Cantor Confusion
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From: mueckenh on 26 Oct 2006 03:59 Tony Orlow schrieb: > > How would you construct an actually infinite set? Pair, power, union? > > They all stay in the finite domain if you start with existence of the > > empty (or any other finite) set. Comprehension or replacement cannot go > > further. So, how would you like to achieve it? > > > > Regards, WM > > > > Inductive subdivision of the unit continuum? We certainly seem to be > able to specify, or approximate arbitrarily closely, some values with > infinite strings of digits. It seems obvious that any finite interval in > the continuum has more than any finite number of points within it. So, > isn't that an actually infinite set, albeit with linear finite measure > and bounds? Sorry Tony, you are in error. We cannot approximate sqrt(2) arbitrarily close. We can visualize it by the diagonal of a square and we can name it. But we cannot approximate it better than to an epsilon of 1/10^10^100. It woud be nice if we could, but assuming we can manage it, only because otherwise mahematics becomes too difficult, is a bit too simple. Regards, WM
From: mueckenh on 26 Oct 2006 04:02 MoeBlee schrieb: > > Cantor laid the foundations of sets theory. If modern set theory would > > not accept his definition, then it should not call itself set theory. > > It's called Z set theory. If you don't want it to be called 'set > theory' just because it has refinements not in Cantor's work, then I > guess we could call it 'zet zery' or whatever. But that's missing the > point. Upon Skolem's refinement of Zermelo's set theory, we have a > formal set theory. Why did Skolem not like it? > This and other formal set theories are what we mean > by set theory for about the last eighty years. We are not obliged to > remain with Cantor's seminal work. > > > But this same definition is the *only* one defined in ZFC: There exists > > a set such that x u {x} is in it if x is in it, abbreviated according > > to Peano: if n is in it, then n+1 is in it. No other than this > > "continue in this manner"-definition can be claimed correct in modern > > set theory. > > You're the one who wants to use "continue in this manner" as part of an > argument, not me. In set theory, we prove the existence of certain > functions if we want to use a 'continue in this manner' type of > reasoning. > > > If you claim that what you call modern set theory has a deviating > > definition of infinity, then I am not interested in your theory. > > This is not a matter of defining 'is infinite'. And if you're not > interested in Z set theory, then fine. But then you don't claim that > your argument about trees in Z set theory? I take it that your argument > about trees is in your informal understanding of pre-formal Cantorian > set theory. And I am not interested in your informal understanding of > pre-formal Cantorian set theory as if your informal understanding of > pre-formal Cantorian set theory has anything to do with formal > mathematics. I proved that there are not more real numbers than a countable set has elements. I proved it in a manner which everybody with moderate mathematical knowledge can understand. And, what is important, in a manner completely independent of your special language. This means that any theory stating the uncountablility of the reals is erroneous. There may be hundreds of different theories and different languages expressing and "proving" this error, including your pet-theory. I am not going to investigate them all in detail as I am not trying to learn anything about creationist sects. I am not going to learn all their languages, as I am not going to learn how to compute astrological horoscopes. It is clear and proven *from outside* that their results are wrong and, therefore, uninteresting for me. Regrads, WM
From: mueckenh on 26 Oct 2006 04:04 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > Yes, just as I said, the discussion is about the philosophy of > > > mathematics and set theory (and, I should add, about informal concerns > > > and motivations), but there is not, WITHIN the set theory discussed > > > there, a definition of 'actually infinite' and 'potentially infinite'. > > > > Does the study of formal languages really make incapable of > > understanding plain text? > > No, but, things being equal, a formalized theory is preferable to an > unformalized one. > > > What is written above means: "INFINITY" IN > > SET THEORY IS ALWAYS "ACTUAL INFINITY". This could be translated as > > completed or finished infinity but usually is not, because that would > > deter new students from studying this matter. And most of them never > > get a grasp of that fact. > > Yes, as I said, in an informal description, set theory takes infinite > sets as objects that are infinite as opposed to "unended". And, again, > as I said, in the formal theory, there are no predicate symbols for 'is > actually infinite' or 'is potentially infinite'. You are so much caught inside your theory that you are unable to look at it from outside. In the formal theory, there is the only predicate infinite used for 'is actually infinite' and there is no use of 'is potentially infinite'. But, alas, if there appear contradictions, then the interpretation of the predicate infinite is quickly adjusted. Regards, WM
