Hesperus is Bosphorus

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What is Wrong with Cantor’s Diagonal Argument?

with 40 comments

Cantor gave two proofs that the cardinality of the set of real numbers is greater than that of the set of natural numbers. According to a popular reconstruction of the more widely known of these proofs, his diagonal argument, Cantor randomly tabulates the real numbers in the interval [0, 1) in an array. I will use a binary version of Cantor’s table for ease of exposition and give Table 1 as an example:

1    0.10111011000 …
2    0.11010111000 …
3    0.11010001110 …
4    0.01101110001 …
5    0.10000100011 …
6    0.00111111111 …
7    0.00101110001 …
8    0.11000110100 …
9    0.11101100101 …
10  0.01111011100 …
11  0.01010101010
.
.
.
d = 0.11000110100 …
i = 0.00111001011 …

           Table 1

Cantor asks us to assume for reductio purposes that this table is “complete” in the sense that it lists every real number in the interval [0, 1). The digits on the diagonal of the list (boldfaced) give us the infinitely long number d. He then constructs a new number i from the diagonal of the table by substituting every 0 after the decimal point in d by 1, and every 1 in d by 0. He asserts that this constructed number is a real number in [0, 1) which cannot be found in the putatively complete list, because i is guaranteed to differ from each number in the list as the number’s digit that falls on the diagonal is changed in i. Since no list of real numbers (in this example the reals in [0, 1)) to be constructed in the same fashion will manage to include all of the real numbers in [0, 1), it follows, according to Cantor, that no such table will have the items in it being enumerable by natural numbers. In other words, there can be no bijection between the set of natural numbers and the set of real numbers. His conclusion is that the size, or cardinality, of the set of reals is greater than that of the set of naturals.

Let me point out an immediate puzzle with Cantor’s argument. I will call two binary numbers in [0, 1) “inverses” of one another if one number has 1 in a certain location after the decimal point, the other number has 0 in the corresponding location; and if it has 0 in that location, the other number has 1. Thus the numbers on lines 3 and 7, for example, are inverses of each other. So are the numbers on lines 5 and 10. Crucially, d and i are also inverses of each other. If we are to assume this list is complete, every number in the list must have its inverse also included in the list. Cantor says i is not included in the list. But how come? The number ’s inverse, namely d, is in the list—it is on line 8. Since its inverse is in there, i must also be in there, like every other pair of inverses.[1]

I see two flaws in Cantor’s diagonalization procedure. These flaws make it infeasible to construct the number i and thereby invalidate his proof.  Let me illustrate the first one on Table 2.[2] It is obvious that, since d is a real number in the interval [0, 1), d must be included in the list—if we are to assume the list is complete. Suppose it is on line 8.

1    0.10111011000 …
2    0.11010111000 …
3    0.11010001110 …
4    0.01101110001 …
5    0.10000100011 …
6    0.00111111111 …
7    0.00101110001 …
8    0.1100011?100 …
9    0.11101100101 …
10  0.01111011100 …
11  0.01010101010
.
.
.
d = 0.1100011?100 …
i = 0.0011100?011 …

         Table 2

The n-th digit of d after the decimal point comes from the n-th digit, after the decimal point, of line n in the table. Thus the first digit of d, namely 1, comes from the first digit of line 1. The fourth digit of d, namely 0, comes from the fourth digit of line 4, and so on. It will be noticed that the 8th digit of d (marked with ‘?‘ on line 8) is indeterminate: unlike the other digits of line 8, the 8th digit of line 8 does not come from any other line than line 8 itself. But then there is nothing to determine whether the 8th digit of line 8 must be 0 or 1.

Thus, if d is included in the supposedly complete table, d will fail to satisfy a necessary condition for being a real number in [0, 1), viz. having all determinate digits. This of course entails that the value of the 8th digit of ’s inverse i is also indeterminate, and hence i too fails to be a real number in [0, 1). The upshot is that Cantor’s supposition that he can unproblematically take the inverse of the diagonal of the table is false.

Let us forget about the first flaw for now and assume that there is a determinate diagonal of the Cantorian table. (Thus I assigned 0 to the 8th digit of line 8 in Table 3 below, for illustration’s sake.) The second flaw arises from the following phenomenon: the diagonal and its inverse clash at some digit. Suppose the inverse i of the diagonal were on line 10 in Table 3:

1    0.10111011000…
2    0.11010111000…
3    0.11010001110…
4    0.01101110001…
5    0.10000100011…
6    0.10111111111…
7    0.00101110001…
8    0.11000110100…
9    0.01110011100…
10  0.001110011!1…
11  0.01010001010
.
.
.
d = 0.110001101!0…
i = 0.001110010!1…

           Table 3

If the diagonal’s 10th digit (marked with ‘!‘ on line 10) is 1, ’s 10th digit has to be 0, and if the diagonal’s 10th digit is 0, ’s 10th digit has to be 1. In other words, since the diagonal’s and ’s 10th digits coincide, we get the result that their shared 10th digit cannot consistently be either 0 or 1. Hence inclusion of i in Cantor’s table leads to paradox! But as we have argued earlier, i must be included (assuming d could) in a table that claims to be complete—complete even if only for reductio purposes.

It is obvious that these two flaws arise from the fact that Cantor lists real numbers in the form of a table.  Such a table turns out to be a deceptively erroneous way of displaying the totality of real numbers.[3]

——————————————–

[1] If i were not to be in the list although its inverse is, we could not assume the list was complete to begin with.

[2] I disregarded the first flaw when setting up Table 1 earlier, for convenience of exposition.

[3] My much more detailed criticisms of Cantor’s arguments and my way of showing that reals cannot be more numerous than naturals can be found at: https://www.academia.edu/26887641/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_

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Written by Erdinç Sayan

July 31, 2016 at 1:55 pm

40 Responses

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  1. Couple of questions:

    – What about other proofs that the reals and the naturals cannot be put into 1-to-1 correspondence? Do you think there is no sound proof of this?

