Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



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


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 ultrafinite matters.
All the above 4 objections where actually at a level that is prior to the argument of Cantor's.
The following are intraargument 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 finitehood 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 nonextended 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 wellknown 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



