JJ: when based on strings, each character in any of [Erdinc’s] tables must be either a ‘0’ or a ‘1’. There simply isn’t any way for any other character to appear.

KS: Totally agree. But this is exactly the same as me saying “the supposed list of all strings is a *GIVEN & FIXED list. If you accept that concept then his argument falls apart immediately”. Erdinc is just wrong if he thinks that a given and fixed list can have some of its bits “become” indeterminate. As I keep saying, Erdinc’s situation of finding the flipped diagonal in the list and thus having a conflicted and thus indeterminate bit – *just does not arise* if the list is a priori Given & Fixed. And it doesn’t matter whether he interprets the strings as numbers or not.

KS: The problem is that induction may indeed generalize the result

No, induction only applies to the elements of the set, and maybe its limit when there is one (there isn’t in our discussion). Not to the properties of the infinite set itself.

KS – Hey JJ – you left off the rest of my statement;

“induction may indeed generalize the result – but ONLY to *any and all* finite length strings. In other words, the relation for the actually infinite case has not been established”. I have no idea why you felt the need to say “No”. Induction can apply to any proposition, but it concerns only finitely indexed instances in a sequence – and the infinite string case is not in that sequence. I don’t see anything contentious about that. I think we agree.

Cheers

Ken

While you may disagree with its importance, that is not a strawman argument on my part. Nothing is being misrepresented. It is a strawman argument on Erdinc’s part, since he does knowingly misrepresnt Cantor. Using it adds additional layers of complication that helps to hide many of his false claims.

For example, when based on strings, each character in any of his tables must be either a ‘0’ or a ‘1’. There simply isn’t any way for any other character to appear. And Erdinc won’t defend how he supposes there could be any other character. Could it be that he thinks the conversion between numbers and strings allows it? I DON’T KNOW. But if he used strings, I wouldn’t have to guess.

Did you notice that even though I point out that Erdinc uses numbers only, I continue to argue based on both numbers AND strings? What I want most, is to see Erdinc acknowledge, and correct for, an argument that isn’t his.

KS: Normally intelligent people do best at understanding when they can “see” the objects of discourse and the arc of logic in their heads “all at once”.

Merriam-Webster: “Intelligence: the ability to learn or understand or to deal with new or trying situations.” If these people you refer to are repeatedly told how their concept is wrong, and refuse to even try to understand why, they are not demonstrating this concept.

KS: Erdinc establishes a finite cardinality relation between node count and strings that applies to finite-length strings and then supposes that this relation still holds for string lengths that are not finite.

Exactly. And there is a perfect analogy: The cardinality of each set formed by a sequential application of the Axiom of Infinity, is a finite cardinal number (which have a subtly different definition than natural numbers) equal in value to the last natural number defined. Similarly, each string is Erdinc’s tree has a finite length. But the cardinality of the set |N, whose existence Cantor accepts by axiom, is infinite. As is the length of the strings that Cantor accepts. Alpeh0 is a cardinal number, but not a natural number. (And I’m ignoring references to omega.)

Erdinc wants Aleph0 to be equal in value the natural number found “at the limit” of the sequence 1,2,3,4,…, and an infinite string to be the one found “at the limit” of his tree. That is, at the limit of “counting to Aleph0/omega.”

KS: The problem is that induction may indeed generalize the result

No, induction only applies to the elements of the set, and maybe its limit when there is one (there isn;t in our discussion). Not to the properties of the infinite set itself.

]]>I agree with you that both of the key arguments of Erdinc are flawed.

STRAWMAN

The only argument that I have thought to be strawman is over the use of “string” versus “number”. Thinking of the cantor strings as numbers vs strings does not essentially change the essence of the error that Erdinc makes. Either way, I agree that he gets the logic wrong. And perhaps indeed you were not in fact quibbling over Erdinc’s use of the term “number” anyway. Apologies if I misinterpreted your words that way.

PERSPECTIVE

You might notice that I often come back to ontology and “concept”. It’s not because formal proofs aren’t proper – but because normally intelligent people do best at understanding when they can “see” the objects of discourse and the arc of logic in their heads “all at once”. Formal proofs can also assume an ontology that turns out to be different to that which is implicitly held by someone else. This is why for example, in my argument against Erdinc’s first assertion, I have emphasized that the supposed list of all strings is a *given and fixed* list. If you accept that concept then his argument falls apart immediately. This can be understood conceptually without having a formal training or an easy familiarity with formal logic.