    – I don’t see what warrant you have for claiming that there is ‘no paradox’. You seem to be insisting on (iv) at the expense of (v), but (v) seems impeccable too, so what do you have to say against it? How could it not hold? Until you address (v), I don’t think you can claim that there is no paradox.

    (I write as someone who has philosophical suspicions about the way Cantor’s results are interpreted, but it seems like you are actuall trying to cast doubt on the correctness of the mathematics itself, which seems highly dubious to me.)

    Tristan Haze

    August 2, 2016 at 1:52 am

    • Thanks for the questions.

      – Yes, there are other proofs to the contrary. All I can say is that if my arguments are sound (as I think they are), then those other proofs will have to be rechecked.
      – I have changed the last two paragraphs of my post. As I say in my second footnote, my full criticism of (v) and some other criticisms I level against Cantor can be found at: https://www.academia.edu/26887641/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_

      Erdinç Sayan

      August 2, 2016 at 1:31 pm

  2. Your treatment makes several errors that all Cantor doubters seem to make.

    First, only Cantor’s first proof was about the real numbers. The Diagonalization proof used what I call Cantor Strings: infinite-length combinations of only two characters. And while you can – as you did – treat Cantor Strings as binary representations of real numbers in [0,1), the complete sets are different. The real number 5/8 has two Cantor Strings, 0.1010000… and 0.10011111. It is possible to get around this problem (which is why it is usually taught using real numbers), but that is not what Cantor did. This is a minor point, tho, since you used Cantor Strings without knowing it.

    Second, Cantor did not use reductio ad absurdum. But he did use something similar: contrapositives. We know from logic that if the statement “If A, then B” is true, then so is its contrapositive: “If not B, then not A.”

    Third, and probably most important, Cantor *NEVER* assumed he was working with an array of *ALL* Cantor Strings. What he assumed at the start of the proof, is that he had a countably-infinite set of Cantor Strings. He called it M.

    Fourth, the array implied by this set cannot be called “square.” Yes, it has infinitely-many rows, and infinitely-many columns. So does the array defined by (E,N), where E can be any even number, and N can be any natural number. You can’t compare the lengths in a way that would fit the definition of “square.”

    Lastly, because he never assumed M contained all Cantor Strings, the crux of your argument is incorrect. Even if the string d is in your list (which you didn’t prove, btw; you just used an example where it was), it does from “a in in the list” that “a’s inverse must be in the list.” In fact, the proof shows quite simply that it d’s inverse can’t be.

    What Diagionalization directly proves is “If M is a countably-infinite set of Cantor Strings, then M is not the complete set of Cantor Strings.” The contrapositive is “If M is the complete set of Cantor Strings, then it is not a countably-infinite set.”

    See http://www.logicmuseum.com/cantor/diagarg.htm if you want to read the actual proof.

    jeffjo56

    August 20, 2016 at 4:33 pm

    • Typo: “it does from” in the second to last paragraph should be “it does not follow from”

      jeffjo56

      August 20, 2016 at 4:36 pm

      • Thanks for the long comments.

        Your construal of Cantor’s argument seems kind of different from the widespread construal of it, especially at the point that Cantor’s array of real numbers is assumed by him at the beginning of his reductio argument to be comlete. That’s why all the objections about “completed infinity,” “actual infinity” arose. Let me say that I follow the common construal.

        “Lastly, because he never assumed M contained all Cantor Strings, the crux of your argument is incorrect. Even if the string d is in your list (which you didn’t prove, btw; you just used an example where it was), it does not follow from ‘a is in the list’ that ‘a’s inverse must be in the list.’ In fact, the proof shows quite simply that d’s inverse can’t be.”

        d is a perfectly real number (or, in your terms, a Cantor string), so I don’t see any reason why d souldn’t be in the presumedly exhaustive list. If d were not in Cantor’s list, then his proof would trivialize: d is not in the list, hence its inverse won’t be in the list, hence he has “found” some number (the inverse) which lies outside of the list.

        If d is a real number, I don’t see any reason (though I won’t give any formal proof) why its inverse shouldn’t also be a real number, and hence not be included in an assumedly complete list of real numbers.

        “What Diagionalization directly proves is ‘If M is a countably-infinite set of Cantor Strings, then M is not the complete set of Cantor Strings.’ The contrapositive is ‘If M is the complete set of Cantor Strings, then it is not a countably-infinite set.’”

        This statement sounds analogous to the following: “If M is a countably-infinite set of real numbers, then M is not the complete set of real numbers.” Therefore, “If M is the complete set of real numbers, then it is not a countably-infinite set.”

        Erdinç Sayan

        August 22, 2016 at 2:06 pm

        • I gave you a link that to Cantor’s actual paper in both German and English. I can’t, and do not need to, defend the way it has been misinterpreted in the years since he wrote it. But if you want to claim a disproof, you do need to address the original, and not a misinterpretation.

          “I don’t see any reason why d shouldn’t be in the presumably exhaustive list.” Yes, d should be in a list that is assumed to be exhaustive. In the correct proof, the list ISN’T assumed to be exhaustive, so this point is irrelevant.

          You can make an (incomplete) infinite list from 01000…, 001000…, 0001000…, etc. The d you get from this list is 111…, and it is not in the list. Neither is its inverse. But in general, d may or may not be in the list you make from a countable M. But it is proven that i isn’t.

          And yes, those two statements at the end of your reply are very similar. But the one about real numbers is not proven as easily as the one about strings.

          jeffjo56

          August 22, 2016 at 3:15 pm

  3. Can you recommend some work published in recent times that adopt your rendering of Cantor’s diagonal argument?

    Erdinç Sayan

    August 22, 2016 at 9:51 pm

    • Why? Do you think being more recent makes it a more accurate representation? In my experience with this theorem, being more modern usually means it contains three errors that were introduced to make it easier to explain the theorem to school children.

      The first is that they use numbers. School children are only familiar with infinite strings when they are used to represent numbers. They can understand that 0.333… represents 1/3, but they usually don’t grasp the full significance of the string being infinite. The fact is that Cantor developed this proof specifically to avoid using numbers. But like I said, this error can be dealt with – it just is usually ignored.

