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Posted by PositiveBear on 1st Oct 2020, see comments on the page Lebesgue Measure
This is the full content of a comment written by PositiveBear on 1st Oct 2020. It was put here instead of the comments page since it is very long, and it is extremely hard to read since the mathematical terms are not properly formatted, but in plain text.
The proof here rests on the idea that every irrational in [0,1] must be the endpoint of some complete interval or contained in the complete interval (see the statement “…make the bizarre attempt to get around the contradiction by claiming that there must be other points in the set B as well as the endpoints of the complete intervals of A.”)
And that is the fallacy. We demonstrate that this is a fallacy by constructing an irrational number that does not lie in any complete interval of A, nor is an endpoint of such a complete interval.
Let us have an enumeration of the rational numbers in (0,1), q_1, q_2, …, q_n, … The set A is defined by the construction laid above of taking a 1/10^n open interval around q_n and taking the union of all such intervals. We label each open interval as B_n = I(q_n, 1/10^n) (i.e. the open interval around q_n of radius 1/10^n).
As Mr. Meyer nicely pointed out, all the complete intervals can be enumerated. So let us enumerate them as I_n, with endpoints a_n and b_n, where a_n <= b_n. Note that I_n is a subset of A, so every element of I_n must be an element of B_k for some k.
Aim: to construct a number r in (0,1) which is neither equal to a_n, nor equal to b_n, nor belongs to B_n, for any n. Implication: if such an r exists, then it follows that this number does not belong to any complete interval I_n, nor is one of the endpoints a_n or b_n, and thus, the measure of the set B is being undercounted in Mr. Meyer’s argument.
Construction: Expand a_n, b_n, and q_n in decimal and take the n-th digit of each. Call the n-th digit x_n, y_n and z_n of a_n, b_n, and q_n respectively. There are 10 possible decimal digit values, and if we remove the digits x_n, y_n, z_n, z_n -1, z_n +1, 0 and 9 (not all of which may be valid digits) there are at least 3 other digit values left. So pick any one of those values, say w_n, that does not equal x_n, y_n, z_n, z_n-1, z_n+1, 0, or 9.
Now, we create the number r = 0.(w_1)(w_2)…(w_n)…, i.e. the n-th digit is w_n. We show that r is the number we need.
Clearly, r does not equal a_n or b_n, because the n-th digit does not match (by construction).
Let us show that r does not belong to any of the intervals B_k = I(q_k, 1/10^k). That will imply that r does not belong to A, and hence it does not belong to any complete interval.
If possible, let |r - q_k| < 1/10^k for some k. We expand q_k in its decimal expansion, q_k = 0.(t_1)(t_2)(t_3)…. Note that, in our previous notation, t_k = z_k (the digit value).
Let j be the first digit where r and q_k differ. So, | (w_j - t_j)/10^j + (w_(j+1) - t_(j+1))/10^(j+1) + … | < 1/10^k.
Case 1: suppose j = k. By the reverse triangle inequality, |a| - |b| <= |a-b|, |w_k - t_k|/10^k - d < 1/10^k,
where d = |(w_(k+1) - t_(k+1))/10^(k+1) + (w_(k+2) - t_(k+2))/10^(k+2) + …|.
Note that, d <= |w_(k+1) - t_(k+1)|/10^(k+1) + |w_(k+2) - t_(k+2)|/10^(k+2) + … Since we are dealing with decimal digits d <= 9/10^(k+1) + 9/10^(k+2) + 9/10^(k+3) + … <= 1/10^k.
Hence, |w_k - t_k|/10^k < d + 1/10^k <= 2/10^k. implies |w_k - t_k| < 2. But this is absurd because t_k = z_k, and w_k was a specifically chosen digit so that it would not equal z_k, z_k-1, z_k+1.
Case 2: j not equal to k. But we know that r and q_k differ at the k-th digit anyway, so we must have j < k.
So we simplify the beginning inequality, | (w_j - t_j)/10^j + (w_(j+1) - t_(j+1))/10^(j+1) + … | < 1/10^k,
using the reverse triangle inequality as in Case 1, but slightly differently,
|w_j - t_j|/10^j - |w_(j+1) - t_(j+1)|/10^(j+1) - … - |w_(k-1) - t_(k-1)|/10^(k-1) - |w_k - t_k|/10^k - d < 1/10^k,
where d = |w_(k+1) - t_(k+1))/10^(k+1) + … w_(k+2) - t_(k+2))/10^(k+2) + …|.
So, |w_j - t_j|/10^j < |w_(j+1) - t_(j+1)|/10^(j+1) + … + |w_(k-1) - t_(k-1)|/10^(k-1) + |w_k - t_k|/10^k + d + 1/10^k,
Again d <= 1/10^k (as in Case 1). Also, |w_k - t_k|/10^k <= 8/10^k, because w_k does not equal 9 or 0 by construction, and so the difference of the digits |w_k - t_k| has to be less than 9.
On the other hand, |w_(j+1) - t_(j+1)|/10^(j+1) <= 9/10^(j+1), and |w_(j+2) - t_(j+2)|/10^(j+2) <= 9/10^(j+2), and so on.
So making the substitutions, we get |w_j - t_j|/10^j < 9/10^(j+1) + 9/10^(j+2) + … 9/10^(k-1) + 8/10^k + 1/10^k + 1/10^k. implies |w_j - t_j|/10^j < 9/10^(j+1) + 9/10^(j+2) + … 9/10^(k-1) + 10/10^k implies (telescoping the sum from the right hand side) |w_j - t_j|/10^j < 1/10^j implies |w_j - t_j| < 1
But |w_j -t_j| >=1 because they are distinct digits. Contradiction.