From: mueckenh on 26 Oct 2006 04:07 David Marcus schrieb: > Han de Bruijn wrote: > > David Marcus wrote: > > > I don't think so. Bohmian Mechanics is 100% deterministic. All of the > > > uncertainty in the results of an experiment is due to uncertainty in > > > setting up the initial conditions of the experiment. > > > > Sure. Back to the dark ages of Laplacian determinism. > > Don't you mean Newton? I am sure, Han, did not mean Newton. Newton was not a determinist at all. He accused Leibniz, who was a bit more deterministic, of being an atheist. Regards, WM
From: mueckenh on 26 Oct 2006 04:22
David Marcus schrieb: > > Sorry, I do not know what you state of knowledge is. > > I already mentioned several books that you could use. It is not me who always faisl to understand. > > > {2,4,6,...} means > > obviously "all natural numbers". That is the usual notation in > > mathematics. > > What happened to 1, 3, and 5? They aren't natural numbers? I forgot "even". Was it that difficult to recognize? > > > Binary Tree > > > Unfortunately, it was described in a way that I can't understand it. A > > > wild guess on my part is that you mean to set up a correspondence > > > between edges and sets of paths. > > > > I am sorry, but if you need a wild guess to understand this text, then > > we should better finish discussion. > > It isn't really a discussion since you are not speaking in English. We can switch to german, if you think you can understand it better. > > > Observe just how the discussion > > runs with all those who understood it, like Han, William, jpale. > > Perhaps you will step by step understand it. > > I suspect there is nothing to understand. That seems to be you general attitude concerning all knowledge surpassing your narrow cage of formalism. > > > > In standard mathematics, a finite set of natural numbers has a largest > > > element. > > > > Please prove that a set of elements consisting of 100 bits has a > > largest element. > > First you will have to define what the phrase "elements consisting of > 100 bits" means, since I didn't say that. What I said was "a finite set > of natural numbers has a largest element". "100 bits" is ununderstandable to you? To make it easier: Take 100 symbols according to your choice and represent by them as many or as large numbers as you can. > > > > So, what are you actually saying? Are you saying that you don't > > > like standard mathematics? > > > > I am saying that standard mathematics is false. As I have shown there > > are finite sets in mathematics which have no largest elements. > > What does "false" mean in this context? Isn' that an English word? It means "not true". > > > > I've never seen "potential infinity" or "actual infinity" in any > > > textbook I've used. > > > > So you read not the right books or too few. > > Or, you are making things up again. Feel free to tell us what book uses > these terms. > > > > Bohmian Mechanics is a deterministic theory that avoids the measurement > > > problem, satisfies Bell's Inequality (as do all theories of quantum > > > mechanics), agrees with all experiments, and doesn't produce negative > > > probabilities. So, it seems to be a better theory than the one you > > > constructed. > > > > > And, Bohmian Mechanics is non-relativistic. > > > > Therefore I did not adopt it. My theory is even a > > local-hidden-variables theory, which satisfies Bell's inequlities, a > > goal never matched by any other theory. > > Unfortunately, experiments show that nature violates Bell's inequality. You are in error. Read the literature and find out that the inequlities are not violated in my model. > So, your theory may be very nice, but it doesn't agree with experiment. > So, your theory is false. Try to inform yourself before you give your statements. But here is not he place to discuss physical theories. By the way, John S. Bell himself agreed that my theory supplied a formal solution, (although we both did not think that negative probabilities were physically meaningful). You, however, as a formalist with no sense to meaning should be satisfied. Regards, WM |