      The second is that they claim the proof is by reductio ad absurdum. Cantor never made any such claim. It is by contraposition. The difference between contraposition and reductio ad absurdum can be too subtle to explain to school children.

      The third is that there is an assumption that the entire set of real numbers/Cantor Strings can be counted. You’d need that kind of an assumption in reductio ad absurdum, but not contraposition. Which is why error #2 is significant.

      +++++

      Here is a more modern outline of the proof:

      1) Let C be the set of all Cantor Stings.
      2) Let S be any infinite subset, proper or improper, of C.
      3) Assume that S is countable; that is, that a function s(n) exists mapping each natural number in N to an element of S, so that every member of S is represented exactly once.
      4) The function s(n) can be used to define a Cantor String via diagonalization; that is, a Cantor String d whose nth character is the nth character of s(n).
      5) The Cantor String i made by inverting every character in d is not in S.
      6) This proves the statement “If a subset of C is countable, then it is not all of C.”
      7) The contrapositive is also proven: “If a set S is all of C, then it is not countable.”

      QED.

      jeffjo56

      August 23, 2016 at 9:06 pm

      • Actually I wanted to know who else, other than you, presents Cantor’s argument the way you do.

        Thanks for the long explanations.

        Erdinç Sayan

        August 24, 2016 at 9:36 am

  4. Thanks.

    6) This proves the statement “If a subset of C is countable, then it is not all of C.”
    7) The contrapositive is also proven: “If a set S is all of C, then it is not countable.”

    Btw, (7) sounds false. If S is the entire C, i.e. if S=C, then S must contain the inverse of every string. Therefore S must also contain the inverse of d for the diagonal of S.

    If (7) is false then it follows that (6) is false too, which should be the result of Cantor’s procedure being flawed.

    Erdinç Sayan

    August 24, 2016 at 2:22 pm

  5. Gee, 2+2=4 sounds false to me; I mean, it can’t ALWAYS be true, can it? So there must be something wrong with arithmetic.

    This what is called a strawman argument, and it is a fallacy. But it is common, specifically because ““If a set S is all of C, then it is not countable”” sounds suspicious, that so many people try to find a flaw in it. Usually through fallacies, although not often as blatant as the one I parroted from your last reply.

    And it doesn’t help that the proof is almost always misrepresented. Like I said before, I can’t defend the practice. Only point it out. That’s why I balked at providing you with more modern treatments. That, and they will usually present the more formal version about powersets instead of Cantor Strings or real numbers. And they tend to use more formal terms, like “bijection” and “onto,” which I can’t be certain a person I encountered on the internet will understand.

    But I just tripped across this one, looking for something else: https://www.google.com/search?q=modern+version+cantor+diagonal&ie=utf-8&oe=utf-8 . It presents two versions, the less formal one about real numbers, and then the formal one about powersets.

    Note that the “real number” one is wrong in an inconsequential way. Since some real numbers have two decimal representations (1/10 = 0.1000… = 0.0999…), proving that you have an unrepresented decimal is not the same as proving you have an unrepresented real. (To avoid this flaw, you shouldn’t add one to each digit; instead, use a “5” if it was a “4”, and a “4” if it was anything else.)

    But the two proofs do correct the other two errors you made: the set that is diagonalized is never assumed to be complete, and the proof is not demonstrated by contradiction, it is by contraposition.

    jeffjo56

    August 25, 2016 at 12:48 pm

  6. Cantor’s theory fails because there is no completed infinity.
    In his diagonal argument Cantor uses only rational numbers, because every number up to the diagonal digit is rational. How it will be continued is irrelevant for the argument. It is neither a matter of the proof nor of mathematics, because there is no definition.

    But there is a stronger argument: According to set theory the figure
    1
    2, 1
    3, 2, 1

    contains an infinite number of elements, namely all natural numbers.

    No row of the figure contains an infinite number of elements.

    According to mathematics, the figure is an inclusion-monotonic sequence of finite rows. That means: Every union of finite rows is contained in one of the unioned rows. (The principle of construction shows, that this property is independent of the number of finite rows. Otherwise there would be a first finite row that does not contain all elements of its predecessors.) If the contents of the figure is a fixed quantity, then this fixed quantity is in one of its rows. Contradiction. Therefore the idea of the natural numbers being a fixed quantity can be excluded.

    For more arguments see
    https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

    Regards, WM

    WM

    November 4, 2016 at 5:59 pm

    • Thanks for the comment.

      “In his diagonal argument Cantor uses only rational numbers, because every number up to the diagonal digit is rational. How it will be continued is irrelevant for the argument.”

      Interesting observation. As I understand, you are saying that Cantor’s table can be seen as a table containing rational numbers rather than real numbers. Although the diagonal d (and its inverse i) is a real number which would not be in this table of rational numbers, that is irrelevant to what Cantor wants to prove.

      Your book in the link you give is pretty comprehensive. Could you tell me exactly where I can find the arguments against Cantor’s diagonal procedure in the book?

      Erdinç Sayan

      November 6, 2016 at 1:26 pm

      • You can find this argument under “Only terminating entries are applied in the diagonal argument” on p. 273.
        Further you can find the argument that an infinite sequence of digits is not a real number (but only converges towards a real number) under “Sequences and limits” on p. 242.

        Regards, WM

        WM

        November 8, 2016 at 11:12 pm

        • Thanks.

          Erdinç Sayan

          November 10, 2016 at 6:10 pm

    • Every field in mathematics is based on definitions of abstract concepts, and axioms that describe unprovable properties they have. While the abstractness of these concepts varies, none have concrete definitions. There is no such thing, in an absolute concrete sense, as a point, a line, a circle, a set, or even the number “1”.

      Some of the axioms can be debated. Euclid’s Plane Geometry is based on the axiom that for any line, and any point not on that line, there is exactly one line through the point that does not intersect the line. There are other Geometries that assume there are none, or many. These conflicting axioms, and conflicting theorems drawn from them, do not invalidate any of these Geometries.

