An uncountable countable set
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From: Tony Orlow on 7 Sep 2006 09:03 Virgil wrote: > In article <44fe1669(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> I find this last sentence vague because of the word "induction". If you >>>> refer to inductive proof, then I am familiar with the Peano axioms and >>>> the underlying logical construction which makes inductive proof valid. >>> You have only a faint idea. If you actually read a book on the subject, >>> you'd find how much deeper, richer, and rigorous this is. >>> >>>> If you refer to inductive logic, the formulation of rules from instances >>>> of fact, then there are statistical methods coupled with feedback that >>>> make it possible. Which were you talking about? >>> Mathematical induction. Inductive sets, et al. As in mathematical logic >>> and set theory, which is deductive. Not inductive logic as in the other >>> sense of inductive - empirical based inference (or however you want to >>> define it). >>> >>> MoeBlee >>> >> So, what is it you think I DON'T get about inductive proof, sets, and >> recursion? As far as I can tell, very few dare to question the >> predefined rules as set forth, but I have yet to see any valid >> counterexample to my rules regarding inductive proof in the infinite >> case. Where an equality between expressions is proven for all n, it is >> valid for infinite n. > > What standard rule of induction or standard definition does TO claim > justifies these claims? > None contradict it, except for transfinitology. > > >> Where an inequality is proven for all n greater >> than some finite m, and the difference upon which the inequality is >> based does not have a limit of 0 as n->oo, it holds also in the infinite >> case. > > What standard rule of induction or standard definition does TO claim > justifies these claims? None contradict it. There is nothing in the Peano axioms which precludes infinite natural values. > > If neither can be justified by any standard rule, then TO must present > his system in its entirety to get it in, as it is in conflict with every > standard system. It is a suggested rule, a simple extension to standard induction, which conflicts with transfinitology, but not with anything else.
From: Tony Orlow on 7 Sep 2006 09:11 Virgil wrote: > In article <44fe1725(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> And yet, you are saying it's not technically correct. So, 1 is a natural >>>> but not a rational or real? If rigorous formulations come to that >>>> conclusion, then rigor does not ensure correctness. >>> You're hopeless. You didn't understand a thing I said, which may be my >>> fault for not providing adequate explanation in the context of your >>> ignorance of the subject; but it is no my fault that you won't even >>> look at a book on mathematics, such as even a introductory text in real >>> analysis. >>> >>> MoeBlee >>> >> Your whole point here is ludicrous. You are arguing that the naturals >> are not a subset of the rationals, which are not a subset of the reals. > > In which argument he is perfectly correct. That there is an isomorphic > image of the system of naturals in some other systems does not mean that > the original naturals are in those systems. In terms of points on the line, there is no difference. The same point can be reached by increments on 0 (natural), multiplications and divisions on 1 (rational), or Dedekind cuts or Cauchy sequences (real). Are 1, 1/1 and 1.000.... three different points or quantities? Viewed as a set of points, the real line has as proper subsets both the set of rationals and the set of naturals. To say the distance to the origin is different at a given point depending on how you got there makes no sense. > The empty set is, at least in the von Neumann model, a natural number > but it is not a rational number nor a real number, as the corresponding > rational and real numbers are both much more complicated objects. The empty set is nothing. The von Neumann ordinals are a misleading model of the naturals. > >> While each superset may require a more complex construction than the >> subset, all elements of the subset are covered by the superconstruction. >> As points on the real line, they are all real numbers. What you're >> saying amounts to what I said above, which is indeed absurd. So, no, I >> don't understand a thing you're saying, when you say naturals aren't >> reals. Sorry. > > Which statement of misunderstanding only underscores the profundity of > TO's invincible ignorance of mathematics. > > There are two ways of constructing the rationals from the naturals > (1) by first constructing the integers and from them certain ratios of > integers, or > (2) By first constructing certain non-negative ratios of naturals and > from them the signed ratios. > Once one has the rationals, one can construct the reals from them either > via Dedekind cuts or as a quotient ring of the ring of Cauchy sequences > modulo the ideal of null sequences. That's all about specifying points on the line. If you specify the same point, it's the same number, no matter how you got there. > > For someone of TO's talents, learning the details of this would involve > at lest a semester's serious and guided study, but more likely a year > or two. After which he just might understand why the set of naturals is > not directly a subset of either the set of rationals or the set of reals. It's really a pedantic point. Constructions differ for the three, but a point is a point is a point. I could stand to study up on Dedekind cuts and Cauchy sequences, but I'm not unaware of how those kind of definitions differ from those of the naturals or rationals. Have a nice day.