THE LEAP

The standard meaning and understanding of the *great leap* for N is the formation of the “actually infinite set” – a different class of object to any of the natural numbers that are its elements. It is Cantor’s “Second Principle of Generation” as asserted by ZF Infinity. And yes, a property that holds for any of the elements of that set – or that holds for any of the indexed finite convergents of that set – cannot be said to automatically hold for N itself.

THE BINARY TREE

The main problem here is:

Erdinc establishes a finite cardinality relation between node count and strings that applies to finite-length strings and then supposes that this relation still holds for string lengths that are not finite. The problem is that induction may indeed generalize the result – but only to *any and all* finite length strings. In other words, the relation for the actually infinite case has not been established.

Ken Seton

kenseton@gmail.com

Erdinc claims to discredit Cantor in two ways: (1) He claims there is a flaw in Cantor’s argument, and (2) He claims to have found a way to do what Cantor proved to be impossible.

The supposed flaw is that the number/string that Cantor identifies can’t really be number/sting, so it isn’t supposed to be in the list. Erdinc concludes this (previous post) because it would be a contradiction if it were really a number/string, not because he can prove that it isn’t one. And in fact, it can be trivially proven that it is, invalidating Erdinc’s first claim. How is this a strawman argument?

JJ: “Aleph0 cannot be “reached” by counting natural numbers, nodes, or segments.”

KS: Well, obviously. It is but a manner of speech. No-one thinks that one can *count* to omega, or “reach” the cardinality of the natural numbers…

Oh? From page 19 of Erdinc’s paper:

ES: “So, the difference goes to infinity as n goes to infinity, … Hence there will be infinitely many more segments than branches as n goes to infinity. Therefore the branches of the tree in Figure 3 are not more numerous than its segments.”

This is a “limit” argument, as used in calculus. It is a claim that for an infinite sequence of objects, all different values of some set of properties, that you can deduce the “ending” values of that set of properties even though there is no end.

That is not Cantor’s concept of infinity. The “great leap” in the Axiom of Infinity, is that the sequence constitutes a set, with no “end” real or imagined. Not that we can “leap” to a pretend end and get a number we call “infinity” that is in any way similar to the members of |N. The “great leap” is that we can attribute a different set of properties to the *set* |N, that are dissimilar to the properties of the objects in |N.

|N contains only finite, natural numbers. For any finite subset of |N, its cardinality is a natural number in |N. But |N has cardinality Aleph0, which is not similar in any way to the members of |N. In short, the phrase “goes to infinity” does not belong in a discussion of what Cantor did, because it is an attempt to describe what happens when you “reach the cardinality of the natural numbers” through counting.

]]>The classic way to prove proposition P by contradiction, is to first assume ~P. Then you prove, by separate logic paths, that both Q and ~Q follow from ~P. I’ll write this as a set of two logical paths, {~P->Q, ~P->~Q}.

Note that this does not say that either Q, or ~Q, is true. It says Q must be true if ~P is hypothetically true, and that ~Q will also must be true if ~P is hypothetically true. I believe that this is what KS has tried to explain. Since we can’t have the contradiction where both Q and ~Q are true, we can reject the hypothetical without saying anything about Q. What we cannot deduce, is that there is anything wrong with either of the logic paths that started with the hypothetical.

Usually there will be intermediate steps in these logic paths, so what we really prove might be more like {~P->A->~Q, ~P->B->Q}. Noting that B->Q is logically equivalent to its contrapositive ~Q->~B, it is possible that the two paths could have been {~P->A->~Q->~B, ~P->B}. This is still in the classic form, with the contradiction being B and ~B. Reversing A->~Q instead leads to another contradiction based on A.

The point is that ANY contradiction found this way proves that the hypothetical is invalid, not that there is anything wrong (or right) with the propositions A, B, or Q. Yet ES did just that, when he claimed that there must be a non-number in the assumed list of numbers.

ES’s interpretation of diagonalization doesn’t use this classic form. ES claims it is a single-path version that extends the contrapositive reversal in a circular manner: {~P->A->~Q->~B->P}. Some logicians accept this single path as an alternate form (see Wikipedia), but not all. It “just feels wrong” for a statement’s negation to follow from itself. And I’m sure that is the basis for ES’s objections to what he thinks was Cantor’s proof. But this emotional reaction ignores the fact that contradiction is the point of the whole exercise. And that the only conclusion that can be drawn that the hypothetical is false, not that there is anything wrong with A, B, Q, or any part thereof. Yet ES did just that, when he claimed that there must be a non-number in the assumed list of numbers.

