do the same job, a rational following
generally Ockham's
razor.
Answer, the argument is a reductionist argument, "if we can do with
less then what is more do not exist", and this reductionism is not
necessarily truth revealing, it is practical yes, but that doesn't
mean it has the final say on the reality of the matter. When we hold
that are certain theory is true, then this comes from our examination
of the very particulars of that theory, i.e. its axioms, logic behind
it, etc..., and not from a mere general feature of the logic
underlying it like that of having always a countable domain, so if I
say that ZFC is true, then this comes after examining its contents,
especially the axioms, and if there was a justification to believe in
its truth, then this justifies saying that the "intended model" of ZFC
does really exist, and this would be a model that copies to the most
degree its reality, and this would not be countable of course, Now to
believe that ZFC is true and yet not having its intended model is a
strange kind of an idea. Since uncountability of reals is proved in
pretty much very weak fragments of ZFC, actually of second order
arithmetic (formulated
in first order), and since those are generally thought to be true
depending on what their content is speaking of, then it follows
naturally to hold that their intended models are uncountable!
Cantor's argument per se is an argument that comes from the
particulars of the question at hand, while the above argument is
coming from the general feature of first order logic, that is besides
the ascending Skolem theorem tells us that there is no control over
the size of the universe of theories in first order logic, so we are
using a piece of knowledge that doesn't have much say on size concept
and we are giving it a truth value against an argument that springs
directly from the particulars of the issue in question and that
directly answers to size of matters.
Not only that the whole argument gives both first order logic and
Reductionist views (whether through Ockham or not), a reality
revealing status without clear justification, and actually it gives it
the final say on a matter that they are actually lacking any control
over or are biased to (to the less in reductionism). So it is also an
unjustified claim.
However this argument can be paraphrased against Cantor in somehow a
successful manner, like the following:
What has been asked is a PROOF of whether the uncountable exists.
All what we have PROVED is the existence of countable domains of
theories in first order logic.
All theories proved consistent like fragments of second order logic
(formulated in first order) are proved so by constructivist methods
that are linked to "countable ordinals" by ordinal analysis and the
alike, which are all within the countable arena of thought, although
internally some prove the existence of the uncountable, but yet
proving their consistency only came by defining countable models of
them, so the believe in the existence of their intended models needs
to be proved.
So all of what we have is a proof of consistency of those theories, we
didn't prove them TRUE, so that we hold their intended models to be
true in the real world, and even if we prove them true, it is still
the case that it can be argued that such a truth only occurs within a
countable mantle, and thus manifest itself by and only by a countable
model, still there is no proof so to say that the intended model
should exist if it was uncountable. And even if we go FULL second
order logic then we go to a system that doesn't support a proof system
and so can hardly be named as logic in order for us to take its
inferences as valid ones about the truth of the matter.
So Cantor's argument is not proved to be TRUE. It is conjectured to be
the most likely case, but NOT proved.
The answer to this argument is that proving the existence of a
countable model for every first order theories also depended on the
concept of uncountability, so still it was indispensable to reach into
such result in the first place, so all alleged truth status attached
to countability despite what internally those theories say even if
proved true, all of that will be only an empty assertion since the
original
assertion depended on the concept of the uncountable. I got this
response form a well-known set theorist, I'm myself not aware of its
particulars, and I'm taking his word for it. However also he remarked
that to there are programs to get rid of the uncountable altogether,
but he said they didn't succeed so far. However this only strengthen
the point in favor of uncountability.
The ultimate answer is that Cantor's argument is a direct argument at
the heart of the issue, the other argumentation are all involving
concepts that are either frankly erroneous or are not related to the
very issue to be solved.
So the final say is for Cantor on this issue.
There are uncountable sets!
Zuhair
LudovicoVan
Nov 19, 2012, 6:39:11 AM11/19/12
to
"Zuhair" <zalj...@gmail.com> wrote in message
news:83ee674b-f242-4171...@n5g2000yqe.googlegroups.com...
> There are uncountable sets!
Thank you for your misinformed and misguided propaganda.
-LV
news:83ee674b-f242-4171...@n5g2000yqe.googlegroups.com...