Clearly, r is not in B_k for any k. So we have found an irrational number that is neither the end point of nor contained in a complete interval.
My response to this argument
Yet another argument where the person believes that if he ignores a contradiction, then everyone else should also do so and pretend that it doesn’t matter. His conclusion is that there is “an irrational number that is neither the end point of nor contained in a complete interval [of A] ”, which leads directly to the contradiction that there must be two endpoints of complete intervals of A, between which there are no other points of A, but which includes this purported irrational. This is a direct contradiction since between any two reals there exists infinitely many rationals, and by definition, every rational is covered by an interval that is an interval of A.
But anyway, let’s examine the argument to see how the fallacy arises. It’s rather ironic that PositiveBear uses the argument he does, since this is basically the same argument that Cantor uses in his 1874 proof, where, if there is a series of nested intervals (each inside another interval) then while the nesting is infinite and never actually reaches a stage where there remains only one point, there must be a point that is not the endpoint of any of the nested intervals. The difference is that in his paper, Cantor correctly notes that this point is the limit value both of the sequence of left endpoints, and of the sequence of right endpoints. Nowhere does Cantor imply that there is a result of the nesting that consists of this number somehow “flanked” by a number immediately to its left, and to its right, which is what PositiveBear is here inviting us to accept.
The reality is that infinitely nested intervals that are all defined as non-degenerate do not somehow “nest down” to a single point, a single point is only obtained by applying a limit. And the fact is that regardless of how large n becomes, there is no finite n at which an irrational number somehow emerges from PositiveBear’s description. But PositiveBear asks us to assume that his description, which is only in terms of finite n, without the inclusion of any limiting condition, results in a well-defined set - but it is a result that has irrational points are not in the set A, but at the same time, there must somehow be rational endpoints of the defined open intervals of A that are not in the set A. He refuses to accept that that is an absurd contradiction.
When PositiveBear claims that he is “constructing an irrational number”, in fact he isn’t, he fails to recognize that there is a difference between creating a formula that can calculate the nth digit of a real number, and a definition of an irrational number. In terms of base 2, an irrational number is the limit of the sum of all a⁄2n as n increases, where a = 0 or a = 1, or one can also say, the limit of the sum of the values represented by a sequence of digits 0 and 1.
PositiveBear’s description only gives, for any chosen n, a rational number that has n binary digits - but an irrational has limitlessly many digits. PositiveBear invites us to pretend that there is such a thing as the sum of all a⁄2n for infinitely many n, that is, where there are more than any finite n of digits. But as I explain elsewhere, if we suppose that there could be a sum of infinitely many digits, since every fraction 1⁄2n is inherently connected to the initial 1⁄2 by the finite factor n, then there would have to be at least one 1⁄2n where n is greater than any finite number - which is absurd. See Sums of infinitely many fractions: 1 and Sums of infinitely many fractions: 2 for a detailed explanation of why the notion of infinite sums without the application of a limit is contradictory. On the other hand, what we can logically assert is that, given PositiveBear’s description, we can define an irrational number that is the limit of the summation of the fractions that he describes.
The irony here is that points that we can obtain by applying a limit to PositiveBear’s description, are the points that are in the complement of a set that is well-defined, and where the definition includes a limit condition. Rather than an absurd contradiction such as in PositiveBear’s naive description, which entails a contradictory conclusion, the correct application of limits gives a simple and complete resolution of all the contradictions generated by the naive description of the set A, demonstrating an overarching coherent and logical framework.
That PositiveBear and others should summarily reject that framework simply because the result disagrees with their prior expectations, in the face of obvious contradictions, tells us more about human irrational obstinacy than about mathematics.
Posted by Tommy Jensen on 26th Oct 2020, see comments on the page Cardinal Numbers
This is the full content of a comment written by Tommy Jensen on 26th Oct 2020. The post was edited to delete the derogatory aspects that had no logical argumentative value.
10:41 a.m., Monday Oct. 26, 2020
Tommy Jensen (Joined Disqus Oct 23, 2020) wrote:
Definition A: You can believe that this is a proper definition if you have completely misunderstood the treatment of the same subject that is normally given in grade 2 elementary school.
Note B: If you weren’t a complete idiot, and if you were aware of the actual definitions, you would know that it is obvious that there are equivalence classes consisting of those sets that have the same cardinality.
Note C: Completely irrelevant besides being false. Obviously the empty set has no one-to-one correspondence between it and any other set.
Definition D: Meaningless word salad.
You work hard to convince everybody that you are absolutely clueless. Less would have done too.
Posted by Trixie Wolf on 26th Dec 2020, see comments on the page The Courant & Robbins Contradiction
This is the full content of a comment written by Trixie Wolf on 26th Dec 2020. It was moved here because it was written as an attempt to “educate” me, based on presumptions that I am unaware of various matters, and as though I have not written in detail about Lebesgue measure at https://www.jamesrmeyer.com/infinite/lebesgue-measure.html. If a poster can’t be bothered to read in detail what I am saying, they should not expect their posts to be displayed on the main pages. I ask for logical argument. What Trixie Wolf writes is essentially a long winded exposition of the theory of Lebesgue measure. I will not allow my comments sections to be filled up with verbiage that isn’t relevant commenting. If someone wants to point out an error in what I have written then they should do so. As Wilfrid Hodges said, “to attack an argument, you must find something wrong in it. Several authors believed that you can avoid [that] by simply doing something else.”
This time I actually read the thing. I apologize for the earlier post.