      When you say “Cantor’s theory fails because there is no completed infinity,” you are stating an axiom that contradicts one in Cantor’s mathematics in the same way. When you say “It is neither a matter of the proof nor of mathematics, because there is no definition,” you are describing your mathematics, not Cantor’s. Anything you claim to prove in that Mathematics has no bearing on Cantor’s.

      Cantor’s Mathematics includes an axiom that goes something like this: “There exists a set that includes the number 1 and, if it includes the number n, it also includes n+1.” This accepts the existence of a set which is defined to have the abstract property “infinite.” It also allows Cantor’s mathematics to define the property “countably infinite.” If your mathematics does not include this axiom, and so does not include this definition, then you shouldn’t care that Cantor’s proves the existence of uncountably infinite sets. Because you are using a different mathematics. And it is one that is internally inconsistent, since it needs a largest natural number.

      Cantor’s actual proof (as opposed to what is taught to beginning, and therefore naive, math students) does not use numbers. It uses combinations of two characters put into 1:1 correspondence with the infinite set it assumes exists, which invalidates the rest of your argument. The rows are defined to be infinite.

      jeffjo56

      November 7, 2016 at 4:24 pm

      • Cantor’s original proof in fact uses only the symbols WM. It concerns actually infinite sequences. But Cantor’s axiom of actual or finished infinity can easily be contradicted.

        The claim is that aleph_0 > n for every natural number n.

        If there are aleph_0 natural numbers, i.e., more than are in any finite initial segment (FISON) {1, 2, 3, …, n} of |N, then, as a consequence, no FISON contains all. So the natural numbers must be dispersed over many FISONs. That is mathematically impossible by the inclusion monotony of the infinite sequence of FISONs. Further it is disproved by the fact that there are no two FISONs containing more that each of them.

        Regards, WM

        WM

        November 8, 2016 at 11:16 pm

        • “Cantor’s axiom of actual or finished infinity can easily be contradicted.” Only by substituting other axioms for it. Axioms which you fail to provide, which must have more problems of their own. Axioms that produce a Mathematics having no bearing on a Mathematics that does accept completed infinities. (See: Euclidean vs. non-Euclidean geometry.)

          “The claim is that aleph_0 > n for every natural number n.” Aleph_0 is not “claimed” to be a natural number in any Mathematics, so this is a blatant misrepresentation. Natural numbers are the cardinalities of finite sets. The cardinality of an infinite set (whose existence is accepted by the axiom) must be “greater,” in any Mathematics that can define what that means. That definition cannot be the comparison of numeric values you imply here. So the only possible objection to the claim you cite stems from not accepting the existence of infinite sets, and the requisite re-definition how to compare finite and infinite cardinalities.

          If you choose (and it is a choice, not a fact of “mathematics” as you imply) to not accept the existence of infinite sets, why do you quibble about a proof that they can have different cardinalities when they are accepted?

          “If there are aleph_0 natural numbers…” Aleph_0 is the “cardinality” of the set, not the “number of elements” in the set. Please try to understand that difference. They mean the same thing, and are directly comparable, only when the sets are finite. No amount of obfuscation allows you to compare them as numbers.

          jeffjo56

          November 9, 2016 at 3:12 pm

  7. Cantor’s axiom can be contradicted using only the axioms of mathematics. They result in the theorem that the infinite sequence of FISONs is inclusion monotone and has, as the name says, only finite elements.

    Aleph_0 is a cardinal number which is in trichotomy with all cardinal numbers (a fixed quantity, according to Cantor). Set theory “proves” (see any text book of set theory, for instance Hrbacek and Jech):

    Exists m in countable cardinals such that for all n in |N: m > n. This m is usually called aleph_0.
    Therefore we can compare the numerical values of n and aleph_0.

    It is not astonishing that you deny the actual existence of finished infinity in this case, because it leads to contradictions with mathematics. That is the usual habit in set theory. But when it comes to define a real number by an infinite digit sequence, then potential infinity is clearly insufficient, because it never gives a conclusive answer since always infinitely many digits are following. Then you quickly switch to finished infinity including a number of digits that is larger than every finite number.

    It is a pity that the consistency of set theory is based upon blatant deception, always exchanging both notions of infinity in a sleight of hand.

    Regards, WM

    WM

    November 10, 2016 at 12:02 pm

    • “Cantor’s axiom can be contradicted using only the axioms of mathematics.” I looked on the internet, and found no mention of universally applicable “axioms of mathematics.” So please, tell me which set you are using here. But keep in mind my recurring reference to Euclidean v. Non-Euclidean geometry: Theorems in *a* mathematics, that contradict theorems in another, prove only that the axioms are incompatible. Not that either set of Axioms contains an “incorrect” axiom. And claiming otherwise is an example of the logical fallacy called “affirming the consequent.”

      “The infinite sequence of FISONs is inclusion monotone and has, as the name says, only finite elements.” So you have found a way to obfuscate the more simple argument that starts with “All natural numbers in |N, the set of all natural numbers, are finite.” Pardon me if I remind you that Aleph_0 is not in |N.

      I’m going to expand and paraphrase what I think you meant in your next statement: “There exists an m among the countable cardinal numbers, such that for all n in |N we assume that m>n. This m is usually called aleph_0.”

      In English, when two adjectives are used with a noun, we use a hyphen to show whether the first adjective modifies the noun, or the second adjective. So a “small-business man” employs less than 10 people, but a “small business-man” is less than 1.6 meters tall. Your “countable cardinal [number]” is similarly ambiguous, in a way that is critical to this discussion.

      A “countable cardinal-number” is a cardinal number you can count to, since “countable” modifies “number.” A “countable-cardinal number” is the cardinality of a countable set, since “countable” modifies “cardinal.” Aleph_0 is a countable-cardinal number, but not a countable cardinal-number. When I referred to direct comparison, I meant comparison of the values of countable-numbers. You represented these with n. With the Axiom of Infinity, Aleph_0 is not comparable this way because it is not a countable-number. It is the cardinality of a countably-infinite set.