> There are uncountable sets!
Thank you for your misinformed and misguided propaganda.
-LV
forbi...@gmail.com
Nov 19, 2012, 7:21:12 AM11/19/12
to
I'm going to give an objection not in the list. I will not
explain it's validity or lack thereof because I haven't explored
it even though I've routinely ignored it.
Suppose one wanted to apply the Cantor diagonalization method
to natural numbers so rather than moving down in scale to the
digit 10^-n one move up to the 10^n digit and produced a natural
number not in the list. How can that be? The presumption is
that every natural number is in the set of natural numbers.
There's something wrong with the argument or something wrong
with the notion of the set of natural numbers. Why should
Cantor's method of diagonalization be valid but not Forbis's
method of diagonalization?
Many people complain that the missing real is really in the
set some arbitrary distance past the Cantor expansion so far.
Why would that be invalid for Cantor where it would be valid
for Forbis?
explain it's validity or lack thereof because I haven't explored
it even though I've routinely ignored it.
Suppose one wanted to apply the Cantor diagonalization method
to natural numbers so rather than moving down in scale to the
digit 10^-n one move up to the 10^n digit and produced a natural
number not in the list. How can that be? The presumption is
that every natural number is in the set of natural numbers.
There's something wrong with the argument or something wrong
with the notion of the set of natural numbers. Why should
Cantor's method of diagonalization be valid but not Forbis's
method of diagonalization?
Many people complain that the missing real is really in the
set some arbitrary distance past the Cantor expansion so far.
Why would that be invalid for Cantor where it would be valid
for Forbis?
Jesse F. Hughes
Nov 19, 2012, 7:41:38 AM11/19/12
to
Nice to see you're still fighting the man.
--
"Who knows, maybe that may be the only way to settle this crap. It's
not like it'd be that hard for me to go back and get a math degree.
I can penetrate the math social group and then finish the takedown
from inside." -- James S. Harris contemplates a new strategy.
Ben Bacarisse
Nov 19, 2012, 7:43:36 AM11/19/12
to
forbi...@gmail.com writes:
> Suppose one wanted to apply the Cantor diagonalization method
> to natural numbers so rather than moving down in scale to the
> digit 10^-n one move up to the 10^n digit and produced a natural
> number not in the list. How can that be?
It can't be.
> Suppose one wanted to apply the Cantor diagonalization method
> to natural numbers so rather than moving down in scale to the
> digit 10^-n one move up to the 10^n digit and produced a natural
> number not in the list. How can that be?
> The presumption is
> that every natural number is in the set of natural numbers.
> There's something wrong with the argument or something wrong
> with the notion of the set of natural numbers.
description is a little brief) the procedure does not produce the
representation of a natural number.
<snip>
--
Ben.
forbi...@gmail.com
Nov 19, 2012, 8:00:12 AM11/19/12
to
an infinite string of zeros in front of it. Since 0 isn't 8
that digit can be replaced with an 8.
LudovicoVan
Nov 19, 2012, 8:11:44 AM11/19/12
to
"Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
news:87boetu...@phiwumbda.org...
-LV
news:87boetu...@phiwumbda.org...
> "LudovicoVan" <ju...@diegidio.name> writes:
>> "Zuhair" <zalj...@gmail.com> wrote in message
>> news:83ee674b-f242-4171...@n5g2000yqe.googlegroups.com...
>>
>>> There are uncountable sets!
>>
>> Thank you for your misinformed and misguided propaganda.
>
> You are just adorable, you are!
>
> Nice to see you're still fighting the man.
No merit: an answering machine would do.
>> "Zuhair" <zalj...@gmail.com> wrote in message
>> news:83ee674b-f242-4171...@n5g2000yqe.googlegroups.com...
>>
>>> There are uncountable sets!
>>
>> Thank you for your misinformed and misguided propaganda.
>
> You are just adorable, you are!
>
> Nice to see you're still fighting the man.
-LV
Ben Bacarisse
Nov 19, 2012, 8:24:48 AM11/19/12
to
contain infinitely many non-zero digits (8s in your example). No
natural number has such a representation -- all element of N have
finitely many non-zero digits in their decimal representation.