The argument by Courant and Robbins is correct, but it requires additional knowledge of theorems from measure theory (and topology, more generally). Cantor’s original proof, for example, requires knowledge that the real numbers are complete, meaning the supremum (least upper bound) of any nonempty subset of real numbers with an upper bound is also a real number, but the completeness of the real numbers is much easier for people to understand and accept without being shown a proof (which can be done by construction of the reals). In particular, the proof by Courant and Robbins is rather deceptive in that it does not rely on the intuitive idea they appear to be presenting to the reader.
The Courant and Robbins argument relies on an understanding of measure on the real line, which is a simple idea with some nonintuitive consequences. A measure on a set is a function that assigns a non-negative real number (or infinity) to subsets of that set. It is monotone (if A is a superset of B, then A’s measure must be at least as large as B), and countably additive (if you divide A into countably many subsets, the sum of the measures of its subsets must equal its measure).
As these requirements aren’t particularly restrictive, there are, naturally, many measures that can be defined on the same nonempty set. The most common measure used on Euclidean spaces is called Lebesgue (pronounced “luh-beg”) measure, and in our case it is the one-dimensional measure of “length”. Unfortunately, Lebesgue measure cannot be defined on all subsets, but this turns out not to matter for this argument. In particular, certain non-constructable subsets (for which the axiom of choice must be invoked to describe) cannot be assigned a Lebesgue measure, so length is not a total function on the reals. However, you won’t ever encounter non-Lebesgue-measurable sets unless you specifically try to create them with AC, so in this case we can pretend they don’t exist.
Some facts about Euclidean length follow. Any set of reals with cardinality (size) less than the continuum has measure zero. This means the rationals have measure zero, which implies (as you correctly deduced) that the lengths of a set of intervals bounding the rational numbers need not sum to the interval (in fact, you can cover the rationals between 0 and 1 with intervals of arbitrarily small total length). Some sets of reals with cardinality of the continuum also have length 0, such as the Cantor set. You might assume this is because the Cantor set is nowhere dense (it contains no intervals), but a similar construction called the Smith-Volterra-Cantor set or “fat Cantor set” is also nowhere dense, yet has positive length, so sets containing intervals are not the only sets with length. All intervals are Lebesgue-measurable, and their length is defined to be X – Y where X is no less than Y. Thus the length of [0, 1] is 1, even if you remove all the rational numbers from it such that there are no longer any intervals (remember, measure is monotone and countably additive, and the rationals form a measurable set of length 0, so 1 – 0 = 1).
Armed with these facts, the argument by Courant and Robbins now comes into sharper focus. If we take the union of all intervals in the construction, because length is monotone, the union cannot be larger than the sum of the lengths of the intervals, and remember: we’re assuming there are only countably many intervals, so countable additivity applies. Since the union of these lengths is less than the length of [0, 1], we have our contradiction: the Lebesgue measure of [0, 1] is 1, but the total Lebesgue measure of the union of these intervals cannot equal this. The only assumption we made here was that the number of intervals is countable (in which case, being countably additive and Lebesgue measure being sound, the lengths must add up), so that assumption must be false.
The problem with this approach is that Courant and Robbins are leaning hard on what is essentially a false intuition that “you can’t cover everything in [0, 1] because there are too many real numbers to cover”, but this is not going to sit well with students who are intelligent enough to see the presentation isn’t rigorous. I think it’s an interesting and worthwhile argument (and it is correct), but presenting it like this is a mistake.
All that said, this has nothing at all to do with Platonism. Platonism is a philosophical viewpoint on mathematics that cannot be proven or disproven by manipulating mathematical symbols.
Posted by Trixie Wolf on 1st April 2021, see comments on the page The Courant & Robbins Contradiction. My responses are given in green text.
This is the full content of a comment written by Trixie Wolf on 1st April 2021. It was moved here because Trixie Wolf ’s comment was far too large to be in the comments section and filled with fallacious argument rather than logical refutation. I include below in green text my responses to his points.
I very much appreciate you keeping the text of my response, even if you moved it elsewhere. My memory and concentration are awful for medical reasons and I might not have been able to return to the discussion otherwise. I’m happy to address your concerns here in greater depth.
I only ask that you read everything I’ve written below, and give it consideration. It can be difficult to engage in sincere discussion if your motivation is debate rather than understanding, and while I’m doing my best to address what you’ve asked for, the context around your blog entry plays an essential role in addressing any misunderstanding.
The reason I didn’t go point-by-point through your discussion is that it was clear from your arguments that you aren’t following the intended proof (very possibly because Courant and Robbins presented it poorly) due to missing some facts from measure theory. I don’t have access to your textbook, and you provided only a vague outline rather than the text of the proof. Based on what you presented, I assumed you were missing vital information. I believed that elucidating the facts I presented in my previous response would be more helpful, since what you were arguing against was not the proof you intended to discuss, but a misunderstanding of the proof (and again, that may be due to Courant and Robbins). Since what I wrote didn’t help, I’ll walk through your discussion and point out the misunderstandings as you’ve requested.
Before I begin, however, I think at this point it’s important to understand that this isn’t the “Courant and Robbins proof”. The proof Courant and Robbins were trying to present is a proof the field of measure theory, first generated by Émile Borel, which follows largely from the Heine-Borel theorem of 1895. It is related to one of Cantor’s own proofs of uncountability (1879) which relies on similar facts from measure theory. It requires an understanding of basic measure theory, but has nothing to do with mathematical Platonism.
JRM response: Factually wrong.
