An uncountable countable set
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From: MoeBlee on 5 Sep 2006 21:38 Tony Orlow 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." Hey, look at that, I DID finally get through to Orlow on at least one point. This is a point that took nearly A YEAR for him to come to grips with, but finally I did get through. > However, that does not mean they should not have > ANY justification. Some folks will agree and others disagree with that. > As I said below, IFR is justified geometrically given > the graph of a function, very intuitively. N=S^L is based on > combinatorics. Is that pure logic? Somewhere in the statement of an > axiom should be included the intent, as should be the case with > governmental laws. And the axioms of Z set theory have intuitive bases also. > 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? I could, but textbooks such as Enderton's 'A Mathematical Introduction To Logic' do a much better job than I can do. And I very much prefer not to explain in a vacuum, without giving other explanations that come prior in a systematic treatment of the subject. If you ever want to take it from the top, then let me know. > > First order PA doesn't give you real analysis. And the PA axioms are > > non-logical axioms. > > Like I said, my axioms of infinity are constructive axioms, rather than > deductive. There are rules for creating the system and rules which > follow from those regarding how it behaves. Since you've not stated a system, only you would know how it works. MoeBlee
From: Tony Orlow on 5 Sep 2006 21:44 Virgil wrote: > In article <44fda5fe(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> In any >> case, I agree with Leibniz that the unique identity of an object is >> defined by its unique set of properties, and that equivalence between >> two objects is the SAME as equivalence between the entire set of >> properties of each. So, it's not that a->b -> b->a, but that a=b <-> >> b=a. The object IS the set of properties which defines it, 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. > > In Zf, ZFC and NBG, Sets are determined only by what are and are not > members of them. Anything else is irrelevant. > Axiom of extensionality: Two sets are the same if they have the same > members. > >> And what is z besides one of the set of properties which defines the >> sets x and y? The distinction between elements and properties is rather >> tenuous. Is it not a property of y that z e y? > > But is it a property of z, necessary to the identification of z, that > z e y? > Not in ZF, ZFC, or NBG. if zey but ~xey, can it be possible that x=z? No. > >> No, but we can take as axiomatic that a=b = A P P(a)=P(b), > > Not in ZF, ZFC or NBG. > Come out of the cave. The sun won't hurt you. That's only a shadow of a caterpillar. > >> and that P=Q = A a P(a)=Q(a). > > Not in in ZF, ZFC or NBG. Come out and wallow in the Pearl Pond with the swine, O Albino Cave Being. You need a bath. :) > >> Thus properties are defined by the objects to which >> they pertain, and objects are defined by the properties which pertain to >> them. Despite the fact that this statement is not first-order, I see no >> problem arising from it. > > Certain problems arise from the "property" of a set being, or > especially not being, a member of itself. Yes, Russell's Paradox indicates a need to define what kinds of properties are allowed and which are not. That kind of recursion causes problems. > > So unless allowable "properties" are somehow constrained to avoid those > problems, TO's set theory will be self contradictory. Agreed. > > In ZF, et all, such constraints are built in. Oh?
From: Aatu Koskensilta on 5 Sep 2006 21:44 MoeBlee wrote: > Aatu Koskensilta wrote: >> One might also note that addition, >> multiplication and so forth as functions of reals have quite different >> properties than as functions of naturals, which might also make it >> worthwhile to be extra-pedantic in some situations. > > And I think the constructions are worthwhile onto themselves as well. > At least I find worth in them. They are certainly interesting in many respects. I must also confess that I have not been paying any attention to this thread for a long while, and am not aware of the exact context of your statements, whatever they were, specifically. >> As a general observation, it seems that when arguing with cranks people >> often have a habit of preferring overtly formal expositions and >> arguments, stressing somewhat irrelevant technicalities. > > You'd have to go back to the original context of my remarks to see if > the technicality is irrelevent. As said, the above was a general observation, and was not about your remarks in particular, which may have been most pertinent for all I know. >> This is >> certainly understandable - and some people probably just enjoy going >> through the formal details, possibly working out some of the details for >> themselves for the first time > > That might be why some people mention such details, but it is not my > reason. My purpose is to bring to the table the rigorous alternatives > to the incorrect and vague formulations of cranks. But often the rigorous and formal alternatives are not pertinent to the questions under discussion. As an example, to questions about provability of this or that from set theoretic principles it is quite unnecessary to bring in any formal theories - which only invite a whole new set of questions and confusions, driving the discussion in all sorts of strange tangents. Similarly, to the question of the status of definitions in ordinary mathematics it is quite unnecessary to bring to bear the formal theory of definitions in logic. Indeed, doing that only complicates the matter further since one can - and should - then ask about the specific relation of such formal treatments and results to the way definitions actually function in mathematics. Much easier, it seems to me, would be offer certain informal observations about definitions. If these fail to clarify the matter, further elucidations could be offered, and if these too fail, it seems a foregone hope that going formal is of any use. It seems your motivation, if I have understood you correctly, is simply a wish to present the formal treatment as a proof, against the idiocy of the loons you nominally offer them as a reply, of the possibility of handling these questions in a rigorous and precise mathematical way, to whatever spectators these "debates" may have. But, then, again, the question of the relation of these formal treatments to the informal questions - about whether this or that can be proved using these or those set theoretical principles, about how definitions in mathematics are to be understood, ... - arises. I, for one, would love to see such questions raised, discussed and answered in informative ways - indeed, one of my pet complaints about logic texts and discussion about foundations is that even the existence, let alone relevance, of such questions is often overlooked, resulting in an over-simplified account of the role of formalisation in mathematics and conveying the idea that conceptual questions are not important. > Many people, including me, have tried to intellectual contact with > Orlow with many different approaches, including keeping things less > technical (I've tried it too). Nothing works, since Orlow is indeed a > crank. Indeed. What puzzles me is, why then try at all? The sensible course seems to give up. It's not as if there is any danger that an innocent soul not already inclined to crankish ideas be led astray by the incoherent babblings of Orlow, Zick, and the numerous other loons in the news. > So, the objective is not always to make intellectual contact - > either by avoiding technicalities or by adducing them - but rather I > have my own objectives in posting, such as the satisfaction I mentioned > of at least putting on the table the rigorous formulations that I > consider to be an important part of the intellectual achievments of > mathematics. A much more constructive way to go about that would be to write up clear expositions of these achievements, answering common questions and dissolving common sources of puzzlement, published in some other way than as buried in thousand message threads celebrating crankishness, surely? > Ha! Let's see you do it with Orlow. Yes, simple, informal arguments > would be nice. But they're in vain with him anyway. The point of > posting is not always to get through to the cranks. That much is obvious. After all, it is blatantly clear that there is practically no hope of getting through to Orlow, Zick, Finlayson, Easterly, and the others in the current bunch of loons and cranks people have chosen to "debate". Usually the futility of such an exercise becomes apparent after just a few posts - which is why it is somewhat baffling to me that otherwise seemingly reasonable people wish to engage in endless and pointless threads to the inhumane extents they do. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tony Orlow on 5 Sep 2006 21:48 Virgil wrote: > In article <44fdbf78(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> MoeBlee wrote: >>>>> Tony Orlow wrote: >>>>>> MoeBlee wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I >>>>>>>> never thought so. >>>>>>> Those are notations that can be understood precisely only in context of >>>>>>> the particular treatment in which they occur. >>>>>> BLAM!! That's exactly what I'm saying. If a particular treatment cannot >>>>>> distinguish between two elements, then they are the same, >>>>> By a 'treatment' of a formal system, I don't mean the formal system >>>>> itself, but rather an informal presentation of a formal system. For >>>>> example, a treatment would include the English text in a textbook that >>>>> discusses a formal system. So the treatment includes both the formal >>>>> system and the informal presentation of that formal system. For >>>>> example, a textbook, lecture, oral explanation, or even an Internet >>>>> post may be considered treatments or at least fragments of some >>>>> presumable overall treatment. >>>>> >>>>> And your "cannot distinguish" is too vague and broad. We prove equality >>>>> of objects in a theory in very specific ways. We don't just say, "Well, >>>>> no one seems to be able to distinguish them, so they must be the same." >>>>> For example, in set theory, x=y <-> Az(zex <-> zey). >>>> That's assuming x and y are sets. If they are atomic objects, >>>> urelements, then A z not(zex v zey). In this case, neither x nor y has >>>> any elements, but does that mean they are the same? >>> In such a theory, we have a primitive predicate for 'is a set'. Then >>> the axiom of extensionality is: >>> >>> (x is a set & y is a set) -> (x=y <-> Az(zex <-> zey)) and we would >>> have some kind of axiom saying there exists a unique set x such that Az >>> ~zex. >>> >>> That allows that there may be distinct urelements. >> Indeed, or we may NOT allow such things, if we adopt a rule of the form >> x = S: A P P(x) = P e S. That is, every object is a set of values which >> each property has when applied to that object. The truth value of each >> statement that may be made about x is EQUAL to the truth value of that >> property's membership in x. While some restrictions need to be made >> regarding what constitutes a property, this general notion seems sound, no? > > TO presents bits and pieces, each of which, he claims, might be a part > of some sort of theory, but we cannot tell how those parts fit together > towards making a coherent theory without seeing all of them together. > > is. >> That's nice. Is there a difference between an abstract object like a >> set, and its definition? > > In ZF, et all, sets are not defined, they are among the undefined terms > which every mathematical theory must have. > > Whenever one attempts to define everything, one ends, at some point, > with circularity. Can I quote you on that?
From: Tony Orlow on 5 Sep 2006 21:52
Virgil wrote: > In article <44fdc460(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> MoeBlee wrote: > >>> Better would be a=b <-> AP(P(a) <-> P(b)). >> 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. > > "0 through 1"? > > Does TO expect to find any truth values strictly between 0 and 1? Yes, they are called probabilities. > >> I am aware that there are difficulties defining what constitutes a valid >> property in this sense, as Russell's Paradox demonstrates, but I think >> the kind of statement that produce such issues can be identified. That >> would be an interesting discussion.... > > In ZF, predicate definition of sets is limited to defining subsets of > sets which are otherwise known to exist, so that Russell's paradox is > vanquished. > > Absent some mechanism of similar effectiveness, TO's system will crash. Well, that doesn't leave the door open to, as Ross would put it, "a universe in ZFC". However, there is a universe, and sets within it, and a place, somewhere in the theory, for a universe. So, yes, a multidimensional universe of properties defining objects needs proper concoction to work to avoid such problems, but must be possible, given the universe as prime example. :) Well order the universe. :D > >> Ummm.... Isn't each isolated theory "subjective" in terms of the >> properties that it explores? If there is no universal system of cohesive >> mathematics, then this is surely the case. In the example of the >> staircase in the limit vs. the diagonal line, point-set topology cannot >> discern the two objects because it looks only at proximity of >> corresponding points. When one defines the objects using a >> segment-sequence topology, as I suggested, there is a very discernible >> difference between the two objects, namely, as I intuited from the >> beginning of that discussion, in direction of the curve. > > But in TO's case, the objects are not mere sets of points but much more > complicated structures, about which TO asserts many properties he has > not proved which do not hold for mere sets of points. > > > >> Thus, from the >> "subjective" perspective of sets of points, they are equal, but from the >> "subjective" perspective of sequences of segments, they are not. So, are >> they equal? > > The issues are not subjective, but matters of definition. In standard > mathematics representing such objects by sets of points is standard, and > as TO wants them represented is not. > >> Two objects are equal only if there exists no way to distinguish them. > > If they are sets, they are indistinguishable if and only if every > member of either is a member of both and every non-member of either is a > non-member of both. > > If they are TO's constructs and they are anything other than sets, it is > up to him to define what he means by equality. > >> 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. > > That is the trouble with any such property definitions, you never can > tell whether you have considered all properties. > >>> 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. > > The only subject being confused by it is TO > >>> 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 > > But as every "object" in TO's universe is a bit, every bitstring is a > bit also and one does not need any strings of bits separately. > > > >>> Please just read a book on logic. >> Please just think hard. :) > > TO rekes not his own rede. |