--
Ben.
Jesse F. Hughes
Nov 19, 2012, 8:34:37 AM11/19/12
to
insightful, that was.
Shame Han de Bruin doesn't hang around here any more. The two of you
could hold seminars on the capitalist nature of set theory.
--
Jesse F. Hughes
Playin' dismal hollers for abysmal dollars,
Those were the days, best I can recall.
-- Austin Lounge Lizards, "Rocky Byways"
LudovicoVan
Nov 19, 2012, 8:55:54 AM11/19/12
to
"Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
news:874nklu...@phiwumbda.org...
> "LudovicoVan" <ju...@diegidio.name> writes:
>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
>> news:87boetu...@phiwumbda.org...
>>> "LudovicoVan" <ju...@diegidio.name> writes:
>>>> "Zuhair" <zalj...@gmail.com> wrote in message
>>>> news:83ee674b-f242-4171...@n5g2000yqe.googlegroups.com...
>>>>
>>>>> There are uncountable sets!
>>>>
>>>> Thank you for your misinformed and misguided propaganda.
>>>
>>> You are just adorable, you are!
>>>
>>> Nice to see you're still fighting the man.
>>
>> No merit: an answering machine would do.
>
> Right. But calling Zuhair's comment "propaganda"? That was bloody
> insightful, that was.
>
> Shame Han de Bruin doesn't hang around here any more. The two of you
> could hold seminars on the capitalist nature of set theory.
As you on the nature and dynamics of breath wasting.
>> "Jesse F. Hughes" <je...@phiwumbda.org> wrote in message
>> news:87boetu...@phiwumbda.org...
>>> "LudovicoVan" <ju...@diegidio.name> writes:
>>>> "Zuhair" <zalj...@gmail.com> wrote in message
>>>> news:83ee674b-f242-4171...@n5g2000yqe.googlegroups.com...
>>>>
>>>>> There are uncountable sets!
>>>>
>>>> Thank you for your misinformed and misguided propaganda.
>>>
>>> You are just adorable, you are!
>>>
>>> Nice to see you're still fighting the man.
>>
>> No merit: an answering machine would do.
>
> Right. But calling Zuhair's comment "propaganda"? That was bloody
> insightful, that was.
>
> Shame Han de Bruin doesn't hang around here any more. The two of you
> could hold seminars on the capitalist nature of set theory.
Keep up the good work.
-LV
forbi...@gmail.com
Nov 19, 2012, 9:04:17 AM11/19/12
to
for every n in N has a finite number of digits and differs from the
first n elements in the list.
Since Cantor generated a number not in the list of rationals from
a well ordered set of rationals, do you accept that his method was
invalid?
Jesse F. Hughes
Nov 19, 2012, 9:20:53 AM11/19/12
to
How's this?
--
"[I]f I could go back, [...] I would tell myself not to step into a position
where the fate of the entire world could rest in my hands. I would [avoid
this] path to a nightmarish and surreal world, a topsy-turvy world, where
everything changes." -- James S. Harris cannot escape his destiny.
Ben Bacarisse
Nov 19, 2012, 10:05:07 AM11/19/12
to
of the process. For the reals in [0,1] the result is a real in [0,1]
but for the natural numbers, the limit is not the representation of a
natural number.
> Since Cantor generated a number not in the list of rationals from
> a well ordered set of rationals, do you accept that his method was
> invalid?
number, it's about any list. The argument must be general: for any
enumeration of set S, some procedure generates an element of S that is
not "in" the enumeration.
That's Cantor's method, and it does not work for the rationals. Some
enumerations of rationals have a diagonal that is also a rational and
some others have a diagonal that is not a rational. But there is no
general method that produces a missing rational from any enumeration of
the set.
It's up to you if you call that "invalid" -- the method can be applied
to some sets and not to others. It can tells us that the reals can't be
enumerated, but it tells us nothing about the natural numbers or the
rationals. For that, we need other arguments. For example, you can
exhibit an explicit bijection from N to Q to show that the rationals are
countable.