It’s interesting that Trixie Wolf decides that he knows best what the proof was that Courant and Robbins presented, when by his own admission, he hasn’t actually seen it. Courant and Robbins (CR) is a proof by intuition. CR claim that they have what they call an intuitive proof, and that the differences in the measure given by the limit of a geometric series and the total measure is a contradiction. They admit that it’s not a fully worked proof and they say it would need more detailed treatment. But if they’re going to call it a proof and an intuitive proof, I take that at face value. I find it interesting that CR says it is intuitive, and I think that reflects the way that many Platonists think of real numbers as being a “bigger” set than the rationals. When CR says it is intuitive, I consider that CR meant that the casual reader wouldn’t need to know anything about the details of Borel, but could intuitively “see” the result that CR considered to be intuitively obvious, where the (fallacious) intuitively obvious is that there must be “more” real numbers than natural numbers. Of course, CR’s proof is quite informal, and so it’s not surprising that my treatment of it is quite informal also. Trixie Wolf is missing the point here, the reason I included CR was because I consider that it demonstrates a certain sort of Platonist mindset that can result in overlooking some hidden assumptions which can lead to contradictions. And that’s why I give more detailed treatments of such matters elsewhere, for example at Lebesgue measure, Infinity and limits, and SVC sets and measure (PDF) where the contradiction noted by CR is discussed in detail.
In addition, Trixie Wolf doesn’t give a specific reference to the text of the Borel he mentions, so I can’t reference it on my site so that readers can judge whether his claim has any validity.
Simply put: this is a proof that has withstood the test of time. Thousands of the most intelligent mathematicians of the past 120 years, including people entirely opposed to the Platonist worldview, accept the correctness of this proof (although it is a nonconstructive proof, so it is not “accepted” in the strictest sense by nonconstructive mathematics—but this does not mean they see the proof as “flawed”, just not constructively valid, and there exist many constructive proofs of uncountability).
Based on this, I feel it may be useful to approach the situation here with a little humility. Do you truly believe it likely that it would be “embarrassingly easy” to show that this argument is incorrect, when the worldwide community of established mathematicians, including those entirely opposed to Platonism, has universally accepted the argument for more than a century? Common sense suggests it is much more likely that there is a misunderstanding of the proof on your part, and this is something that hopefully you would want to rectify.
JRM response: The tired old Argument by Appeal to Authority (the “that Very Clever Person’s right, therefore you’re wrong” argument)
By that argument, nothing would ever change, and no progress would ever be made. It’s possible that the fact that that argument has been put to me so many times rather than actually pointing out any error in my work might indicate that there are a lot of people out there who don’t like what I am staying but aren’t able to find any error in it, and so they fall back to an appeal to authority.
I’m also a bit concerned because when you point out flaws you have a tendency to do so by insulting and degrading other mathematicians, and you remove people’s comments from view when they contradict your assumptions. Neither of these approaches is productive to discussion or learning.
JRM response: Accusations without supporting evidence.
Trixie Wolf is making very strong accusations here, and to do so without presenting evidence to support those sweeping generalizations is disturbing. Removing comments when they contradict my assumptions? If Trixie Wolf thinks that is the case, seeing as he is not shy of voluminous writing, why doesn’t he provide the instances for evidence?
As for my moderating comments, a comment section is for, unsurprisingly, comments, which should be concise and pithy, not rambling essays. I reserve the right the keep my comments section as it should be, and unless a comment is completely specious, I post it on this moved comments page rather than remove it.
With that in mind, I can address the proof and your discussion. Bear in mind, however, I don’t have access to the text you are arguing against.
1) “It appears that it was Courant and Robbins who originally came up with the proof, although it can be seen mentioned elsewhere.” This is not correct, but I’ve addressed it above.
JRM response: Factually wrong, as noted above.
2) “a logical analysis of the underlying argument shows that the Platonist belief in the independent ‘actual’ existence of limitlessly many mathematical entitles”
Proofs from set theory, including this one, do not assert the “actual” existence of any mathematical entities or objects. They assert only what symbolic manipulation can and cannot do, and rely only upon base assumptions from mathematical logic and axioms (and a couple of undefineables: a thing called “set” and a binary relation on sets called “membership”). The only assumption from set theory regarding infinitely many things is the Axiom of Infinity, which states there exists an infinite set. But this does not mean there exists a real, physical, infinite set. It only means we can manipulate symbols which represent infinite sets of things in a particular way without reaching a fundamental contradiction in the process. Most of us are comfortable that set theory does not contain fundamental contradictions because none have been discovered after an extensive amount of work trying to break its foundations.
JRM response: Argument by selectively ignoring facts.
While set theory may prove some results, the fact is that there are many erroneous claims that set theory proves certain results when that is not the case. For example, in a strict formal system, an non-denumerability proof only proves that there cannot be any one-to-one correspondence between two given sets within that formal system. But conventional mathematics simply assumes, without any supporting proof by the axioms of set theory - as I explain on my pages Proof of more reals than naturals? and Cardinal Numbers - that that implies that there “exist” numbers that cannot be defined in any mathematical system, and that in turn implies that there are infinite sets that are “larger” than other infinite sets because they have these additional “existing” but non-definable numbers. The irony is that if there actually was a proof that there “exist” “indefinable” numbers (there isn’t), then that proof would be proving that some entities “exist”, but that there is no sequence of symbols can define even one of them - so is set theory only about symbol manipulation?