But there is another flaw in ES’s reasoning. Not only did Cantor never say he used proof by contradiction, WHAT ES CLAIMS HE DID DOES NOT FIT EITHER FORM. The hypothetical proposition ~P that ES uses consists of two parts, H1: “there is a list” and H2: “the list is complete.” Proof by contradiction requires that both paths depend on the same hypothetical proposition (or the single-path form circles back to the negation of the hypothetical proposition). The form ES says Cantor uses is {H1->A->B->~H2}, which cannot be a proof by contradiction. It is, however, a proof that H1 and H2 can’t be true together. Which is exactly what Cantor said.

(Note: ES may try to claim that it is {(H1&H2)->A->B->~H2}. This is still not a proof by contradiction since the last proposition is not the negation of the first. Regardless, none of the steps in the chain of logic utilize H2, so that is not what Cantor did. ES is using a classic strawman argument.)

]]>STRAWMAN

My strawman comment concerned *only* the issue of non-strings and non-numbers.

I think it is no big deal that Erdinc has chosen to refer to his objects as numbers / non-numbers, even though for uniqueness and generality I would *much* prefer strings / non-strings. The meaning of “non-object” seems clear to me – an object built or found in the pattern of that which is expected but which has an inconsistency or fails to be satisfactorily complete. If you like, an object that doesn’t make it as the intended object. As I have said, I think his approach is misguided (on ontological grounds) and wrong (you have explained this in a more traditional way). I was certainly not casting aspersions on your explanation of the logic of diagonalization.

REACHING THE INFINITE

JeffjoJJ: “Aleph0 cannot be “reached” by counting natural numbers, nodes, or segments.”

KS: Well, obviously. It is but a manner of speech. No-one thinks that one can *count* to omega, or “reach” the cardinality of the natural numbers, except, in essence, in the way that you also describe. Whether it be Aristotle, Gauss, Cantor or many an-other great, a one-foot-after-another count across the list of natural numbers *never* finds a maximum. Regarding the actually infinite set, I like the way that Paul Cohen puts it:

“The usual axiom of infinity makes a *great leap* by assuming the

existence of one set which contains all the integers. That is, it

asserts the existence of a large set *which cannot be obtained

by the repetition of a certain process*. “

The *great leap* is always there, even if just captured by existential assertion within the apparent simplicity of ZF Infinity. So when one says that the set N contains ALL the natural numbers, one is not suggesting that there is a maximum natural number. And you see, when I say something like …

$$ Doesn’t it just seem a tad synthetic or artificial or (to coin a phrase) “artefactual from definition” that we must regard the limit of the node count *for all n” as a complete set of nodes while the limit of the finite string count *for all n* as a complete set of strings fails to have gone “far enough” over the domain in n to actually contain a string that is *for all n* ?? $$

… I am not at all ignoring the great leap, nor asserting some a natural number maximum. The phrase “fails to have gone far enough” is not meant to imply that if one can just count *enough* then one might complete the count – heaven forfend – but rather – that if the thing being counted can be established as having an equivalence with N then that thing being counted can be said to also *partake* of the same leap ! The *leap* of ZF Infinity is always required.

Cheers

Ken Seton

kenseton@gmail.com

I agree that ES has attempted both in his attempt to discredit Diagonalization. Unfortunately, they are strawman arguments.

STRAWMAN

A strawman argument uses an intentional misrepresentation of the original proposition, which has an easier counterargument (of either form). But how does one argue against a strawman argument? Is it “misrepresenting” the strawman when one uses the correct representation of the original proposition? After all, it is intentionally not what the strawman used. But it IS correct, and it is not used to make things easier. If anything, they become harder. KS, if by “I just don’t like to see you being critiqued on an irrelevant or strawman point” you meant my attempts to make ES understand what diagonalization does, I’d really like to see you explain what you meant.

Cantor did not use real numbers in Diagonalization, and ES insists on using them. This is an intentional misrepresentation. They make it easier for ES to make ambiguous claims about the properties of thes set, such as “they are nonnumbers.” He won’t say what he thinks they are, just what they he thinks they can’t be. That is, he is taking the easy way to a conclusion. While this is the lesser of his strawman arguments, I just want to point out that it is a classic strawman argument.

Diagonlaization is not a proof by contradiction. There’ll be more about how it simply *can’t* be interpreted as one, and how ES’s logic is flawed, in my next installment. This is how the proof actually works: In part 1, It proves statement A: “Any list of the real numbers in |R1 necessarily omits a number in |R1”. In an independent part 2, it considers statement B: “There is an s(*) that lists all real numbers in |R1.” This contradicts statement A, which was proven to be correct. So B can’t be.