(BTW, trouble is likely to ensue if you keep using the term "a well
ordered set of ...". It's better to use a weaker term like enumeration.
There are sets that can be well ordered but can't be enumerated, and
Cantor's argument starts by assuming only an enumeration, not a well
ordering.)
--
Ben.
Rupert
Nov 19, 2012, 11:18:40 AM11/19/12
to
On Nov 19, 11:47 am, Zuhair <zaljo...@gmail.com> wrote:
> (4) Ultrafinistism: Those restrict human mathematical reasoning to
> only feasible length descriptions, so it is more consistent than
> finitism, but yet it is too restrictive that most mathematicians see
> no clear justification for it to be true.
> The mere justification of what is available around us, and the finite
> nature of our abilities, is not a clear evidence of why should the
> universe abide by such inabilities. That's besides the fact that
> actual infinity through set construction is intelligible, so why
> commit ourselves to such a restriction based on some inability that
> the universe and reality around us might not necessarily copy and yet
> at the same time this non copying can still be touched by some of our
> descriptive apparatus though not in full as with ultra-finite matters.
>
You should have a look at the first chapter of Edward Nelson's
> (4) Ultrafinistism: Those restrict human mathematical reasoning to
> only feasible length descriptions, so it is more consistent than
> finitism, but yet it is too restrictive that most mathematicians see
> no clear justification for it to be true.
> The mere justification of what is available around us, and the finite
> nature of our abilities, is not a clear evidence of why should the
> universe abide by such inabilities. That's besides the fact that
> actual infinity through set construction is intelligible, so why
> commit ourselves to such a restriction based on some inability that
> the universe and reality around us might not necessarily copy and yet
> at the same time this non copying can still be touched by some of our
> descriptive apparatus though not in full as with ultra-finite matters.
>
"Predicative Arithmetic", titled "The impredicativity of induction".
Zuhair
Nov 19, 2012, 12:11:02 PM11/19/12
to
On Nov 19, 3:21 pm, forbisga...@gmail.com wrote:
> I'm going to give an objection not in the list. I will not
> explain it's validity or lack thereof because I haven't explored
> it even though I've routinely ignored it.
>
> Suppose one wanted to apply the Cantor diagonalization method
> to natural numbers so rather than moving down in scale to the
> digit 10^-n one move up to the 10^n digit and produced a natural
> number not in the list. How can that be? The presumption is
> that every natural number is in the set of natural numbers.
> There's something wrong with the argument or something wrong
> with the notion of the set of natural numbers. Why should
> Cantor's method of diagonalization be valid but not Forbis's
> method of diagonalization?
>
You need to present your argument in detail, so that one can comment
> I'm going to give an objection not in the list. I will not
> explain it's validity or lack thereof because I haven't explored
> it even though I've routinely ignored it.
>
> Suppose one wanted to apply the Cantor diagonalization method
> to natural numbers so rather than moving down in scale to the
> digit 10^-n one move up to the 10^n digit and produced a natural
> number not in the list. How can that be? The presumption is
> that every natural number is in the set of natural numbers.
> There's something wrong with the argument or something wrong
> with the notion of the set of natural numbers. Why should
> Cantor's method of diagonalization be valid but not Forbis's
> method of diagonalization?
>
on it.
forbi...@gmail.com
Nov 19, 2012, 1:39:40 PM11/19/12
to
On Monday, November 19, 2012 7:05:07 AM UTC-8, Ben Bacarisse wrote:
> forbi...@gmail.com writes:
> > That's the point. The n in N is finite. The natural number generated
> > for every n in N has a finite number of digits and differs from the
> > first n elements in the list.
>
> True, but that's not what diagonalisation is. The diagonal is the limit
> of the process. For the reals in [0,1] the result is a real in [0,1]
> but for the natural numbers, the limit is not the representation of a
> natural number.
What is the limit of the diagaonalization on 10^n if not a natural number?
> forbi...@gmail.com writes:
> > That's the point. The n in N is finite. The natural number generated
> > for every n in N has a finite number of digits and differs from the
> > first n elements in the list.