3) “it shows that that belief leads to a contradiction”
A belief cannot lead to a contradiction in the sense in which we mean the word “contradiction” in mathematics. Logical contradiction is arrived at by symbolic manipulation. Faulty axioms can lead to a contradiction, but axioms are not beliefs, and beliefs about Platonism cannot be expressed in axiomatic form in the mathematical language that set theory is restricted to.
JRM response: Argument by selectively ignoring facts.
A belief system can lead to a contradiction if that belief system leads to an unfounded untenable, probably implicit, assumption, and which is used to prove something, and where the contradiction may not be immediately obvious. As I noted above, Trixie Wolf’s claims re the formal axioms of set theory conveniently ignores the beliefs that are tagged onto such theories, but which are not proved by them, and which lead to contradictions.
4) “At that point they make a classic error of logic. They assume, without providing any logically reasoned argument, that the only possible cause of the contradiction is that some real numbers are not included in the list of real numbers (whatever list might be used).”
The form of a proof by contradiction is to make a single assumption, while everything else in the argument is well-established. If the only assumption made is that the list is countable, then it is a valid conclusion that the assumption is wrong. To show that this may not be the case, you need to point out that the authors made at least one additional assumption in the proof (which I assume you will do further on). But proof by contradiction is a valid form of argument and this is precisely how it works.
JRM response: Factually wrong.
There is an assumption that the calculation of a limiting value for measures of ever-decreasing intervals that all have rational denumerable endpoints (even in the limit), and where the right/left endpoint of an interval is the left/right endpoint of the next interval in the sequence, must also be applicable to the case of intervals that overlap and where the limiting case requires the final endpoints to be irrational.
5) “But that ... does not prove that such numbers cannot possibly be covered by some interval that is centered on a number in the list,”
The use of “prove” here is a little confusing because this statement is not what is being proven. I think you mean they failed to show that some numbers are not covered by any interval as part of the proof, but this follows from proof of the countable subadditivity of Lebesgue measure. Specifically, it can be shown in mathematical logic that Lebesgue measure is countably additive for sets on which measure can be defined, and the Lebesgue measure on real intervals is explicitly defined as b – a.
More specifically, it can be rigorously shown that the Lebesgue measure of a set that is the union of an at-most-countable collection of Lesbegue measurable sets has a single, precisely defined value which is equal to the sum of the measures of all the sets in the collection (or else has infinite measure if one of the sets has infinite measure or if there are countably many sets whose sum does not converge to a finite value). Since the measure of [0, 1] is 1 and cannot have any other value under Lesbegue measure (under the standard definition of measure on the reals), any set of points whose measure does not sum to 1 is guaranteed not to be the interval set [0, 1]. This means that it is sufficient to show a set has a measure other than 1 to prove that the set cannot be [0, 1], which is what the proof illustrates.
(*) JRM response: Argument by selectively ignoring facts.
Trixie Wolf refers to the “rigorous” “mathematical logic” of Lebesgue measure, which, as I have already noted, involves the assumption regarding a limit summation value - as detailed in my previous response above. In terms of logic, of course, anyone can say that a proof is “rigorous”, by which they mean that given:
- the initial assumptions/axioms, and
- the logical rules of inference,
then the result inevitably follows.
That is correct, that is rigorous, but it does not address the question as to whether that system of assumptions and rules of inference is consistent, i.e. that the system is free of contradictions. But Trixie Wolf’s glib claim that it is a rigorous proof simply ignores this elephant in the room - the inherent contradiction - as I point out on my site at Lebesgue Measure.
Reject the causative assumption and the elephant - the contradiction - disappears.
6) “[But that] does it[not?] prove that a collection of such numbers - each of which is a single isolated point - can constitute a length.”
This is a recurrent theme in your discussion which I think is the largest source of error in your arguments. You aren’t defining what “length” means, but rather, appear to be working from an intuitive definition of what may “constitute a length” in your mind, as though the essence of the word is up for argument. That is not mathematics. When we say “length” here, we mean Lesbegue measure, and the Lesbegue measure of any collection of intervals is well-defined.
JRM response: Argument by assertion without any supporting evidence
Trixie Wolf says that my “largest source of error” is that I don’t define “length” but unless Trixie Wolf actually shows where in my writings this is, or generates an error, that is meaningless verbiage. For an individual interval the length or measure is simply the difference in the value of the two endpoints. A calculated result that is a length/measure is simply that which, according to the theory and method of calculation, corresponds to the length/measure of a single interval. Where’s the problem? If Trixie Wolf wants the term length/measure to only mean what Lebesgue measure describes, that is Trixie Wolf’s choice, but it doesn’t mean that everyone else has to meekly follow Trixie Wolf’s diktat. There are other ways of defining theories of measure.
The fact that the intervals are themselves composed of sets of points is not a problem in the definition and leads to no contradiction. The proof merely illustrates that any countable collection of points has measure zero. (This follows already from the fact that b – a = 0 when a = b, and Lesbegue measure is provably countably additive.)
JRM response: Factually wrong
Trixie Wolf asserts that there is no contradiction in Lebesgue measure. I demonstrate the inherent contradiction on my page Lebesgue Measure but Trixie Wolf’s doesn’t even start to try to explain where he thinks that demonstration of a contradiction might be wrong.
7) “we can apply the same argument as in the proof above to the fractions between 0 and 1”
And in doing so, you correctly demonstrate that the rational numbers have Lesbegue measure less than 1. (In fact, they have Lesbegue measure zero, as evidenced by the fact that you can make the total sum of the intervals arbitrarily small.)
Some uncountable sets of points have Lesbegue measure zero as well, such as the Cantor set. Other uncountable sets of points, like the “fat Cantor set” have positive measure—and in that case, despite containing no intervals at all! And despite having no intervals, both sets also have no isolated points: for every point and every distance, there are infinitely many points of the set within any given point.