      What is astonishing, is that you deny the *role* of the Axiom of Infinity. You implicitly substitute for it an “Axiom of No Infinity” which you feel is part of “universal mathematics.” And then you claim that there is a concept of the “greatest cardinal number” in this “universal mathematics” that contradicts the Axiom of Infinity, when all it does in contradict your Axiom of No Infinity. If you want to pity a lack of consistency that is based on blatant deception, start there.

      JJ

      jeffjo56

      November 11, 2016 at 4:02 pm

  8. “’Cantor’s axiom can be contradicted using only the axioms of mathematics.’ I looked on the internet, and found no mention of universally applicable “axioms of mathematics.” If you are able to understand German you may consult my book W. Mückenheim: “Mathematik für die ersten Semester”, 4th ed., De Gruyter, Berlin 2015.

    “Pardon me if I remind you that Aleph_0 is not in |N.” There are aleph_0 natural numbers in |N. That is contradicted. The reason is very simple: Set theorists confuse union and limit. If there is |N, then it is not the union of all FISONs but the limit of their sequence.

    The set of countable cardinal numbers includes the natural numbers and aleph_0. That is the common use in set theory from Cantor on. Please inform yourself about the trichotomy and comparability of aleph_0 and all naturals in set theore (for instance Hrbacek and Jech).

    Further I do not deny the axiom of infinity (although I would deny a wrong axiom like: there are two even prime numbers or there is a circle with three corners in mathematics). I have proven only that the interpretation as finished or completed or actual infinity is in contradiction with mathematics.

    Regards, WM

    WM

    November 12, 2016 at 12:25 pm

    • “If you are able to understand German you may consult my book…” I don’t understand German, but it should be trivial to reference the set of Axioms you think apply to “all mathematics.” With all due respect, the fact that you avoided this question, combined with the fact that there is no such set of axioms, indicates that you are basing your conclusion on your own intuition, and not a formal mathematics.

      “There are aleph_0 natural numbers in |N.” This is an incorrect statement; or at least, one based on incorrect definitions. “There are X elements in Y” is a statement that is defined, in any mathematics that deals with sets, for finite sets only. The cardinality of an infinite set, in any mathematics that accepts the existence of infinite sets, is not the same thing as “the number of elements in the set.” This is a direct result of the fact that an infinite set always has a bijection with a strict, infinite subset of itself. And claiming otherwise is further evidence that you are not working within a formal mathematics. You are using your own unstated set of Axioms that includes an Axon of No Infinity, so that you can make elements of |N and aleph_0 directly comparable.

      “The set of countable cardinal numbers includes the natural numbers and aleph_0.” I thought I expressed my understanding that Aleph_0 is the cardinality of a countable set such as |N, but not a cardinality that can be described by “counting” (i.e., it is not an element of |N). So the set of comparisons indicated by {} requires a stronger definition when alpeh_0 is one of the arguments, than when only members of |N are.

      “I have proven only that the interpretation as finished or completed or actual infinity is in contradiction with mathematics.” Again, there is no universal “mathematics” that you can claim is contradicted. There is only one where you treat infinite sets as though they have a cardinality in |N. Specifically, your claim that “no FISON contains all [natural numbers]” is merely an obfuscated re-statement of “every natural number n is finite, and so n<alpeh_0." There is no statement *in* *modern* *set* *theory* that this contradicts, only one in you unfounded "universal mathematics."

      jeffjo56

      November 12, 2016 at 3:17 pm

  9. If you cannot accept or don’t know universal mathematics including the existence of inclusion-monotonic sequences then further discussion is useless. But perhaps you can find the buf in set theory from the following:

    The sequence of points (n-1)/n with limit 1 is a strictly monotonic increasing sequence, i.e., according to analysis it does not attain its limit 1 as a term. The points with coordinate (n-1)/n lie on the real axis in the interval [0, 1). When connecting every point geometrically by a line to the origin 0 we get the sequence of intervals [0, (n-1)/n]. Obviously these connections have no influence on the limit, such that the limit of this sequence is [0, 1]

    The “set-theoretical limit” is the union of all closed intervals, namely [0, 1). However, this is not the least upper bound because every x < 1 is contained in infinitely many closed intervals [0, (n-1)/n] of the sequence together with infinitely many y such that x < y < 1.

    We see here a tiny difference between the analytical limit and the set-theoretical limit. This is usually acknowledged. "In the infinite" however, where this gap grows to a yawning chasm, it is neglected. If the sequence of FISONs (finite initial segments of naturals) is considered then their analytical (improper) limit omega with |omega| = aleph_0 is identified with the union |N of FISONs (or natural numbers) with cardinality less than aleph_0

    As a result we can state: The union |N of FISONs is not the (improper) analytical limit omega of the sequence of FISONs and does not contain aleph_0 (i,e, more than every finite number of) FISONs (or natural numbers).

    Regards, WM

    WM

    November 14, 2016 at 11:36 am

    • Achilles is faster than Tortoise, so in a race he lets Tortoise have a head start of X1. In order to catch Tortoise, Achilles must first reach point X1, by which time Tortoise is at X2. Then Achilles must reach point X2, by which time Tortoise is at X3. This sequence can be continued as a strictly monotonic increasing sequence, just like yours. Each sequence has a limit which, according to you, cannot be attained. In my sequence, that limit is where Achilles catches Tortoise. So according to you, Achilles cannot catch Tortoise.

      Set Theory says there is a completed set {X1,X2,X2,…}, even if the sequence can never be completed. So Achilles can catch Tortoise. By claiming that the limit cannot be attained, you deny the Axiom of Infinity, and so are using Set Theory inconsistently.

      I know of, and can accept, several different sets of axioms (or fields of mathematics). Some axioms are shared between the fields. Some are just similar. Some are in only one field. Some even contradict others. But no field is called “universal.” All I’m asking, is that you identify which set you are using, and use it, and it alone, consistently.

      If you want to compare your approach to a consistent one, find the more formal proof that deals with Power Sets. Your FISONs are just a limited version of Power Sets, so the same principles should apply.

      jeffjo56

      November 15, 2016 at 3:37 pm

    • You described FISONs (starting on page 183 of your book, the only part I read) with several questionable uses of Set Theory. Whether you called them curly brackets, parentheses, or braces, the symbols “{” and “}” are notational only. They are not operators that are “inserted” to “change” anything. So “1,2,3,…,n” is a collection of numbers, and the FISON F(n):={1,2,3,…,n} is a set.