>
> True, but that's not what diagonalisation is. The diagonal is the limit
> of the process. For the reals in [0,1] the result is a real in [0,1]
> but for the natural numbers, the limit is not the representation of a
> natural number.
Does this also apply to the enumerated set of natural numbers?
> > Since Cantor generated a number not in the list of rationals from
> > a well ordered set of rationals, do you accept that his method was
> > invalid?
>
> I think you misunderstand the argument. It's not about one list and one
> number, it's about any list. The argument must be general: for any
> enumeration of set S, some procedure generates an element of S that is
> not "in" the enumeration.
>
> That's Cantor's method, and it does not work for the rationals. Some
> enumerations of rationals have a diagonal that is also a rational and
> some others have a diagonal that is not a rational. But there is no
> general method that produces a missing rational from any enumeration of
> the set.
irrationals. If that were not the case then the rational produced
would have to differ from itself in its indexed position.
It seems easy to prove every finite set of rationals has a bijection
that is also a rational.
> It's up to you if you call that "invalid" -- the method can be applied
> to some sets and not to others. It can tells us that the reals can't be
> enumerated, but it tells us nothing about the natural numbers or the
> rationals. For that, we need other arguments. For example, you can
> exhibit an explicit bijection from N to Q to show that the rationals are
> countable.
>
> (BTW, trouble is likely to ensue if you keep using the term "a well
> ordered set of ...". It's better to use a weaker term like enumeration.
> There are sets that can be well ordered but can't be enumerated, and
> Cantor's argument starts by assuming only an enumeration, not a well
> ordering.)
and the rest are successors to elements in the set. What is your
definition of a set and a well ordered set such that it cannot be
enumerated?
Virgil
Nov 19, 2012, 5:35:16 PM11/19/12
to
In article <k8d5p4$96j$1...@dont-email.me>,
It is considerably less "misinformed and misguided propaganda" than what
LVmanages to produce.
At least in the opinions of almost everyone except LV himself
(or herself as the case may be).
--
LVmanages to produce.
At least in the opinions of almost everyone except LV himself
(or herself as the case may be).
--
George Greene
Nov 19, 2012, 7:13:35 PM11/19/12
to
On Nov 19, 5:47 am, Zuhair <zaljo...@gmail.com> wrote:
> (3) Argument of Finitism: Only finite sets and finite processes exist,
> nothing else. So Cantor's argument is flying high up in imaginative
> thinking far away from the grounds of reality.
It CAN'T be VERY far.
> (3) Argument of Finitism: Only finite sets and finite processes exist,
> nothing else. So Cantor's argument is flying high up in imaginative
> thinking far away from the grounds of reality.
Even if you think that only finite things exist, i.e., even if you
think
that every set has a finite natural number as a cardinality, the
problem
becomes that there are AN INFINITE number OF THOSE.
If some things are cardinalities, and some are not, or if
cardinalities
are encoded as sets (in 1st-order ZFC we typically encode
every cardinal as an initial ordinal), then you need some
justification
for claiming that there is NOT a set of all and only the cardinals.
The point is, you can't credibly claim that there "exist only" finite
things when THE NUMBER OF such finite things IS PROVABLY infinite.
> The problem with that is that there is no clear justification of why
> should the notion of "finiteness" be given so much credit over
> "infiniteness"
we STILL
wind up with an INFinite number OF THINGS!!
> if we say that everything in our world is finite and
> deem infinity as being at best a logically consistent imaginative
> ideal,
cardinalities are EACH AND EVERY AND ALL *concretely*IN* our world.
Our concrete world ITSELF contains INFINITELY many CONCRETE things.
So YOU CAN'T credibly or consistently banish "infinity" to the realm
of "imaginative ideal". The universe of discourse as a whole, GIVEN
THAT EVERYTHING IN IT IS FINITE AND CONCRETE, IS NOT imagined or
ideal.
IT TOO MUST be concrete.
> then the same can be exactly said about finitism also, since it
> accepts large finite sets that we may not happen to even touch in any
> finite way, like numbers that we cannot describe using all of our
> abbreviation capacity,
a given abbreviation capacity, THERE IS A DIFFERENT abbreviation/
notation THAT DOES describe it.
