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Objections against Cantor
Posted:
Nov 19, 2012 5:47 AM
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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
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