An uncountable countable set
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From: Virgil on 5 Sep 2006 18:57 In article <44fdc5c9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> I am NOT accepting the axioms. The axioms are artificial statements > >> which can be made to work together, but which do not follow from > >> elementary logic the way they should. Every axiom should be justifiable > >> based on fundamental concepts. I don't see that here. > > > > You keep avoiding what I've told you several times. To get an adequate > > amount of mathematics, you have to adopt axioms that are not derivable > > from pure logic alone. So whatever axioms you produce will be what you > > call "artificial statements" just as you call the axioms of set theory > > "artificial statements." > > Perhaps I ignore that statement because you have not given any > justification for it. I am not convinced that it's necessary to invent > rules which do not follow from more elementary fundamentals. Those more elementary fundamentals will themselves be expressed as rules which then would require existence of yet more elementary rules, and on ad infinitum. You cannot get anywhere without having place to start from. > Inadequate why? N=S^L is inductively provable From what? One needs the entire arithmetic of the natural numbers before it even make sense, and then only makes sense for N,S and L all being natural numbers. > and inductive proof is derived directly from logic by forming an > infinite loop. An inductive proof involving a statement about natural numbers requires the natural numbers at the very minimum. If N, S and L cannot be numbers, what does "N=S^L" mean? > IFR is justified primarily geometrically Which, even if true, would presume a priori the entire structure of the real number system, and/or the geometry of the real line. So that neither "N=S^L" of "IFR" can be an axiom independent of all those many other axioms necessary for either to have any meaning at all. if they are to hold at all in any system, they must be theorems, not axioms. > The axioms of internal and external infinity are > constructive axioms like Peano's, and therefore somewhat "artificial" While I am not sure I recall them, I suspect that, like TO's "N=S^L" of "IFR", those "axioms" must be theorems, not axioms in any system in which they are to hold true.
From: Virgil on 5 Sep 2006 19:06 In article <44fdca3b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <1157367467.816725.158560(a)i3g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > > > > > Then such numbers like 1/3 do not exist (in that representation). > > > > > > > > Indeed, in your tree with terminating edges, such numbers do not > > > > exist. > > > > > > > > > > > > > > But > > > > > the same holds for Cantor's list. > > > > > > > > I do not see how you can jump to that conclusion. > > > > > > Cantor's list has as many lines as my tree. The diagonal has as many > > > digits as each path of my tree. The only difference is that the paths > > > split while the diagonal does not. > > > > And the difference is that given the list we can immediately state what > > the n-th digit of the diagonal is for arbitrary n. You can not state what > > the n-th path is for arbitrary n. > > The H-riffic number system indeed can. :) In your dreams, TO! > > > > > > > > More to the topic: Do you know how Cantor's diagonal is constructed > > > > > from a list of reals? And how this list is constructed? > > > > > > > > Again, the second proof was *not* about the reals. > > > > > > Please spare your nonsense. Cantor did not consider anything else than > > > the reals. If today the proof concerns some wider range then this is > > > not due to Cantor. > > > > Please spare your nonsense. The second part of that article is > > *definitely* > > not about the reals. It uses the reals only "beispielsweise". > > Then later it was applied to the reals to show them uncountable. But not by Cantor. > It > confuses symbolic combinatorics with the real continuum, which is a mistake. That TO calls it so does not make it so. There are many legitimate ways of extending Cantor's actual diagonal proof to show that the reals are uncountable
From: Virgil on 5 Sep 2006 19:12 In article <44fdcaf1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > > Dik T. Winter schrieb: > > > > > > > But if you wish, provide a definition of "number". As far as I know, > > > > there is not one in mathematics. > > > > > > A number has only one of the following properties: It is larger than or > > > smaller than or equal to any natural number. > > > > So omega and aleph-0 are numbers. They satisfy the definition. Thanks. > > So, they are numbers which are larger than any finite number? Aleph_0 is a cardinal greater than any natural as cardinal. Omega is an ordinal greater that any natural as ordinal.