      The elements of your set |F (I can’t represent it better in text, the “|” distinguishes it from its elements) are sets, not numbers. So your “insertion of braces” into |N *notationally* removes each number n from it, and replaces it with a corresponding set F(n). All that you stated it that there is a bijection between |N and |F, but the corresponding elements of each are different things.

      Because of your improper usage, I can’t be certain how to interpret “The union |N of FISONs.” I think you mean the union of the infinite number of FISONs. I have no idea why you think this needs to include Aleph_0, since Aleph_0 is not in |N. But |N is equivalent, in every way, to the union I just described. What do you think I described incorrectly?

      Aleph_0 is similar to a natural number in many ways, but it is not a natural number. So it is not “missing” from |N. In exactly the same way, |N is not a FISON, even though it is similar because it is a set of consecutive integers. Neither |N, nor a properly-described predecessor of |N in your “insertion of braces” process, is “missing from |F,” or “destroyed” in any way.

      jeffjo56

      November 16, 2016 at 5:08 pm

  10. jeffjo56 “By claiming that the limit cannot be attained, you deny the Axiom of Infinity, and so are using Set Theory inconsistently.”

    I am using mathematics consistently. It shows that set theory is inconsistent. For every n in |N: FISON(n) is the union of all its predecessors. There is no FISON violating this theorem (like there is no natural nunber being infinite). Therefore the claim of more natural numbers than are in any FISON is wrong in mathematics. If it is true in set theory, you gave the choice: Either prove un in mathematics or believe in the counterfactual axiom

    jeffjo56 “All I’m asking, is that you identify which set you are using, and use it, and it alone, consistently.”

    I use the axioms of natural numbers which yield the potentially infinite sequence |N, the axioms of an ordered field and the completeness by limits of equivalence classes of Cauchy sequences. These axioms yield the sequence of FISONs and show that in the arithmogeometrical figure
    1
    1, 2
    1, 2, 3

    by shifting all numbers vertically into the first row, without changing their column, never more natural numbers than can be in any FISON are in the first row.

    jeffjo56 So “1,2,3,…,n” is a collection of numbers, and the FISON F(n):={1,2,3,…,n} is a set.

    The curly brackets destroy the order. Therefore it is advisable to use parentheses as Cantor did: (1,2,3,…,n). But when I write 1, 2, 3 in the figure above, there is no difference to (1, 2, 3) or {1, 2, 3}.

    Regards, WM

    WM

    November 16, 2016 at 9:12 pm

    • “I am using mathematics consistently.” There is a bijection between the set of natural numbers |N, and the set of FISONs |F. It is defined by FISON(n) = {1,2,3,…,n}. You described this bijection. But you did so poorly, and do not seem to recognize it as a bijection. On page 183 of your book, you say “while no natural number is removed [from |N], |N is lost [from |N].” But |N is not claimed to be a member of |N in any mathematics. Nor is it described by the bijection you implied. Yet you claim it was a member, and is removed by the bijection. You are using mathematics inconsistently.

      “Inconsistent” means a Mathematics contradicts itself, not that it contradicts something you derive in a different Mathematics. So your claim is equivalent to saying non-Euclidean geometry is inconsistent, because it contradicts results in Euclidean geometry.

      “I use the axioms of natural numbers which yield the potentially infinite sequence |N.” And this statement means you deny the Axiom of Infinity, which yields that |N is not just a “potentially infinite sequence,” it is an “actually complete set.” Because you deny this, your conglomeration of different Mathematics-es produces the result that Achilles cannot catch Tortoise. It also produces the result that Achilles catches Tortoise at X=VA*X1/(VT-VA). So it is your conglomeration that is inconsistent *with* *itself*.

      “Never more natural numbers than can be in any FISON are in the first row.” I can’t understand that sentence. But I defined your FISONs more formally. I suspect that this definition, combined with the acceptance of |N as a complete set so |F is also a complete set, makes whatever it is you thought you meant incorrect.

      “The curly brackets destroy the order.” Sets do not have order. {1,2,3} and {3,2,1} represent the same set. Claiming that order is destroyed is being inconsistent.

      Curly brackets are just one way to represent sets, and have no inherent meaning. “The positive integers less than 4” is another way to represent the same set, without curly brackets. Attaching more meaning to them is being inconsistent.

      Cantor use parentheses to mean the same thing you mean by curly brackets.

      jeffjo56

      November 17, 2016 at 3:44 pm

  11. Ordered sets have order. Cantor uses curly brackets for wsets and parentheses for ordered sets. See:
    Beiträge zur Begründung der transfiniten Mengenlehre. [Math. Annalen Bd. 46, S. 481-512 (1895); Bd. 49, S. 207-246 (1897).]

    In order to understand the argument about FISONs note that every set of FISONs, when unioned will yield one of the FISONs. This holds for all FISONs independent of their abundance. In an infinite set of natural numbers there is not infinite natural number. Likewise in an infinite set of FISONs there is no FISON violating inclusion monotony.
    For further information see p. 182 of
    https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf

    Regards, WM

    WM

    November 18, 2016 at 11:10 pm

    • “Ordered sets have order.” Neither modern set theory, nor Cantor’s Diagonalization proof, is based on anything having to do with “ordered sets.” But this is one of the reasons I keep asking you to identify the set of axioms you think define your “universal mathematics,” and use it consistently.

      “Cantor uses curly brackets for wsets and parentheses for ordered sets,” Not in the proof in question. And regardless, that is just a notational convention. One you do not support in your book, so this is irrelevant.

      “In order to understand the argument about FISONs note that every set of FISONs, when unioned will yield one of the FISONs,” And any two natural numbers, when provided as arguments to a “maximum” function, yields a natural number. This is a perfect analogy, and holds for all natural numbers. Again, you argue with inconsistency, and irrelevancy.