Note that part 2 isn’t very formal. Diagonlaization was meant to be a visualization of his more formal “power set” proof. We know that Cantor knew how to use Proof by Contradiction, because he said he was using it there. He did not say he was using it in Diagonalization. When isolated from part 1, you could consider part 2 to be a proof by contradiction. Or a proof by contraposition. Or just an explanation of what is obvious. The point is that the parts are independent, and presented in series. The assumption in part 2 is not a part of the argument in part 1.

ES has even admitted that Cantor didn’t make the assumption in part 1, but insists that the two parts must be viewed not only in parallel, but combined. This is a misrepresentation of the proof. What I keep trying to do, is establish that statement A can be proven without using proof-by-contradiction, or making any questionable logic or assumptions. So what I do cannot be called “strawman,” because it is providing the correct interpretation of the Diagonalization proof we are both addressing. In doing so, I discredit ES’s arguments in both of the ways I mentioned at the start of this installment: I point out ES’s error, and I show how to get the correct conclusion.

]]>KS: A first question for JJ: What is the cardinality (or how can we define the cardinality) of all the segments on a binary tree ?

First, there is no significant difference between the set of segments, and the set of nodes. You can replace each “segment” with the node on the right side of its line, and get the exact same set.

The set of nodes is (trivially) countably infinite. So are the segments. No one has doubted this.

You can define cardinality in two parts (formal treatments use more, but they can be understood from these two): If a bijection exists between set A and set B, they have “the same cardinality.” If an injection from A to B exists, but a surjection can’t, then B has a greater cardinality that A.

Please note that cardinality is not defined as a number. But if there is a bijection between A and what I call a “counting set” – a set of consecutive natural numbers starting with 1 – then the same comparative relationships, that exist between A and any other set B, exist between that counting set and B. So if a set “has the same cardinality as the counting set {1,2,…,n},” we often use the shorthand that the set “has cardinality n,” or even “has n members.”

KS: Perhaps you think that a transfinite cardinality is never realized ? In which case the first transfinite cardinality (Aleph0) of the number of bits present in 1/2pi would also never be realized ! If you think that the question does not make sense or is not well-framed, please explain with care.

For finite sets, n is the highest number in the counting set we use this way. For infinite sets, THERE IS NO HIGHEST NUMBER. So the shiorthand seems to fail.

But the set of all natural numbers, |N, exists axiomatically. It can be put into a bijection with many other sets, all of which BY DEFINITION “have the same cardinality as |N.” We can’t use a natural number in this shorthand, so Cantor defined “transfinite numbers” to use in its place. They cannot ever be reached when you count a set. They are **NOT** “potential infinity”, which is the imaginary (I mean a figment of his imagination, not sqrt(-1)) limit that ES used in his paper.

Aleph0 cannot be “reached” by counting natural numbers, nodes, or segments. It is the value we define to be the cardinality of the infinite set |N, not a member (real or a figment as ES ties to use it) of |N.

KS: You never know, Jeffjo might teach us something.

I hope I have.

]]>JJ: What both of you keep ignoring, is that “denumerable” means “can be put into an infinite list (or mapping).”

KS: Hey Jeffjo – you are second-guessing my opinion on this. At no time have I ignored what denumerable means.

Then let me clarify what I meant: it is irrelevant how ES thinks his “segments” work, because his conclusion is that |R1, the real numbers in [0,1] (or really Cantor Strings, there are problems with using real numbers), is denumerable.

If you defend (which does not imply acceptance) that conclusion, then you are ignoring that “denumerable” means there is a function whose domain is |N and whose co-domain is the subject set. There is a trivial proof, that does not assume the function is a surjection (or even an injection), that any function, whose domain is |N and whose co-domain is |R, necessarily misses at least one member of |R1.

]]>KS: Don’t get too excited. I have previously acknowledged your argument and how you so-to-say obtain “nonstrings”. I just don’t like to see you being critiqued on an irrelevant or strawman point. But I still think your interpretation is nonsense, for (other) reasons I have previously described. And the essence of this is, to reiterate …

The conflict concerns the meaning and usage of “ASSUME”. In common language:

– You take it to mean that a priori the list must be treated as definitively and actually complete such that it HAS TO contain the bit-string corresponding to its own flipped diagonal.

– What it actually means is that ANY list AS GIVEN (any unchanging *fixed* list of bit-strings) is conjectured to contain all bit-strings. But we find that the flipped diagonal can never be met in ANY given list. So the issue of an indeterminate / conflicted bit never arises and we conclude that the list is incomplete. We differ on this because you put ontological priority on “Assume Completeness” and I put ontological priority on “the (any) GIVEN FIXED list”. This doesn’t muddle the issue, it just identifies the actual location of our difference and respective obstinacy.