My point here is that working with infinite sets can be counterintuitive, which is why we must rely on definitions and logic rather than mere intuition. Much like learning upon being introduced to negative numbers that a negative times a negative is a positive, we find that some intuition must be abandoned when we jump from the finite realm to the infinite.
JRM response: Exactly, which is why one shouldn’t make intuitive assumptions when calculating measure, such as in (*) above!
8) “But even the abstruse machinations of Lebesgue measure theory cannot rescue Courant and Robbins.”
There’s nothing abstruse or “machination”-like about Lebesgue measure, and it isn’t a theory, it’s a definition. Measure theory is a theory.
JRM response: Argument by word play
This is word play that doesn’t advance Trixie Wolf’s argument. If Trixie Wolf wants to say that a Lebesgue measure value is given by a theory of measure along with Lebesgue definitions, that’s fine, but I consider that a combination of a certain set of axioms, definitions and rules is a theory, and I call that Lebesgue measure theory. If Trixie Wolf doesn’t like that appellation, that’s Trixie Wolf’s problem, not mine.
Lebesgue measure is quite simple, and not mired in the sort of strangeness you seem to be projecting upon it. For measurable sets, it’s just the outer measure: the minimum value you can get by summing the measures of a collection of intervals covering the set. That’s literally all there is to it. The only definition you need is the definition of the measure for an interval, which for the real line is defined as b – a. (Here, measurable sets means sets which (roughly put) when you split the set in two pieces those pieces always sum to the same value.)
JRM response: Factually wrong.
As I have already pointed out, that isn’t simply “all there is to it”, see (*) above.
9) “Furthermore, they even claim that by choosing a smaller and smaller initial fraction, the total length of the purported single points remaining becomes closer and closer to 1!”
This is true. Convergent infinite series with smaller terms do indeed sum to smaller values, and there is no lower limit on how close to zero a series of positive terms can sum to.
10) “The Platonist view assumes that all real numbers are some sort of ‘actual’ things that ‘exist’ in some sense in some ‘reality’,”
There’s a remarkable irony lurking here.
This proof has nothing to do with “real objects”, but your arguments do. You keep referring to measure using intuition rather than symbolic definitions, and it’s the fact that you are projecting Platonism onto a proof that has none in it that you are seeing contradictions in your mind.
JRM response: Factually wrong.
Trixie Wolf quotes “The Platonist view assumes that all real numbers are some sort of ‘actual’ things that ‘exist’ in some sense in some ‘reality’,”. That is what a Platonist is. If Trixie Wolf thinks that someone who assumes that all real numbers are some sort of ‘actual’ things that ‘exist’ in some sense in some ‘reality’ should have a different name, that doesn’t affect the argument, where the point is that a Platonist approaches things and views things differently to a non-Platonist.
A proof is a symbolic manipulation, period, end of sentence. This proof isn’t saying anything about “measuring” something with the ruler in your kitchen, it’s a manipulation of symbols under axiomatic assumptions that produces a result which, in this case, is well-established not to be contradictory. Using certain very specific and rigorous rules, we are able to derive one sentence in a theory from other sentences in the theory under assumptions given only by axioms, which can be kept very simple.
A mathematical proof is not at all what you seem to think it is. It has nothing to do with your notions of “existence”.
JRM response: Egregiously factually wrong. No. No. No.
The fact is that some proofs are fully worked purely symbolic manipulation, but most are not. Most are informal and assume that there is a corresponding fully formal equivalent. Sometimes that assumption later turns out to be wrong. Trixie Wolf simply claims that there is no contradiction in Lebesgue measure, and so completely ignores the obvious contradiction that I pointed out. Ignoring a contradiction doesn’t somehow make it disappear. See the page Mathematical Proofs which includes several relevant links.
That said, Borel’s proof of uncountability is rigorous and correct by the rules of symbolic logic, and no contradiction in its framework (measure theory, and more generally set theory) has ever been found.
JRM response: Irrelevant, as already noted above.
And as I have to continue to point out, there is an inherent contradiction. Simply ignoring it, as Trixie Wolf does, does not make it magically disappear. And even if Borel were relevant, it is yet another appeal to authority (the “that Very Clever Person’s right, therefore you’re wrong” argument).
11) “we would be careful to note 1⁄9 is a limiting value that the values never actually reach”
You are manipulating symbols that have specific meanings and applying other meanings to them non-rigorously. The fact that the rules for decimal notation would require, were you not to specify “this value repeats”, an infinite expansion has nothing to do with the argument. If you write it out in base 3, one-ninth is 0.01. What you’re discussing are limititions and quirks of the rules for using decimal notation, and the meaning for decimal notation comes from rules on manipulating symbols that you are not using or interpreting correctly.
JRM response: Argument by assertion without any supporting evidence
This is rambling verbiage that doesn’t even begin to reveal any error in what I have written, or how I am supposedly using certain symbols incorrectly.
I hope you read this. I really, really do. However, after reading some of your other discussions and responses, I’m not optimistic. I’m certain you’ll skim this and throw it away, because the alternative is too difficult: slowing down and considering what I’m saying might make a dent in your belief that you are smarter than tens of thousands of people, the very best in the world, who have spent their lifetimes carefully and rigorously applying rules which you disregard from ignorance regarding their essential importance and a mistaken belief that mathematical proofs are somehow statements about the “existence” of objects found only in your imagination.