      Nobody claims the set of natural numbers should contain an infinite number. Similarly, there is no reason to think an infinite set of FISONs should contain an infinite FISON, or a “FISON violating inclusion monotony.”

      jeffjo56

      November 19, 2016 at 3:55 am

  12. If there is no infinite FISON, then the union of all FISONs (or all natural numbers) has not an actual infinity of elements. We have to distinguish between the union of all FISONs and the limit of the sequence of FISONs. Set theory fails to do so.

    You can understand this better in a comparable finite case: In set theory the sequence of closed intervals [0,(n−1)/n] has as its limit the half open interval [0,1). Using analysis we obtain 1 as the limit of the right-hand side borders (n−1)/n. Connecting every point (n−1)/n to the origin 0 cannot change the limit. Therefore the analytical limit is the closed interval [0,1].

    Apply this to the limit omega of the sequence and to the union |N of all FISONs, then you might see the fundamental mistake which set theory is based upon.

    Regards, WM

    WM

    November 19, 2016 at 2:29 pm

    • WM started: “If there is no infinite FISON, …” Your acronym “FISON” stands for “Finite Initial Segments of Naturals” (page 187 of your book). If you want to continue this discussion, and be considered seriously, please try to understand what consistency means. Specifically, that “infinite Finite Initial Segments of Naturals” is self-contradictory, and therefore can’t exist. So you can’t try to use them as an example.

      But, to address the point you think you were making: There is no infinite natural number. But the set of all natural numbers is infinite; i.e., it has an infinite cardinality. THIS IS NOT A CONTRADICTION IN ANY MATHEMATICS. Any mathematics that includes the Axiom of Infinity takes it as a given. Any mathematics that doesn’t cannot express the concept of something being infinite.

      There is a bijection between natural numbers and FISONs, where each natural number n is uniquely paired with the FISON F(n) whose cardinality is n. By definition, every natural number is included in this bijection. By definition, no FISON is missing from this bijection. So, just like every natural number is finite, every FISON has a finite cardinality.

      Any two sets in a bijection with each other have the same cardinality, The set of natural numbers |N has infinite cardinality, but contains no member that is infinite. The set of FISONs |F has infinite cardinality, but contains no member with infinite cardinality.

      If you want to try to rebut this, please say which of the preceding statements you think is wrong.

      WM continued: “… then the union of all FISONs (or all natural numbers) has not an actual infinity of elements.” There is no infinite number n, but the Axiom of Infinity says the union of all natural numbers has an “an actual infinity of elements.” So your claim that the premise “Set X has no infinite elements” leads to the conclusion “set X has not an actual infinity of elements” is an example of the logical fallacy known as “affirming the consequent.” The conclusion does not follow from the premise.

      Again, any set of Axioms that accepts an Actual Infinity says the opposite of your conclusion is what is correct; and any set that does not accept an Actual Infinity cannot express the premise or the conclusion. So again, and more specifically, I ask you: What set of axioms are you using that allow you to express the concept of actual infinities?

      Wm also said: “Using analysis … the limit …” What axioms support saying the limit is part of the interval?

      jeffjo56

      November 19, 2016 at 3:54 pm

  13. JJ> So your claim that the premise “Set X has no infinite elements” leads to the conclusion “set X has not an actual infinity of elements” is an example of the logical fallacy known as “affirming the consequent.” The conclusion does not follow from the premise.

    The conclusion follows from mathematics. Set theory requires that the set of all FISONs has more natural numbers than any FISON. That is contradicted by mathematics, in particular by the well-ordering of FISONs.

    Definition: A set of small FISONs is a set of FISON the union of which is one of the FISONs. Example {{1}, {1, 2, 3}, {1, 2, 3, 4, 5, 6, 7}} is a set of three small FISONs.

    If there are sets of FISONs that do not obey this rule, then there must be a first FISON violating it. Find it! Or stop claiming that the set of all FISONs can have a union greater than every FISON.

    JJ> I ask you: What set of axioms are you using that allow you to express the concept of actual infinities?
    Actual infinity is tantamount to the claim quoted above: The union of all FISONs is |N which is a larger set than every FISON.

    JJ> Wm also said: “Using analysis … the limit …” What axioms support saying the limit is part of the interval?

    In analysis the limit of a strictly monotonically increasing sequence is never a term of the sequence. The union of an inclusion monotonic sequence is always a term of the sequence.
    Therefore:
    The set theoretical limit of the sequence of intervals is its union: lim[0, 1 – 1/n] = [0, 1).
    The analytical limit of the sequence of intervals: lim[0, 1 – 1/n] = [0, 1].
    Both differ like:
    The set theoretical limit of the sequence of FISONs lim(F_n) = U(F_n) = |N.
    The analytical limit of the sequence of FISONs lim(F_n) = oo or omega or aleph_0.

    In the latter case set theory claims |N = omega or Card|N = aleph_0.
    This is obviously wrong because the union of FISONs cannot be an actual infinity like the union of intervals [0, 1 – 1/n] cannot be [0, 1].

    Regards, WM

    WM

    November 20, 2016 at 6:04 pm

  14. “The conclusion follows from mathematics. Set theory requires that the set of all FISONs has more natural numbers than any FISON.” This is one of the assertions where you are blatantly inconsistent.

    A FISON is not a natural number, it is a set of natural numbers. Set Theory does not “require that the set of all FISONs” contain any natural numbers, only that it contain sets.

    I think you mean “the *union* of all FISONs, has more natural numbers than any FISON.” This is no more a contradiction than “The set of all natural numbers has cardinality greater than any natural number.” In fact, the two statements are equivalent, and both are accepted by the Axiom of Infinity. Withit, there is no contradiction here.

    What you are ignoring, it that it is your unspecified, and non-existent, conglomeration of different fields of mathematics that has the contradiction you keep finding. Specifically, there is one derivation that shows that Achilles catches Tortoise at X=X1*VA/(VA-VT). There is another that says he catches up only at the “greatest” member of the set {X1,X2,X3,…}. This is a monotonically increasing sequence that, according to you, can have no greatest member.