ES: I hope you don’t think I am the inventor of infinite binary strings. They are all over the place. Google it. KS: You should be just a little more circumspect in your condescension. Then again, sometimes a little chutzpah or cheek is not a bad thing.

KS: “What is the cardinality (or how can we define the cardinality) of all the segments on a binary tree ?” ES, I know what you think. I know what I think. But I asked the question *not* to you, or for myself. I asked it of JJ – in order to *tease out* his conception of the binary tree. Maybe he doesn’t think it is a valid mathematical object. Maybe he does. Read my question. You never know, Jeffjo might teach us something. && I have just seen his recent posts and agree with much there. &&

FINITE STRINGS

I have to side with JJ with at least one aspect of his reference to “finite strings”.

It is not that the binary tree represents only finite strings BUT that the *relationship* between the cardinalities of segments versus the cardinality of strings (that is, the way chosen to observe / measure aspects of the binary tree object) concerns only *finite* instances.

Let me elaborate:

The ratio Segments / Strings concerns finite length strings. Induction shows that this ratio stays above 2 (but approaching 2) for every finite string length. In terms of any transfinite cardinality that might be assumed at totality over all nodes and strings, this might be as good as equivalence. Now it is critical to understand that induction can only ever make a pronouncement regarding finite instances. JJ is right that set theory will *not* accept that a property that holds for every finite case of an object necessarily holds for the actually infinite case of that object. A classic example: Every convergent of an infinite bit-string real is rational but the actual real is irrational. JJ is quite right that omega and aleph0 are concepts concerning infinite collections beyond finite instances. Omega is not a natural number and aleph0 is not a finite cardinal. Therefore induction cannot show that a property holds for an element so-to-say “at omega” or for a set that has become an infinite totality – it can only show that a property must be true (or false) at each *indexed* step or instance. And it is in this sense that someone can say that you “are only looking at finite strings”.

JJ is right that *at each instance* you relate segment (or node) count to a count of *finite strings*. I think you cannot not argue with that. The real question is whether the observed relation can be projected to the infinite case. The problem here is that being able to specify some cardinality relation for each finite instance does not automatically permit us to project that same relationship to the actually infinite case. How do we know that the cardinality property holds for the infinite case ? We don’t. We might think it *should* hold but cardinality properties are notoriously *unable* to be so projected. Just think of the ratio of perfect squares to naturals which projects to zero in the infinite case. But in fact by Cantor’s definition of equivalence, they both have the same (transfinite) cardinality. Does this make a mockery of the equivalence concept ? Maybe, but that is a very different and separate argument. Here, we go with his definition.

JJ: ”What both of you keep ignoring, is that “denumerable” means “can be put into an infinite list (or mapping).”” Hey Jeffjo – you are second-guessing my opinion on this. At no time have I ignored what denumerable means. In fact I fully agree with you. Yes, I said that the binary tree is not a mapping – meant to mean (perhaps clumsily) that it is not a (supposed or otherwise) denumeration of infinite strings. I think we all agree with this. Of course it can be interpreted as containing various mappings as soon as we do things with it, as soon as we carry out some analysis.

ES has to agree that the finite node / string ratio sequence does not of itself establish the infinite case. His demonstration that the infinite strings of the tree cannot have a cardinality greater than the number of nodes relies an ability to project the finite cases to the infinite case. It’s not that the binary tree fails to represent all infinite strings but that the cardinality relation he shows for the finite cases concerns finite strings and has not been shown for the case of infinite strings.

Interestingly, the accumulation of the set of segments (or nodes) can be projected to the infinite case using the set theoretic limit. As can the corresponding set of finite strings. And I would think generally most would assert that the cardinality of these projections (limit sets) is Aleph0. So where is the problem ?

INFINITE STRINGS

The problem is that we want to get a handle on the accumulation of nodes and strings for the *actually infinite* case. So concerning this, for consideration:

For the strictly increasing sequence of sets, the set theoretic limit has a domain “For all n”.

Doesn’t it just seem a tad artificial or (to coin a phrase) “artifactual from definition” that we must regard the limit of the node count *for all n” as a complete set of nodes while the limit of the finite string count *for all n* as a complete set of strings fails to have gone “far enough” over the domain in n to actually contain a string that is *for all n* ?? This terrible nexus is at the root of almost all set theory puzzles and paradoxes, real and imagined.