An uncountable countable set
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From: Tony Orlow on 5 Sep 2006 15:07 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? Why then do we not consider an inductive proof of the form E y e N A x>y P(x) not to prove P(aleph_0) or P(omega)? Is it not true that A y e N aleph_0>y and omega>y? Tony
From: MoeBlee on 5 Sep 2006 15:12 Tony Orlow 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. 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. > >> Yes, I am working on that, or was trying to until I got stuck with child > >> care duty the last several weeks. It's not far off, but it's not at all > >> like ZFC. I have tried to present several axioms, such as N=S^L and IFR, > >> and the axioms of internal and external infinity. > > > > Those are not statements of pure logic, hence they're "artificial > > statements". Or, if they are pure logic, then they're inadequate. > > Inadequate why? N=S^L is inductively provable, and inductive proof is > derived directly from logic by forming an infinite loop. 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. > That's all very > elementary. IFR is justified primarily geometrically, and is also very > straightforward. Together they provide means for measuring the relative > sizes of infinite sets of the symbolic and the quantitative variety, > respectively. The axioms of internal and external infinity are > constructive axioms like Peano's, and therefore somewhat "artificial", > but justified as a construction. Were Peano's axioms inadequate? First order PA doesn't give you real analysis. And the PA axioms are non-logical axioms. MoeBlee
From: Virgil on 5 Sep 2006 15:16 In article <44fd9b91(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > What the set of all finite numbers makes infinite? The axiom of infinitys > > states that the set of all finite (natural) numbers does exist. From that > > is is easy to prove that that set is not finite, hence infinite. > > Yes, according to the Dedekind definition of an infinite set, but that > definition leads to some conclusions that offend some people's > sensibilities. The conclusion that the earth is not flat offends some peoples sensibilities. But we nevertheless conclude it. > > > > And indeed, when you go step by step you will not get at the end. > > Right, you will always have gone a finite number of steps and have > arrived at a finite natural. Beyond which there are always infinitely more that one has so far reached. > Well, that's what I'm trying to do, to provide an alternative > perspective and system. A "system" which we are still waiting to see. >I certainly reject the Dedekind definition of an > infinite set as being inconsistent with notions of infinite measure when > it comes to the finite naturals. Since no finite natural has infinite measure, TO's alleged inconsistency is imaginary. > By viewing the situation in terms of density and range Which requires first that one construct the reals, which TO cannot do in vacuo, and has not yet done at all. , > > Yes, much consternation, I can understand that. So apparently you are > > accepting the axioms, but not what follows from the axioms. What kind > > of logic are you using? > > 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. Axioms by definition do not follow from anything, That is why we call them axioms. TO seems to feel that there must be some ur-truths which are true in the mathematical world without benefit of any assumptions at all, but has not been able to state what they are. > Every axiom should be justifiable > based on fundamental concepts. I don't see that here. What TO calls his "fundamental concepts" are merely HIS set of axioms, which he will not elucidate. > > > > > > Rusin had the gall to tell me that if I don't accept that there are an > > > infinite number of finite naturals, then I will join JSH and others on > > > his "kill list". I don't claim that my conclusions are derived purely > > > from ZFC or NBG, but that there are more fundamental concerns which > > > contradict both, and that some other prioritization of principles needs > > > to happen. Proper subsets are smaller. The addition of a single element > > > needs to be reflected in the size of the set. Infinite values are larger > > > than finite values. Things like that. > > > > Well, you are stating such. So provide a framework. Either within the > > accepted axioms, or without it. But if you want to remain within the > > accepted axioms, you should also accept what follows from these axioms. > > And if you want to go outside, provide your set of axioms. > > Yes, I am working on that Come back when done, TO, but not til then. > , or was trying to until I got stuck with child > care duty the last several weeks. It's not far off, but it's not at all > like ZFC. I have tried to present several axioms, such as N=S^L and IFR, > and the axioms of internal and external infinity. The boys start school > Thursday. Halleluiah. All the things TO has presented as potential "axioms" for his system have been a good deal less reasonable than those of ZF. The axioms, informaly 1) Axiom of extensionality: Two sets are the same if they have the same members. 2) Axiom of regularity (also called the Axiom of foundation): Every non-empty set x contains some member y such that x and y are disjoint sets. 3) Axiom scheme of separation (also called the Axiom scheme of comprehension): If z is a set and f(x) is any property which may be possessed by elements x of z, then there is a subset y of z containing those x in z which possess the property, y = { x \in z: f(x)} . 4) Axiom of pairing: If x and y are sets then there exists a set containing both of them. 5) Axiom of union: For any set S there is a set A containing every set that is a member of some member of S. 6) Axiom scheme of replacement: Every formally defined function whose domain is a set has a codomain which is also a set, subject to a restriction to avoid paradoxes. 7) Axiom of infinity: There exists a set x such that the empty set is a member of x and whenever y is in x, so is S(y). 8) Axiom of power set: For any set x there is a set y that contains every subset of x. And for ZFC, 9) Axiom of choice: For any set X there is a binary relation R which well-orders X. This means that R is a linear order on X and every nonempty subset of X has an element which is minimal under R. Kunen also includes a redundant axiom saying that at least one set exists. The existence of a set follows from the axiom of infinity. The axiom of pairing can be deduced from the axiom of infinity, the axiom of separation, and the axiom of replacement.