From: Virgil on 5 Sep 2006 19:35 In article <44fdcf67(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> The difference between = and <-> disappears when logical truth values > >> are quantities from 0 through 1, so I don't see that as any better, but > >> equivalent. > > > > You say, in the absence of having specified a syntax for a language in > > which this all happens. > > > >>>> 'equality' would be a better word than 'equivalence' here, I think. > >> I suppose, though the same applies to "equivalence classes" doesn't it? > >> No matter. > > > > No, that is the point. There is a difference between members of an > > equivalence class and the equivalence class itself. > > I didn't say that all objects within a class are EQUAL, but given some > criterion for distinguishing objects, one can form CLASSES where a given > property is the same for all members of any given class, ignoring all > other properties. That raises the issue of which classes are also sets and which are "proper" classes. In ZF, ZFC and NBG, one can avoid the issue of equivalence classes entirely by having a unique representative for each of what would otherwise require an equivalence class, and comparing arbitrary objects to the representative objects. This is the principle used in NBG using the von Neumann naturals as representatives of both finite cardinality and finite ordinality. So that there no equivalence classes need be posited. > > All I was saying is that if the set of property values IS the object, > then the object IS the set of property values. Is that so difficult to > understand? Since that requires that classes of properties must all be sets, it raises all sorts of problems, as well as being a breeding ground for Russell's paradox and other anomalies. > > > > A theory is a set of sentences closed under entailment. Theories are > > not made subjective for our reasons for interest in them. The > > subjectivity is in our deciding to study one theory and not another, > > but as a set of sentences closed under entailment, the theory itself is > > not affected by whether we are interested in it or not or by our > > reasons for interest or disinterest in it. > > You must need another cup of tea. I am not talking about psychological > subjectivity, but the fact that any normal theory only addresses certain > properties of the objects is discusses, and therefore may not have > distinctions that are available in other theories. That presumes that the 'properties' in one theory are even available in another theory, or that, even if they are, that the objects they apply to are the same in both. TO's list of axiomatic (unverifiable) assumptions is growing by leaps and bounds. > > > > >> Two objects are equal only if there exists no way to distinguish them. > > > > See, that is what is subjective (or epistemological). We don't define > > equality by "way to distinguish" but rather by FORMULAS. Actually "equality" in any theory is by definition. In ZF, ZFC, NBG, equality of sets is determined exclusively by what objects are members of the sets being compared. Any other "properties" are irrelevant. > If there are only finitely many primitive predicate symbols, then there > are only finitely many properties being addressed by the theory. For > instance, set theory only uses 'e' and '=', and misses most properties > of sets. Which properties are mostly, if not entirely, irrelevant to set theory. > > An OBJECT is a set of its defining properties, and a set is a collection > of objects which share one or more properties. Not in ZF, ZFC or NBG, nor any related set theory. And there is no coherent set theory in which things "are" their properties. > > Yes, If I just think hard enough, in a blaze of enlightenment I'll see > > that you and you alone have the answers. > > There's lots of people working on answers in the face of this kind of > dismissal. :) And a lot of people issuing this kind of dismissal to the garbage those people are proposing. Others have been there and done that and learned the hard way that it doesn't work very well, if at all. First thing you know, TO will reinvent a theory of types, or some such anachronism.
From: Virgil on 5 Sep 2006 19:39
In article <1157487601.440694.181380(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > Cantor's "diagonal" proof did not even concern them. It was others who > > later applied it to the set of reals. it was originally about the set of > > all lists (functions with domain N) or strings from an "alphabet" of two > > "letters". > > A set of elements E = (x1, x2, ..., x???, ...), which depend of > infinitely many co-ordinates x1, x2, ... x???, ... where each > co-ordinate is either m or w. In somewhat more modern terms, the set of all functions f:N -> {"m","w"} or equivalently the set of all functions g:N -> {0,1} > > > > If one considers the alphabet of {"L","R"} for left branch and right > > branch, Cantor's original proof essentially proves that the number of > > paths in an infinite binary tree is uncountable. > > > > > > > Give my a tree of infinite paths consisting of 0's and 1's, and I show > > > that there > > > are not less edges than paths. > > > > Cantor shows less edges than paths. In a choice between a proof by > > Cantor and a proof by "Mueckenh", I will choose Cantor every time > > even when Cantor's proof is wrong and WM's proof is correct. > > Regards, WM |