Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 25 Jul 2005 13:24 Robert Low said: > Tony Orlow (aeo6) wrote: > > Daryl McCullough said: > >>Not quite. "Uncountable set" means "set with a larger cardinality > >>than the set of naturals". > > Yes, that seems to be the Cantorian definition. Does it actually follow that an > > infinite set larger than the naturals can't be enumerated? Personally, I don't > > see the connection at all, > > The connection is fundamental. Set A is bigger than set B if there > is no surjection from B to A. If set B is the integers, then > saying that set A is bigger than set B is saying that there is > no surjection from the integers to A, which implies that there > is no bijection between them, and hence that there is no enumeration > of A (by definition of enumeration). > > That's how the game works. We agree on the basic definitions > of words like 'bigger', 'enumeration' and so on, and then > work out logical consequences of them. > > Now, until you give a definition of 'bigger', and explain > why my first paragraph above is rendered irrelevant, we'll > all be very grateful. Or at the very least, very > surprised. > Your definition of "bigger" works fine for finite sets. When it comes to infinite sets, however, it gives a lot of false positives in the form of equivalences. Sizes of such sets need to be built upon the properties of their members, so the method differs a little for whole numbers vs strings, although all of these sets are related in ways. An infinite set of numbers, for instance, can be compared to another infinite set of numbers by looking at the functions which describe them; if one is defined by a function that is always larger than the function describing the other, then it is a smaller set. Sets of strings are measured using N=S^L, so we could say, for instance, that all decimal numbers which include only 1's and 0's would be 2^(log10(N)) in size. A more precise notion of size for infinite sets requires a slightly more complex method than bijection, as far as I can see. -- Smiles, Tony
From: imaginatorium on 25 Jul 2005 13:28 Tony Orlow (aeo6) wrote: > Robert Low said: <skip mud...> > Sure, that sounds like good mud. The set of all finite integers is a poorly > defined set, an "indeetrminate" set, with no clear boundary. I can see, > intuitively, that a set that contains arbitrarily large finite values must > include an infinite value, although i am not sure what their proof relies on. Ah. So do I understand then, that despite ranting endlessly at people who have studied normal mathematics, and have proofs of things, you don't actually have a proof of this bit of Orlovian math, merely a hunch that somehow it has to be like this. How is work on the axioms progressing? Will there be an axiom of indefiniteness, that sort of blurs things when there would otherwise be a contradiction? How about your axiom of indeterminateness of set membership? After all, you can't let (e.g.) the (infinite) set of naturals have a clearcut subset of members which are "infinite", since if you did, we would ask you about its complement. And group theory? There _is_ going to be an infinite cyclic O-group, is there? > Now, what do you not understand about N=S^L. The number of binary strings of > length L is 2^L, so you cannot have an infinite set of binary strings unless L > is allowed to be infinite, in which case the binary value is infinite. There > can only be a finite number of finite strings with a finite alphabet. For any particular value of L, yes, of course. I wish you'd stop repeating this trivial and obvious fact, and concentrate on understanding why no-one but you thinks it helps your (circular) "proof".
From: Chris Menzel on 25 Jul 2005 13:05 On Wed, 20 Jul 2005 19:40:06 GMT, Stephen Montgomery-Smith <stephen(a)math.missouri.edu> said: > You do come across as sincere in your differing opinions. I can only > suppose that in some strange manner that your brain is wired > differently than ours are. What seems completely logical and sensible > to us, seems to be nonsense to you, and conversely, what seems to be a > proper argument to you, is so weird and strange to us that we seem > unable to even know where to start it trying to disuade you from your > point of view. Oh please. There isn't just a difference in points of view here. TO is making demonstrable, elementary mistakes of both logic and mathematics. It would be one thing if he were to point out clearly some particular principle (Axiom of infinity? Power set? Excluded middle?) that he disagreed with -- we might then be able just to acknowledge a difference in intuitions, in which case the idea of "different wiring" might have some purchase. But TO has at least implicitly acknowledged such things as the existence infinite sets and cardinals, and the power set axiom, and he reasons using excluded middle, so his rejection of the theorems these principles entail shows that his views are at least implicitly contradictory. In addition, however, he has explicitly made many elementary mathematical errors and committed numerous logical howlers, all of which have been pointed out to him very clearly. That may indeed be a matter of wiring, but that would be a shame; I would rather hope it is a curable combination of ignorance and rather appallingly shameless hubris. Chris Menzel