    This is your contradiction. All forme you have found can be shown to be equivalent to it. It is in your ill-described mathematics, no matter how you obfuscate it. The mathematics that excludes the Axiom of Infinity. The Axiom that says what you call a contradictory statement is, in fact, an accepted truth.

    jeffjo56

    November 21, 2016 at 6:05 pm

    • JJ> I think you mean “the *union* of all FISONs, has more natural numbers than any FISON.”

      No, I don’t. If the union of all FISONs has more naturals than any FISON then these naturals must be in the FISONs to be unioned already. Therefore your distinction is nonsense.

      JJ> This is no more a contradiction than “The set of all natural numbers has cardinality greater than any natural number.”

      No, you have not yet understood. Try again: If all FISONs (or the unions of all FISONs) contain more naturals than any FISON, then inclusion monotony is violated. Hence mathematics is violated.

      Regards, WM

      WM

      November 21, 2016 at 6:18 pm

      • MW> If the union of all FISONs has more naturals than any FISON then these naturals must be in the FISONs to be unioned already. … you have not yet understood.

        No, it is you who doesn’t understand.

        |N is an inductive set, defined by two rules. (1) It includes the number 1, and (2) if it includes the number n, it also includes the number n+1. The second part defines every member of IN after 1, even though you can’t apply it, pairwise, to define every member. Any set you have tried it on implies one more element you haven’t.

        Another way to say that is that the definition process continues toward *POTENTIAL* infinity, but can’t reach an end (which is what “potential” means here.) But the Axiom of Infinity allows us to treat the set as *COMPLETED* by this inductive step. I call the the principle of induction: even though you can’t reach the end of what it defines, it still defines all possible elements.

        This principle applies to any pairwise operation. We can’t pairwise-add every term in a converging sequence like 1/2+1/4+1/8+ … .But the sum *OVER* the entire set can be determined. AND IT IS A VALUE THAT CAN’T BE FOUND BY PAIRWISE SUMMATION.

        The maximum *OVER* a converging sequence like that in Zeno’s Paradox, my {X1, X2, X3, …}, also exists. You claim, *CORRECTLY*, that this maximum isn’t a member of the set itself. But the Axiom of Infinity still says that the maximum *OVER* the set exists. Any mathematics that denies this existence is self-contradictory, since it also says that Achilles catches Tortoise at that maximum, and that it happens at X=X1*VA/(VA-VT).

        Your |F is also an inductive set, defined by two very similar rules. (1) It includes the FISON {1}, and (2) if it includes the FISON {1,2,…,n}, it also includes the FISON {1,2,…,n,n+1}. The pairwise union of elements always produces a FISON that is equivalent to the largest FISON used so far. But the union *OVER* the *ENTIRE* set, which can only defined by accepting the Axiom of Infinity and the principle of induction, is defined to be the set |N, which is not a FISON.

        THIS IS NOT A CONTRADICTION. It is a logical consequence of accepting the Axiom of Infinity, and the principle of induction. BUT, any mathematics that does not accept them does have a contradiction, as described by Zeno of Ilea.

        jeffjo56

        November 21, 2016 at 11:37 pm

        • Mathematical arguments appear to be useless in your case. But I know that every student (of many hundreds that I have taught) understands that the union of all rows of the ArithmoGeometrical figure
          1
          1, 2
          1, 2, 3

          cannot be longer than all rows. This is simply prohibited by definition and is independent of any axioms or other conditions. 99 % of all intelligent humans understand that by shifting all numbers of all infinitely many rows into the first row this first row cannot become longer than all contributing rows which are finite.

          Knowing that this proves my case I will stop here. Remain happy with your delusions. They will not cause damage because set theory is total nonsense and therefore without any effect to real life.

          Regards, WM

          WM

          November 23, 2016 at 11:05 pm

  15. Do the same set of students you refer to think that there is a largest number in |N? And that it is a member of |N? That is, can the maximum of all rows in this figure

    1
    2
    3

    be “bigger” than any row? Or – as is the case – is there a problem defining “the maximum over all rows” and “the union of all rows” when the set of rows is infinite? And so does using the expression require you to step outside of your very narrow box, and accept that there is no finite definition for either?

    But I see that simple mathematical arguments about infinite sets, because they are based on principles in mathematics that you do not want to accept, are useless in your case.

    jeffjo56

    November 24, 2016 at 9:29 pm

    • Define a measure M(x) on a set |X such that, for any x in |X, M(x) is a natural number.

      Define the class of functions |P to be any function P(x) that has these two properties:
      (1) If x is in the range of M(*), then so is P(x).
      (2) M(P(x))=M(x)+1.

      The Principle of Induction says that any such function, with at least one object in its range, defines an infinite set. The issue here is whether this principle should be accepted. The inconsistencies WM claims to find are the direct result of comparing a Mathematics that accepts it, and makes logical inferences from it, to one that denies it.

      The “final result” of applying the P(x) that defines Zeno’s paradox, to all of the intervals in the set it defines, must exist. This can be proven in any mathematics that can define the intervals (notice I don’t claim a “universal” Mathematics, as WM does). That is, Achilles *CAN* catch Tortoise. But it can also be proven that this “final result” cannot be a member of the set itself. So a consistent Mathematics must be able to define a “final result” of P(x) for an infinite set, that is not in the set.

      According to WM’s argument about rows, if there is a “final result” of applying the P(f) that defines the set |F of all FISONS, then it must be a FISON. But if it is a FISON, then inclusion monotony is violated.Therefore, he claims, any mathematics that produces such a “final result” is inconsistent.

      WM’s conclusion “it must be a FISON” simply doesn’t follow, because he does not define what “final result” means for an infinite set. The inconsistency he finds is an inconsistency in any Mathematics that does not have such a definition. That is, WM’s “Universal Mathematrics” is inconsistent because it does not include Set Theory’s Axiom of Infinity. The contradiction he claims is actually proof that he needs to supply a definition, like the one Set Theory defines. The one where Aleph_0 is the “final result” of applying P(n)=n+1 to all of |N, but is not in |N.

      jeffjo56

      November 26, 2016 at 4:08 pm


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