From: Tony Orlow on 7 Sep 2006 09:14 Virgil wrote: > In article <44fe17fb(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <1157471262.974912.294510(a)b28g2000cwb.googlegroups.com>, >>> mueckenh(a)rz.fh-augsburg.de wrote: >>> >>>> Virgil schrieb: >>>> >>>>> In article <1157367209.318653.75760(a)p79g2000cwp.googlegroups.com>, >>>>> mueckenh(a)rz.fh-augsburg.de wrote: >>>>> >>>>> >>>>>> Counting is the most primitive version of addition. >>>>> Counting is not addition at all. >>>>> >>>> So you cannot even count, in your infinite unphysical mind? >>> Counting precedes adding, in that one may count when unable to add. >>> >>> But how is one able to add without being able to count? >>> >>> A primitive version of counting, from which the word "calculus" >>> allegedly derives, is Greek shepherds in classical times counting their >>> flocks by pairing off their sheep with pebbles (calculi). >> Yes, counting does precede addition. Addition is repeated increment, >> multiplication is repeated addition, exponentiation is repeated >> multiplication, tetration is repeated exponentiation... >> >> Increment is fundamental. It's not just successor, but successor with >> measure, and this is what generates the natural numbers. > > Except that successorship does not involve any measure of anything, TO > is, for a change, nearly right. Increment is successor with measure, which implies successor doesn't have measure, so you are saying exactly what I said, which apparently, in your opinion, what entirely correct. That's good, because I was backing up what you said for a change. Isn't that nice? > > Set successorship only involves the union of a set with the singleton > set which contains it as a member. > The successor of any set, x, is (x \/ {x}) That's not the only way to describe it, but that's the von Neumann model, anyway. Tony
From: Tony Orlow on 7 Sep 2006 09:28 Virgil wrote: > In article <44fe1d38(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: > >>> You ignore instead of asking me for justification. The justification is >>> in the incompleteness theorem, which is even STRONGER than what I >>> mentioned. Aside from incompleteness, if you knew even just a little >>> bit about the subject you'd understand the sense in which you can't get >>> adequate mathematics (such as, say, enough to do calculus) from just >>> logical axioms. >> Listen, in retrospect, I agree with the statetement, "To get an adequate >> amount of mathematics, you have to adopt axioms that are not derivable >> from pure logic alone." However, that does not mean they should not have >> ANY justification. > > The axioms of set theory whether ZF, ZFC or NBG or some other variation > has a long and serious history of development, and the reasons why those > and not other axioms and axiom systems have survived are in that history. > > So that if TO wants justifications, they are available, but they are > often too technical for someone of TO's level of understanding to > comprehend, and even if not, tend to be buried in papers that are > otherwise highly technical in precisely the ways TO object to all the > time. > In other words, you justify the viability of the axioms by the fact that they have gone through a long evolutionary process and survived in their current form, rather than having died out. The same can be said for the platypus. Quack! ;) > > >> As I said below, IFR is justified geometrically given >> the graph of a function > > On the contrary, there are lots of "graphs" for which it is nonsense. Not of invertible functions, which are required for bijections between sets of reals in the first place. > > >> very intuitively. N=S^L is based on >> combinatorics. > > But is only valid for naturals, even when properly interpreted. > And is often misinterpreted. > No, it is quite applicable in the infinite case, for instance, to binary trees or digital systems. > >> Somewhere in the statement of an >> axiom should be included the intent > > The intent is discussed in detail in all those books that TO declines > to read. Uh huh. > > >>> Oooh boy. You don't even have a logicistic system, logical axioms, nor >>> rules of inference, and yet you're telling me that you can prove your >>> mathematical axioms, let alone your notion of inductive proof as >>> provable by logic by forming an infinite loop shows you really have no >>> idea what induction is. >> If you say so. What is it, then, in a nutshell? I mean, you can't define >> "mathematics", but maybe something a little more restrictive could be a >> good place to start. So, why don't you explain induction? > > It has been defined quite clearly any number of times for TO, but he has > such a weak memory that nothing sticks. > > Induction says: Given a subset S of the set of all naturals,N, > if the first natural is in S and the successor of every element of S is > also an element of S, then S = N. > > And that is all it says. Is that so difficult to remember, TO? That's not all it says, or it wouldn't be applicable to proofs. P(1) ^ (P(n)->P(n+1)) -> A neN P(n), for any property or statement P regarding n. It's used to prove statements regarding the naturals. The mistake in the standard treatment is that it's never taken to be valid for infinite n.
From: Tony Orlow on 7 Sep 2006 09:33
Virgil wrote: > In article <44fe2642(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: >>> In article <44fd9eba(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Dik T. Winter wrote: >>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> >>>>> writes: >>>>> Your axiom uses things that are not defined. What is the *meaning* of >>>>> "x<z"? >>>> Geometrically it means that x is left of z on the number line. >>> >>> And for someone standing on the other side of the number line would x be >>> on the right of z? >>> >>> And does the line stay horizontal as one moves around earth? Which way >>> is larger if the line ever goes vertical. And how does the "larger" work >>> at antipodes? >>> >> Silly questions. > > In response to a silly definition. >>> >>>> It means >>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it >>>> needs to, wouldn't you say? >>> Not hardly. >>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) >>> is a bit better but still insufficient. >> True, I should have specified y<>x and y<>z. I guess it's usually done >> using <= for this reason, eh? >> >>>>> > > That is not a definition, because it makes no sense. "The set of >>>>> > > naturals >>>>> > > is as large as every natural"? >>>>> > >>>>> > It is not larger than all naturals >>>>> >>>>> That is something completely different again. >>>> It's not LARGER than every finite. >>> Which natural(s) is it "not larger" than", in the sense of not being a >>> proper superset of that natural or having that natural as a member? >> ....11111 binary (all bit positions finite) > > Unless that string has only finitely many bit positions as well as only > finite bit positions, it is not a natural at all, as it is then neither > the first natural nor the successor of any natural, and every natural > has to be one or the other. It is the successor to ....11110. Duh. I've already proven that this is a finite value, given that all bit positions are finite, and that therefore no place in that string can achieve an infinite value, and that any such number has predecessor and successor. The cute thing is that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) >>>> If I say it's larger than all naturals, how do you read that? >>> As its being a proper super set of that natural and containing that >>> natural as a member. >> Which natural? The "all" natural? > > Every natural. Only TO believes in an "all" natural. Is "organic" already used as a mathematical term? > >>>> Nowhere has the set ever become infinite in count, as long as you only >>>> count finite units. >>> For the definition of cardinality, that is not an issue. The only issue >>> is the possibility of bijection with other sets. >> Yes, cardinality ignores many issues. > > Only those irrelevant ones that TO homes in on, to the exclusion of all > the relevant ones. Uh huh. |