JRM response: Appeal to authority
Yet again, Trixie Wolf relies on the tired old Appeal to authority line, it’s getting very tiresome.
JRM response: Ad Hominem attack
And finally he resorts to the ad hominem attack (“belief that you are smarter than ...”), picking that out of thin air without providing any evidence that I have such a belief. Perhaps some people might think that such ad hominem attacks indicate that the person making them does so because they feel that their other arguments aren’t good enough?
I never claimed that “mathematical proofs are somehow statements about the ‘existence’ of objects”. What I did claim is that much of conventional mathematics is based on assumptions that arise from a belief in the independent non-physical existence of mathematical entities, and this affects the consequent construction of the mathematical system used. As I have stated on this site, a mathematical proof should be a series of statements that logically follow from one another from an initial set of assumptions, where the logical rules and initial assumptions cannot result in a contradiction. That is nothing to do with the general notion of the existence or non-existence of objects - to claim that I assert that a mathematical proof is a statement about the existence of objects (in general, as opposed to proving that a certain mathematical entity exists within the system, which is what most proofs do) is a cheap shot which completely misses its target. It’s rather ironic that were there actually a proof that there “exist” “indefinable” numbers (there isn’t, see Proof of more real numbers than natural numbers?) then that proof would be proving that some “indefinable” entities “exist”, and that those entities somehow belong to the domain of certain variables, i.e., those entities somehow can satisfy the definition of the domain of the variable, even though in a fully formal system, the domain of a variable is simply a defined set of symbol sequences.
Posted by Eagle on 14th April 2022, see comments on the page Is Mathematics Unreasonably Effective?
I will try to explain myself in a more detailed manner.
I simply say that mathematical statements are of the form "given certain definitions and axioms, we get certain conclusions". Once such a statement is obtained, this is forever. However, the fact that such a statement is deemed very relevant in applications, moderately relevant or irrelevant, well this assessment can evolve over time. Maybe we do agree on this point. Truth (seen as intern to mathematics) is forever while relevance (when connecting ideas with reality) may fluctuate as science progresses.
My parallel between the situation for Banach-Tarski and that of polynomials was simply an analogy to explain that, when studying a topic, given some intuition we have, it is possible that we first rigorously prove a statement going against the intuition without it being the end of the story: the door is still open for a second statement that would go in favour of the intuition. The only way this can happen, if mathematics is consistant, is that we are speaking of two different things in these two steps. Both try to model the same situation but they do so with subtle differences, and they do not give the same answers to all questions. Once again, what I had in mind being clarified, it may be that we agree on certain things here.
The situation of the paragraph above can appear in theory, but can it appear in practice? Well, I claim it appears in the case of defining the volume of all parts of 3D space. But since it is technical to get to details, I prefer to explain that a scenario as in the paragraph above is not fully unreasonable by relying on such a scenario that is fully elementary. I continue to consider this example as mathematically enlightening but it is possibly not relevant from the point of view of history of science. I also admit that this is a statement about "really mathematical objects" (polynomials) while the study of volume tries to model the volume of real 3D objects.
I came up with this analogy with polynomials because Olivier Leroy himself came from algebraic geometry and considered his solution to Banach-Tarski via locales as something analog to "getting from affine geometry to projective geometry": (https://www.pourlascience.fr/sd/mathematiques/grothendieck-un-heritage-mathematique-fertile-9237.php)
Trying the make this switch understandable for a student of 1st year of university rather than 3rd year, I decided to mention a more basic switch simplifying a mathematical theory (and not unrelated to algebraic geometry), namely the question of counting roots of polynomials.
What is meant by "in a better way"? I am no expert of the field (even though I read and understood the paper of Simpson) but locales have initially been studied by people very far away from measure theory (algebraists, abstract topologists, logicians) with their own criteria in mind (I guess "having a category with nice properties", a bit like before when we wanted the good way to count roots). I find it remarkable that, even though locales were not at all designed for measure theory but only for topology, it turns out to enable us to measure all parts of space!
It is also interesting to notice that, from the point of view of locales, no need for Baire theorem: in the sense of locales, the intersection of ALL dense open subsets of R^n is itself dense (it does not contain any points, but it intersects any nonempty open set).
From the point of view of science, I find locales interesting as they consider that the primitive notion is that of opens, rather than that of points. A topological space is a set of states endowed with a topology ; a locale is "the lattice of opens". If you consider that we you access in experimental physics is not the state of objects but results of measurements, it is interesting to axiomatise a general notion of "properties that can be checked by an experiment", which yields the definition of a topology. See https://mathoverflow.net/a/19384 . Locales pushes this one step further by not only keeping a strong focus on opens but forgetting about points (which has nice consequences, as the one of the previous paragraph).
Finally, I should clarify which point of footnote  I was in disagreement with. It was "if you restrict your working by some convoluted maneuvers so that you are not invoking a part of your set theory", which I interpreted as either "For locale theory to work, we need to close our eyes and not use some part of mathematics, otherwise we get a contradiction" or "Locale theory is essentially throwing away the bad sets of R^n to get the problem under the carpet". I do not know if you had any of these interpretations in mind or not, and I am interested to know what you had in mind.
Meanwhile, I explain why the two interpretations above are incorrect. For the first one, locale theory is as the rest of mathematics: as long as we respect the laws of mathematical reasoning, all proofs are obviously welcome. It is just that we can model 3D space in two different ways, leanding to two mathematical statements: R^3 as a topological space and R^3 as a locale.