From: Tony Orlow on 25 Jul 2005 14:00 Dave Rusin said: > Bored today, I peek in again at this year's Tiresome Poster of The Year > winner and find, to my surprise, that there has been a tiny bit of > progress. Something like actual definitions have been offered. Imagine! > > In article <MPG.1d4863d52071fde5989f51(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > >> Tony has any number* of "proofs" that an infinite set of natural > >> numbers must include "infinite naturals", but these are generally > >> circular. The one from "information theory" says that since there can > >> only be a finite number of strings of finite length (even if the length > >> has no limit), then to get an infinite set of numbers, you must include > >> some that are infinitely long. The bit after "since" is a restatement > >> of what he purports to prove, but he ignores people pointing this out. > > > >Ahem! That is another misrepresentation. The bit after "since" is a statement > >about symbolic systems, and is a fact outside of the natural numbers. > > True! (The last bit, I mean) -- maybe even truer than you think. Whatever that means.....probably meant to be insulting. It's a fact in the area of symbolic systems, not quantitative systems. > > >Given a set of symbols of size S, > > (To be precise, it's the SET which has size S, not the symbols themselves. > Here "set ... of size S" should mean that S is a cardinal; I don't know > whether Master Orlow is going to use a _finite_ set of symbols or not, but > in fact what he's about to say is TRUE even for non-finite sets S.) Thanks for the clarification. I sure am glad no one thought I was talking about symbols that all have some certain size..... But I should have known you would find some way to misinterpret things. > > >one can construct a set of all strings of length L, > > True! (But this time "set ... of length L" does NOT mean the _set_ has > "length" L; this time it's the strings IN the set that have length L. > Isn't "natural" language a bear? Everyone says things ambiguously > this way, but only T.O. is actually tripped up by the ambiguity.) Yeah, natural language is a bear when you deliberately make it that way. Who ever talked about sets having lengths? People don't talk about the length of a set. You are going out of your way to try to make me sound stupid, but all it proves is that you can't read. Please try to be constructive. > > We'd better be clear what "strings" are. It's not enough to have a > bunch of symbols; the symbols have to come in a certain order, e.g., > "1A$" is not the same string as "A1$". So unless you want a very > generous notion of what "strings" are, you'll have to interpret > "length L" as meaning that L is an _ordinal_, or at the very > least that L is a linearly ordered set (not necessarily > well-ordered, I suppose). Yes, strings are ordered sets of symbols which need not be unique. L is a number, a quantity, representing the number of symbols in the string. It is not ordinal. > For strings of finite length, this is > unnecessarily fussy since the finite ordinals and finite cardinals are > identical, and we can just say "...of length 3" without much fuss and > bother. On the other hand, there is a very clear difference between > what one might call "strings of length 1+omega" and "strings of length > omega+1" -- one of them has a zeroth term, then a 1st, 2nd, 3rd, etc; > the other instead has a 1st term, then a 2nd, 3rd, etc. AND a last term. > Failure to notice the distinction has led our hero to spout all kinds > of gobbledygook. Actually, that distinction is artificial. Numbers are numbers, really, whether finite or infinite. Ordinals are simply numbers that denote position in a discrete coordinate system, whereas cardinals are number representing size or number. What major difference do you see between starting with an index of 0 vs. 1? The origin of the system is arbitrary. > > But again, yes, for any ordered set L one can indeed construct the > set of all L-strings of elements of S. (It's isomorphic to, or by > some definitions equal to, the set of functions L --> S .) > > >and the set of strings has size S^L. > > True again! Even when S and L are not finite (assuming "... has size ..." > means "... has cardinality ..."). Indeed, this is the usual _definition_ > of what cardinal exponentiation is. As a bonus, we get to wink at the > subtleties of the last section, since if L and M are two ordinals > of the same cardinality, then S^L = S^M. > > >This is a fact, > > True! > > >which > > [mal-formed sentence alert: we've got a subject to a sentence for which > no main verb follows. I think what was intended was that the verb "proves" > was to follow "...(S is finite)," below. Or something like that.] > > >when combined with > >the fact that digital strings are strings on a finite alphabet (S is finite), > > True! -- sort of. I'm not sure who uses the term "digital strings". > Do they have in mind strings of a _particular_ finite length? Of arbitrary > but finite lengths? Or what? Hmm, we'll let this slide for a moment. > > >S^L can only be infinite if L is infinite. > > True again! Indeed, ( S^L is finite ) iff > ( ( S is finite AND L is finite ) or ( L is empty ) or ( S is empty) ) . > > >Therefore, an infinite set of > >digital numbers MUST contain numbers with infinite numbers of digits. > > Aha! Tony wins! Yes indeed, it is true -- IF "set of digital numbers" means > "set of strings in S^L, where S = {0, 1, ..., 9}" (I'm guessing here), > then indeed, such a set cannot be infinite if L is finite. That's > absolutely correct! If we specify _A_ finite set L and consider the > set of digit strings of length L, then the set of all of those is finite. > Sure enough! Someone actually understood what I was saying???? Miracle of miracles. Dave wins! > > BUT --- what has that got to do with the set of natural numbers? > No one has ever said that each natural number is a digit string of some > _single_ finite length L. (Indeed, no one but hacks ever says that natural > numbers are digit strings in the first place, but never mind that now.) Are you saying that not all natural numbers can be represented as strings of binary or decimal digits, or that not all unique infinite strings of digits represent unique whole number values? > > It is true that EACH natural number n (individually) can be represented by a > digit string of SOME finite length L(n). But (duh!) there is no single > finite ordinal L which is at least as large as every one of these, > in other words, bigger numbers need more digits. Duh. > > If you want to consider all the natural numbers at once as being members > of a single set S^L so that you can apply the previous line of reasoning, > you'll have to use an infinite L . You're welcome to do so, and you > may choose for example to use the ordinal omega , so that today's date > would be > ...0002005 - ...0007 - ...00020 > > That's pretty cumbersome but not wrong. It's damning for your line of > reasoning because now the "L" in your arguments is not a finite set, so > you can't conclude N is finite. Huh? Firstly, L is not a set, but a string length, and we are considering how the size of the set of strings varies with the length of strings we include, given a certain set of symbols of size S. I don't see anything damning to my argument, especially because you have just drawn the conclusion I drew a long time ago. If L is infinite then so is the set of strings, and vice versa. The finiteness (not the size) of the set of strings is the same as the finiteness of the string length. > It's also a potently misleading notation > because S^L will include other things besides natural numbers -- things > like ...01010101 which are most emphatically not natural numbers. But they are whole numbers, which is really the point, and are required in the infinite set. > And yes, you can use other, larger ordinals too, embedding the natural > numbers into S^(omega+1), for example --- a set which includes not only > the previous non-numbers like ...010101 (which I would now assume is > shorthand for 0 ...010101) but also things like 1...000000 . > This last bit of freedom allows you to make an even bigger fool of > yourself by continually changing your choice of ordered set L, > thus changing the set of things about which you make your wild claims. Again, L is not a set. And you were doing so well.... > (For example, even using L = omega+1 does not allow an interpretation > of what "101...0101" means. In which S^L does this string lie, grandmaster?) In which S^L? Are you asking what number that is? It's 5N/7+1. Treat the infinite whole numbers as fractions of N and you'll suddenly see how well this all works. > > Summary: To his credit, Tony has actually said some correct things about sets > of the form S^L . But he mistakenly assumes that the set of ("finite") > natural numbers is a subset of S^L for some finite L, and based on > some of his other screed seems to believe that the natural numbers > coincides with S^L for some infinite-but-never-quite-specified-and- > indeed-of-time-shifting-value ordered set L. So his correct > statements about S^L have no bearing on questions about the set > of natural numbers. You mean there is no bijection between finite digital strings and natural numbers? You really think that is a valid dismissal? It's not even remotely true. The set of finite naturals is equivalent to the set of finite digital strings. Claiming you have an infinite number of finite whole numbers is equivalent to claiming you have an infinite number of finite strings, which as I understand it, standard analysis also claims. However, you have just agreed that one CANNOT have an infinite number of ANY kind of finite-length strings, if they are constructed from a finite set of symbols. So, despite your "infinite-but-never-quite-specified-and-indeed-of-time-shifting-value ordered set", whatever that means (I think it was meant to make me feel stupid), you have not pointed out how you can represent all infinity of these finite naturals in a digital system without using infinite strings, or how you can require infinite strings without representing infinite values. You have simply erroneously attributed some senseless claim to me and dismissed it using an equally senseless response. So, please clarify your position. Is there a 1-1 correspondence between digital strings and whole numbers? Can you have an infinite set of strings without infinite lengths? How can you have an infinite set of digital whole numbers, without having infinitely long strings representing infinite values? > > HTH! HAND! Yeah mecca lecca hey mecca hiney ho! > > dave > -- Smiles, Tony
From: Tony Orlow on 25 Jul 2005 14:20