> Actually the infinite seems much simpler and easily touched by human
> imagination than most of large finites.
already insisted on the concrete relevance of EVERY finite
cardinality, no
matter how large. If you are going to disavow some of those, then
nobody
has anything to say except "why?". That they can't be referred to or
imagined
is frankly obviously false. Given world enough and time, each and
every one of them could in fact be written out in unary. The fact
that individual human brains are too finite to cope with that is
SURELY IRrelevant.
Seriously, do you REALLY want to claim that Ack(6,6) DOES NOT EXIST??
Ben Bacarisse
Nov 19, 2012, 8:57:35 PM11/19/12
to
forbi...@gmail.com writes:
> On Monday, November 19, 2012 7:05:07 AM UTC-8, Ben Bacarisse wrote:
>> forbi...@gmail.com writes:
>> > That's the point. The n in N is finite. The natural number generated
>> > for every n in N has a finite number of digits and differs from the
>> > first n elements in the list.
>>
>> True, but that's not what diagonalisation is. The diagonal is the limit
>> of the process. For the reals in [0,1] the result is a real in [0,1]
>> but for the natural numbers, the limit is not the representation of a
>> natural number.
>
> What is the limit of the diagaonalization on 10^n if not a natural
> number?
The limit is not defined -- the sequence is unbounded.
> On Monday, November 19, 2012 7:05:07 AM UTC-8, Ben Bacarisse wrote:
>> forbi...@gmail.com writes:
>> > That's the point. The n in N is finite. The natural number generated
>> > for every n in N has a finite number of digits and differs from the
>> > first n elements in the list.
>>
>> True, but that's not what diagonalisation is. The diagonal is the limit
>> of the process. For the reals in [0,1] the result is a real in [0,1]
>> but for the natural numbers, the limit is not the representation of a
>> natural number.
>
> What is the limit of the diagaonalization on 10^n if not a natural
> number?
> Does this also apply to the enumerated set of natural numbers?
undefined? If so yes.
(The limit can be defined is we so with but that involves inventing a
new entity, oo maybe, which is not a natural number).
>> > Since Cantor generated a number not in the list of rationals from
>> > a well ordered set of rationals, do you accept that his method was
>> > invalid?
>>
>> I think you misunderstand the argument. It's not about one list and one
>> number, it's about any list. The argument must be general: for any
>> enumeration of set S, some procedure generates an element of S that is
>> not "in" the enumeration.
>>
>> That's Cantor's method, and it does not work for the rationals. Some
>> enumerations of rationals have a diagonal that is also a rational and
>> some others have a diagonal that is not a rational. But there is no
>> general method that produces a missing rational from any enumeration of
>> the set.
>
> I take it all enumerations of all rationals have diagonals that are
> irrationals. If that were not the case then the rational produced
> would have to differ from itself in its indexed position.
> It seems easy to prove every finite set of rationals has a bijection
> that is also a rational.
something else but I can't work out what and sentence before is not
helping me. Any example maybe?
>> It's up to you if you call that "invalid" -- the method can be applied
>> to some sets and not to others. It can tells us that the reals can't be
>> enumerated, but it tells us nothing about the natural numbers or the
>> rationals. For that, we need other arguments. For example, you can
>> exhibit an explicit bijection from N to Q to show that the rationals are
>> countable.
>>
>> (BTW, trouble is likely to ensue if you keep using the term "a well
>> ordered set of ...". It's better to use a weaker term like enumeration.
>
> I will accept this term though I have resevations.
>
>> There are sets that can be well ordered but can't be enumerated, and
>> Cantor's argument starts by assuming only an enumeration, not a well
>> ordering.)
>
> My definition of a well ordered set is one where there is a first element
> and the rest are successors to elements in the set. What is your
> definition of a set and a well ordered set such that it cannot be
> enumerated?
definitions. Let's say I take ZFC to be the basis of my sets and,
because of the C (meaning the axiom of choice), all sets can be
well-ordered. This includes the real, and I presume you accept Cantor's
theorem that the reals can't be enumerated.
--
Ben.
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