From: Virgil on 5 Sep 2006 15:22 In article <1157471413.829226.11330(a)d34g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1157368118.735322.299590(a)74g2000cwt.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David R Tribble schrieb: > > > > > > > > > I was not aware that abstract mathematical concepts (e.g., sets) > > > > had any physical constraints. > > > > > > Then you should learn it. It you are unable to physically (i.e. in > > > written form or in your mind) distinguish all the elements of a set, > > > then the set does not exist. > > > > While "in one's brain" may be considered physical, > > "in one's mind" is distinctly not physical. > > The mind is like the soul a present from God? Not quite.
From: Tony Orlow on 5 Sep 2006 15:26
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. > >>>> So, it's not that a->b -> b->a, but that a=b <-> >>>> b=a. >>> That part seems messed up. a=b <-> b=a is just the symmetry of >>> identity. >> Yes, it's that simple. If the object IS the unique set of logical values >> applied to all properties, then each unique set of logical values for >> each statement about an object IS a unique object. :) > > Whatever that means, I doubt it is the principle of the symmetry of > identity, which is that a=b <-> b=a, which makes no mentions whatsoever > of "logical values" or "properties". 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? > >>>> and the >>>> inability to discern two objects by their properties makes them equal, >>>> at least until some property is discovered which can discriminate >>>> between the two. >>> That's going to make the theory subjective - depending on discoveries. >>> Why don't you look at how different mathematical theories handle >>> identity? >> Ummm.... Isn't each isolated theory "subjective" in terms of the >> properties that it explores? > > 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. > >> 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. Formulas are a fine way to distinguish objects. For instance, I distinguish a vastly greater number of different infinities than cardinality simply by ordering formulas on a unit infinity. Good suggestion. > >> How do we know if this is the case? By enumerating all possible >> properties of each. Can we do that? No. We can only say that, given the >> set of properties under discussion in any given theory, the two are not >> distinguishable, within that theory. We cannot say that they are >> absolutely the same object. > > No, we may do better than that in theories in which there are only > finitely many primitive predicate symbols, such as set theory. I told > you all about that already. 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. > >>>>> It depends on the specific theory. In a first order theory with >>>>> infinitely many primitive predicate symbols, we have no theorem schema >>>>> for doing what you suggest. But set theory has only two primitive >>>>> predicates (one if you take equality as defined) so we can state such a >>>>> theorem schema. However, we don't need to do that since the axiom of >>>>> extensionality allows us to prove x=y merely by proving Az(zex <-> >>>>> zey). >>>> And what is z besides one of the set of properties which defines the >>>> sets x and y? >>> That's a confused view of the axiom of extensionality and the role of >>> variables. >> The perception of confusion would appear to be a subjective and rather >> relative phenomenon. > > True. You might not really be confused about the axiom of > extensionality and the role of variables, but rather only pretending to > be. And you might not be confused over the nature of objects and properties, over inductive logic vs. inductive proof, between element count and value in the naturals, or any of a number of things, and maybe just want to be a devil's advocate. Hard to tell sometimes. > >>>> The distinction between elements and properties is rather >>>> tenuous. Is it not a property of y that z e y? >>> For any PARTICULAR z, it's a property of y that z is or is not a member >>> of y. That doesn't entail that y IS the set of properties that y has. >> Consider each object in the universe to be a bit. Does each unique set >> correspond to a unique bit string, where each object's bit posiiton is a >> 1 if the object is a member, and 0 if it is not? > > I thought a 'bit' is a 0 or 1. In that case, in set theory, it is not > the case that each object is a bit. Sorry, a bit position. Number the objects in the universe starting from 0. Every unique set is therefore a unique bit string representing which elements are members. > >> The set y IS the set of >> objects which are members of y, no more, and no less. > > That's correct regarding set theory, and it conflicts with your notion > that a set is the set of its defining PROPERTIES. A set is the set of > its MEMBERS, and it is not the set of defining PROPERTIES. An OBJECT is a set of its defining properties, and a set is a collection of objects which share one or more properties. > >>> In second order, it would be a=b <-> AP(P(a) <-> P(b)), and P=Q <-> >>> Aa(P(a) <-> P(a)). >> If you prefer, you may use <-> instead of =. >> >>> But those don't ential that, for example, b = {P | P is a defining >>> property of b}. >> Really, they do. |