Daryl McCullough said: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > >Daryl McCullough said: > > >> Not quite. "Uncountable set" means "set with a larger cardinality > >> than the set of naturals". > >Yes, that seems to be the Cantorian definition. Does it actually follow > >that an infinite set larger than the naturals can't be enumerated? > > Yes, it certainly does follow. The definition of "enumerated" means > "put in one-to-one correspondence with the set of naturals (or a subset > of the naturals)", which implies "having the same (or smaller) cardinality > as the set of naturals". Okay, well that is only because you view all "countable" sets as equivalent. In my book, the set of multiples of 1/2 is twice the size of the whole numbers. It seems perfectly countable. And, the powerset of a countable set, as far as I can see, is also countable, as demonstrated by denoting each subset by a unique binary string, which also represents a whole number. The power set is larger than the root set, but that doesn't mean onecan't enumerate all the subsets in a set linearly. > > >Personally, I don't see the connection at all, and view it as a > >conflation. > > I don't see how you could fail to see the connection. Maybe the above will help you. > > >That's interesting, since the mathematiicians here seem to > >like to haggle over words that they themselves can't define, > >like "infinite". > > That's false. Nobody likes to haggle over the definition > of "infinite". It has a perfectly good definition. No, "infinite set" seems to have an established "definition". No one seems able to define "infinite". > > >I use Bigulosity, to distinguish my measures from cardinality. > > Fine. Give a definition of Bigulosity. What *I* mean by > a definition is a rewrite rule so that any sentence involving > the word "Bigulosity" can be rephrased into an equivalent > sentence involving standard mathematical and logical concepts: > > existential quantification, > universal quantification, > addition, > multiplication, > equality, > set membership, I am not redefining any of the above concepts, except for equality as regards the sizes of infinite sets. > > >Try "unending". Or, as I requested, give ANY synonym, or > >definition of the word itself. > > For mathematical purposes, a word is defined if you are > able to rewrite any sentence involving that word into > an equivalent sentence involving only standard concepts. Gee, if I stick to standard concepts, then don't I just come up with standard analysis? There's already a million of you hard at work toeing that line. You don't need me there. > > >> Nothing of any importance about mathematics would change > >> if we substituted different words for the basic concepts. > > > >Then you shouldn't be having a word problem with me, right? > > I have no idea what you are talking about when you use > the words "infinite", "finite", "larger", "smaller", > "size", etc. You don't know what an infinite number, set, string, tree, or process are? They go on forever, without end, sometimes in more than one direction. Finite ones have an end, which is where the word comes from. Smaller means closer to zero, and larger means further from zero. For set sizes, smaller means closer to the null set and larger means further from it, with "more" elements. Strictly speaking, a set is larger if one or more elements is added, and smaller if one or more elements is removed. > > >My arguments have NOTHING to do with terminology. > > Then rephrase them without using the terminology "infinite", > "finite", "size", "larger", "smaller", "without end" etc. > Rephrase it using only *mathematical* concepts. > > When I say "There are infinitely many natural numbers" I > mean exactly "There exists a function f from naturals > to a subset of the naturals". I mean that there is no end to the set. > > >If I say that the set of evens is smaller than the set of naturals, > > Don't say that. Say what you mean mathematically. For example, maybe > you mean > > The set of evens is a subset of the set of naturals. > > I agree with that. But don't use the word "smaller" because > that word doesn't mean anything definite. Yes it does. It is a proper subset. Elements have been removed. Therefore it is smaller. This is not the only criterion for size, but in the case of a proper subset, the relationship is obvious. We could say most generally, that if we can tranform one set into another by adding and removing elements, then the first is larger if we remove more elements than we add, and smaller if we add more than we remove. > > >I think you all know what I mean > > No, I don't. > > >Cardinality is supposed to be a measure of set size, > > >> As a challenge, see if you can express your claims about > >> infinite sets, or infinite naturals, or set size, or whatever, > >> *without* using the words "infinite", "larger", "size", etc. > >Yeah sure, and you describe your fluffy pink flying elephant > >without using the words "fluffy", "pink", "elephant" or "flying". > > If I'm talking about fluffy pink flying elephants, then I'm > not talking about mathematics. When you talk about "infinite objects" > without defining them, then you aren't talking about mathematics. No, but you want me to talk about them without using the words that denote them. > > >How do you expect me to talk about infinity or infinite > >sets without using the word "infinite"? > > If you can't, then you aren't talking about mathematics. The word has been defined. If you don't understand the definition as unending, then i recommend contemplating Turing machines, trees, number systems and symbolic systems, as well as spece and time. I can't define "end" for you if you don't already know what that means. I am not getting into infinite regress as a distraction. Hopefully I clarified things a little for you above. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony |