An uncountable countable set
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From: MoeBlee on 25 Oct 2006 16:27 Ross A. Finlayson wrote: > MoeBlee wrote: > > Lester Zick wrote: > > > >I don't insist that conversations must be about set theory. However, > > > >when various incorrect things are said about set theory, then I may see > > > >fit to comment, as well as I may see fit to mention set theoretic > > > >approaches to certain mathematical subjects. In the present case, a > > > >poster mentioned certain things about set theory, then I responded, > > > >then he asked me questions, then I gave answers, then you commented on > > > >my answers. That hardly presents me as insisting that the scope of this > > > >newsgroup be confined to set theory. > > > > > > Technically set "theory" is only a method not a theory since it can't > > > be proven true or false. > > > > By a 'theory' I mean a set of sentences closed under entailment. That's > > what *I* mean when I use the word in such contexts as this one. I deny > > that there are other senses of the word; I'm just telling you exactly > > what I mean in this context. > > > > > Moe, you get all bent > > > out of shape and claim they're spouting nonsense on the internet for > > > no better reason than they don't follow the party line on the subject. > > > > No, I have no problem with someone rejecting the axioms of any > > particular theory, thus to reject the theory. What I post against is > > people saying things that are incorrect about certain theories. Saying > > that one rejects the axioms or even that one rejects classical first > > order logic is fine with me (though, I'm interested in what > > alternatives one offers). But that is vastly different from saying > > untrue things about existing theories. > > > > MoeBlee > > Eh, ZF is consistent with itself. That's because anything can be > proven in an inconsistent theory, except where it would conflict with A > theory of course, because there are only and all true statements in A > theory. What benefit, what satisfactions do you derive from typing nonsense such as above? (The only correct thing you said above is that an inconsistent theory proves every statement (that is, every statement in the language of the theory)). > Hey, did you find that counterexample in the book? I already asked you for the page number of the specific result you have in mind. If you give me the page number, I'll look at it next time I'm in the bookstore (supposing that the store still has a copy). MoeBlee
From: MoeBlee on 25 Oct 2006 17:00 Lester Zick wrote: > You've already rejected the alternative I offer although I don't know > why. What alternative? You referred me to a post you made. That post doesn't include mathematics for real analysis. > Well see the problem here, Moe, is that you and standard set analysis > have no demonstrable basis for truth. So "untrue" things cannot be > said of a standard analytical method for sets which has no method of > demonstrating "truth" except in reference to its own assumptions of > truth the first among which would seem to be that what it assumes is > true is true. I'm not talking about whether the axioms and theorems are true. I'm talking about whether a theorem is indeed a theorem of a certain set of axioms or what a particular formulation is or is not (not what it is understood to say, but rather the question of the brute fact of just what the formulation is as a sequence of symbols). Such questions, again irrespective of the question of the truth of the axioms and theorems, are ones that can be settled, at least in principle (and to a great extent, in practice) by direct inspection of sequences of symbols. > In other words the only standard of truth in standard > set analysis would seem to be the lack of inconsistency with its own > assumptions of truth. That is incorrect. The method of models defines 'true' in a way that is not just reducible to consistency. > And when someone like me comes along and suggests the analytical > techniques of standard set analysis are flawed you get all bent out of > shape because you insist not that I accept individual assumptions but > because I reject the techniques which standard set analysis insists > are somehow "true" and if I don't I'm therefore saying things about > standard set theory which are somehow "untrue". No, I call you on the mistatements you make about what the actual formulations of set theory are and on your mischaracterizations of set theory that accrue from your ignorance on the subject. I have little desire to convince anyone that the axioms and theorems of set theory are true. And I welcome, as opposed to desiring to squelch, alternative theories. I have a long range agenda of eventually learning as much alternative mathematics as I can. But that does not entail that I should abandon calling people such as you on misstating, mischaracterizing, and utterly misunderstanding what the actual formulations of set theory and mathematical logic are and calling certain other people on the fact that their proposed theories are - for the lack of a specified mathematical language, logic, primitives, and axioms, and not even shown to be amenable to such specification - lacking as rigorous or (in certain sad cases) even coherent mathematics. Hey, as to set theory and mathematical logic, IF you know ANYTHING about it, then you should critique it to your heart's desire! But your grumblings about set theory and mathematical logic are irrelevent since they are not about set theory or mathematical logic, but rather about what you, as a function of ignorance on the subject, merely IMAGINE set theory and mathematical logic to be. MoeBlee
From: MoeBlee on 25 Oct 2006 17:22 Lester Zick wrote: > So all you're really doing is regressing your appraisal > of standard set analysis and techniques to a group of terms which only > appear to have meaning within standard set analysis. No, all the definitions revert to one undefined non-logical primitive (two undefined non-logical primitives, if we take equality as undefined.) That one non-logical primitive is 'is a member of'. The logical primitives are 'for all', 'not-both' (or we could use 'neither nor', or we could use more than one sentential connective as primitive), and denumerably many variables. > Which doesn't even address the problem of whether the primitives you > argue from are actually true. They're just assumptions. No, primitives are individual symbols. They're not even statements nor assumptions that can be appraised for truth. > >x is equinumerous with y <-> Ef(f is a bijection from x onto y). > > > >There is no assumption of having defined 'cardinality' prior to the > >above definition. > > So if you don't actually say the word "cardinality" it isn't there? Yes, the term 'cardinality' is not in the above definition. > So > if you say the word "equinumerous" instead we can all go home and rest > easy that there is no "primitive" implication I don't refer to a "primitive implication", and I have no idea what you mean by it. > than things which are > equinumerous don't bear a cardinal relation to one another to begin > with? To begin with in the "real world" (whatever you take that to be)? The definitions are of symbols of the language, which are rendered for convenience by English nicknames. The ordering of the definitions is not ensured to match some consensus of concepts of real world ontological or metaphysical relations and dependencies, whatever that might even be. Look, again, I have to say, it is just not productive for me to try to explain this in the vacuum of your knowledge of anything on the subject. After this post, I really need to execise some self-discipline by not wasting my time composing explanations for you that do no good since they presuppose at least some familiarity with the subject matter, which you don't have. Or, if you like, I can start at page 1 with you. But I know you're not interested in that. So, if you're not going to start at page 1, then it is just not suitable for me to try to explain to you what's going on at page 100. MoeBlee
From: MoeBlee on 25 Oct 2006 17:41 Lester Zick wrote: > So your arguments > for establishment views I am interested in learning ZF and related theories, but I do not claim that ZF is superior to any other theory. In my arguments with cranks I do not claim that ZF is the best theory (I have no opinion really on a "best theory", especially since at this stage I am just trying to learn, not make definitive normative judgments about "best"), but rather my points usually are to correct mistatements about what the actual formulations and theorems of set theory are, and to point out that the crank alternatives are not formal theories and not even within a thousand miles of being amenable to being a formal or EVEN coherent theory. > It's all just a smoke > screen as far as I'm concerned, so much jargon and verbiage used to > simulate a sophisticated technical mathematical edifice where there is > none. And, since I have studied the subject, I know that you are wrong on that matter. The systems and terminology are given precisely (or can easily be made PERFECTLY precise with a bit of work on the reader's part, if perfect precision is required, as I do happen to require it of set theory). You can claim a smoke screen all you want, but you won't even look at the actual performance of the mathematics you call a smoke screen. So of course, there is no possibility of convincing you that the system and terminology is precise. I could tell you that there exists a portable hard drive that carries more than a gigabyte in a device smaller than a pack of gum, and you can say forever that no such thing exists, since you won't even look at a showing of such a device. I can say that set theory is a completely rigorous system that axiomatizes the usual theorems of real analysis, and you can say forever that it is not rigorous, just a smoke screen, forever, since you won't read a book that shows just such a rigorous axiomatization. MoeBlee
From: David Marcus on 25 Oct 2006 18:28
stephen(a)nomail.com wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > The vase problem violates Tony's mental picture of a vase filling with > > water. If we are steadily adding more water than is draining out, how > > can all the water go poof at noon? Mental pictures are very useful, but > > sometimes you have to modify your mental picture to match the > > mathematics. Of course, when doing physics, we modify our mathematics to > > match the experiment, but the vase problem originates in mathematics > > land, so you should modify your mental picture to match the mathematics. > > As someone else has pointed out, the "balls" and "vase" > are just an attempt to make this sound like a physical problem, > which it clearly is not, because you cannot physically move > an infinite number of balls in a finite time. It is just > a distraction. As you say, the problem originates in mathematics. > Any attempt to impose physical constraints on inherently unphysical > problem is just silly. > > The problem could have been worded as follows: > > Let IN = { n | -1/(2^floor(n/10) < 0 } > Let OUT = { n | -1/(2^n) } I think you meant Let OUT = { n | -1/(2^n) < 0 } > What is | IN - OUT | ? > > But that would not cause any fuss at all. I wonder. Does anyone reading this think | IN - OUT | <> 0? -- David Marcus |