As for the second point, it is quite the opposite: any classical subset yields a sublocale (in an injective manner), but their are additional sublocales that do not come from classical subsets. So locale theory is more deploying/unraveling the classical point of view than constraining/restricting it. Contradiction is avoided because when classical subsets are considered as sublocales, their intersection may not be classical! It will be a sublocale whose points are exactly the points of the classical intersection, but as sublocales are not uniquely defined by their points, this does not suffice to say that this sublocale is the one associated with the classical intersection of these sets (and it is simply false in general). Actually, locale theory guarantees that the exterior measure becomes sigma-additive (in a suitable sense). While usually, it satisfies only one inequality. It is a bit like with polynomials where, without multiplicity and complex roots, we only have one inequality: we need to unravel the situation to find "where goes what is missing".
Locale theory somehow makes sense of the topological glue that goes "between points in space and glues them together". And it says that when we remove all singeltons from space, there is still a lot of glue... and that this glue should be given mass (I mean "volume", "Lebesgue measure").
Posted by Jorgen Harmse on 1st July 2022, see comments on the page Cardinal Numbers
"... not simple and straightforward." That's fair enough: defining cardinality in plain ZF requires ordinal numbers and the constructive hierarchy. However, you seem to have some problem with cardinality theory that goes beyond technical difficulty. If you admit that cardinal numbers are well defined then we have fewer disagreements than I thought.
"I didn't say that the definition depended on the continuum hypothesis." I see: you link to some point you make in your discussion of the continuum hypothesis. However, it seems that you are now denying that cardinal number is well defined. It's very hard to discern what position you are taking.
"... laughable in its pretence that the meaning is not obvious ..." It's obvious to me that the set of integers is not limitlessly large because it has zero Lebesgue measure and its cardinality is less than that of the real numbers. Apparently the opposite is obvious to you for some reason. Saying that a concept is obvious is not a substitute for a definition.
The definition of cardinal number was obvious to Cantor: he had defined equipotence, and a cardinal number was simply an equivalence class under equipotence. He realised a few decades later that the arbitrary collecting of objects into sets can lead to paradoxes. (This is why plain ZF needs something like Scott's trick and von Neumann set theory distinguishes between sets and other classes.) You ask us to be rigourous and give actual definitions (which we have done), but then you pretend that 'limitlessly large' is obvious.
"... "the notion of a limitlessly large set that can be smaller than some other limitlessly large set" serves to highlight the inherent absurdity ..." Now you're adding the undefined term 'smaller'. Presumably these terms could be defined in a reasonable way that implies that a limitlessly large set cannot be smaller than any other set, but so what? It would then turn out that either there are no limitlessly large sets or your notion of 'smaller' is unconnected to cardinality.
"Yet more evasive word play." What? Your whole article is evasive wordplay.
"Extending that concept to infinite sets necessarily requires that the cardinality of an infinite set is also a property that is a measure of the quantity of its elements, and where two cardinal numbers always differ by a definitive measure." What do you mean by 'quantity of its elements'? Mathematicians had assumed until Cantor's day that all infinite sets have the same quantity of elements, but Cantor provided a definition to make the notion precise. You could give your own definition and create your own cardinality theory, but I expect that it will not be as powerful, and mathematicians will be annoyed if you use the term 'cardinal number' since that term has for a century been used for something else.
"I didn't say that there was anything wrong with doing so." Thank you for that, but then why mention the difference? It seemed that you were looking for something to criticise, but now I don't know what your point is.
"... resort to any further absurdity ..."? I thought you agreed that cardinal numbers are well defined. What then is the absurdity?
There's a problem with the text interface, so I'll just note that your last paragraph is a misrepresentation.
Posted by Jorgen Harmse on 9th July 2022, see comments on the page Cardinal Numbers
"... definition of cardinal number in conventional set theory is not simple and straightforward." That would be OK, but the whole tone of your article suggests much bigger problems. It was not at all clear that you were just commenting on technical difficulty.
"I didn't say that the definition depended on the continuum hypothesis." I see - you were just pointing to something you mentioned in that article.
"Your criticism that the term "limitlessly large" as applied to a set is undefined is laughable in its pretence that the meaning is not obvious: an infinite set, a set that has no limit to the quantity of its elements." It's obvious to me that the set of integers is limited in quantity in that it cannot be put into one-to-one correspondence with the reals. Our understanding of a term that you consider obvious is apparently different, so we need a definition. Piling on undefined terms may be amusing, but it doesn't show a problem with cardinality theory.
"Yet more evasive word play." Your whole article is evasive wordplay.
"Extending that concept to infinite sets necessarily requires that the cardinality of an infinite set is also a property that is a measure of the quantity of its elements ..." 'Quantity of elements' is another vague term. Cardinality theory provides a way (maybe not the only or even the best, but the only one I know) to make that precise for an infinite set without specifying a well ordering. Pretending that you already know without giving a definition is pretentious bullshit.
"I didn't say that there was anything wrong with doing so." So what's the point of your article? You seem to be saying that there is something unrigourous about cardinality theory.
"... looking desperately for something to criticize." A strange complaint from a writer of polemics.
"... resort to any further absurdity in an effort to prop it up." What absurdity? This sort of writing seems to me like the last gasp of someone who has run out of arguments.
"... at the same time pretend that a difference in the cardinal numbers for two infinite sets cannot indicate a difference in the numbers of the elements in the sets." 'Number of elements' is vague. If you ask how many elements a set has then I would usually answer with a cardinal number, so I would if anything assert the opposite of what you suggest.
If all you have to offer is the claim that the cardinality of a set is not a measure of the number of elements of that set ..." I didn't say that at all. If all you can do is misrepresent then perhaps I should move on.