Cantor Confusion
in [Math]
|
Prev: Pi berechnen: Ramanujan oder BBP
Next: Group Theory
From: Virgil on 22 Oct 2006 16:58 In article <1161518008.776999.238550(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Summarizing the original question: You need the limit omega to > construct the irrational numbers. Every irrational square root of a positive integer can be constructed, in several ways, as a recurrence relation.
From: Franziska Neugebauer on 22 Oct 2006 17:48 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Franziska Neugebauer schreef: >> Han de Bruijn wrote: >>> Franziska Neugebauer wrote: >>>> Han de Bruijn wrote: >>>>>Franziska Neugebauer wrote: [...] >>>>>>x = y :<-> Az(z e x <-> z e y) >> >>> >> >>>Of course. Because all members are sets. But I think this is an >> >>>infinite recursion with the equality definition. Does it end >> >>>somewhere? >> >> >> >> Where do you spot recursion? >> > >> > Two sets are equal if they have the same members. >> >> The "two sets" are one set if they have the same members. Let >> A und B refer to the "two sets". If A = B then the "two sets" >> are one set. >> >> > Two members are equal if they have the same members ... Right? >> >> Two variables x and y refer to the same member (set) if x = y. Once >> again: Where do you spot recursion? > > Or rather call it a vicious circle definition. "x = y :<-> Az(z e x <-> z e y)" is *not* a "circular". There is no reference to "=" on the rhs. > See the response to this by Virgil. Message id? F. N. -- xyz
From: Virgil on 22 Oct 2006 18:31 In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Therefore lim [n-->oo] {1,2,3,...,n} = N. That "limit" is undefined. > > > I gave sometime ago a definition of the limit of sets that is (in my > > opinion) workable, but Mueckenheim did not allow that definition. The > > reason being that under some formulations of the vase problem that > > definition would make the vase empty at noon. > > So it must be wrong and needs no further attention. Why? > Because the contents of the vase increases on and on. Such a process > cannot lead to emptiness in any consistent system - independent of any > "intuition". Except that "Mueckenh", while claiming such inconsistency, has not been able to demonstrate it to the satisfaction of anyone but himself.
From: Virgil on 22 Oct 2006 18:37 In article <1161533640.504465.93700(a)k70g2000cwa.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Virgil schreef: > > > In article <b008d$453887ef$82a1e228$31075(a)news1.tudelft.nl>, > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > Virgil wrote: > > > > > > > In article <6cf73$45387e07$82a1e228$27759(a)news1.tudelft.nl>, > > > > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: > > > > > >>Ha! Mathematicians can't even define their most frequently used symbol, > > > >>which is the equality ' = '. And that is a prerequisite for their "is". > > > > > > > > The equal sign, "=", has many meanings, which differ depending on > > > > context, so that there cannot be one Procrustian meaning that fits all > > > > contexts. > > > > > > > > For sets A and B, A = B means that for > > > > all x, x is a member of A if and only if x is a member of B. > > > > > > Good! Now given two such members x and y. What does x = y mean? > > > > In ZF it means > > "for all z, z is a member of x if and only if z is a member of y". > > That's precisely what I thought. You need the machinery of infinity > even for a basic thing like a = b : two things being equal. What "machinery of infinity" is that? A = {} iff for all z (z e A <-> z e {}) does not require {} to have any elements. t does not even require that any z, except {}, exist.
From: cbrown on 22 Oct 2006 19:00
David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: > > David Marcus wrote: > > > cbrown(a)cbrownsystems.com wrote: > > > > I took HdB's statement as "it is not possible to have a theory that is > > > > empirically supported and states 'A and not A, simultaneously' ". > > > > I should state up front that I don't actually claim that "A and not A" is a good ontological description of entanglement; I was mostly just tweaking HdB. > > > > The fact that there are empirically supported theories which state "it > > > > is not possible for A and ~A to be true simultaneously" doesn't negate > > > > the fact that there are equally empirically supported theories that > > > > state the opposite. > > > > > > Perhaps. Although, I suppose it depends on whether you think the > > > empirical evidence really supports the theory or the people claiming it > > > does are just confused. > > > > Until an experiment which distingushes between two similar but distinct > > theories is proposed and perfomed, there's no scientific way of knowing > > whther adherents to either theory are confused or correct. > > But, if one theory is vague or ill defined, then it can be hard to say > whether an experiment really supports it. I donlt quite get what you are implying here. I assume you have some particular physical experiment in mind where this is the case as regards QM. Could you elaborate? > > > > However, many > > > physicists seem to prefer the illogical "Copenhagen" explanations of > > > Bohr. > > > > I would claim that the Copenhagen interpretation is not illogical /on > > its own terms/ - it's simply inconsistent with other interpretations > > (including "common sense"). > > The "terms" of the Copenhagen interpretation seem to be that anytime you > try to ask a logical question, you are told you can't. > If I understand correctly, very loosely, Bohm sees the probablistic results of QM as being a result of the fact that the hidden variable regarding position, by virtue of its being inherently unmeasurable beyond some arbitrary degree of accuracy, introduces errors in our computations which are essentially chaotic (in the mathematical sense). So we see results that "look" probabilistic, when in fact they are fully determined. In my admittedly uninformed opinion. the Copenhagen interpretation takes probalistic results it to be evidence that, for classical quantities such as position, it is the case that (like my hometown Oakland, CA) "there is no there there". Our assumption that there is a specific "element of reality" called "the object's position" is simply a classical intuition, like the intuition that it shoudl be possible to state unambiguously that "A and B occur simultaneously". Instead, it posits as its "elements of reality" more complex mathematical objects (self adjoint operators). I understand that there are philosophical attractions to both explanations; but I don't see why you claim that there are certain 'logical' questions which cannot be asked in the latter interpretation. > > > > The main limitation I can see in his theory (from my exhaustive 30 > > > > minute study :-) ) is that it seems to rely on the assumption of > > > > non-locality, in a theory that isn't relativistic. > > > > > > > That seems a /lot/ easier to swallow than it would be in a relativistic > > > > theory. > > > > > > I'm not quite sure I follow what you mean. > > > > Non-locality isn't a particularly vexing issue in a non-relativistic > > setting, because the question of what we mean by "event A is > > simultaneous with event B" is perfectly clear. > > > > In a relativistic setting "event A is simultaneous with event B" is > > much more complicated; so when we say "non-locality implies that some > > event A simultaneously affects event B", this is much more complicated. > > I might be wrong, but isn't that the point of the EPR experiment? > > It wasn't the point Einstein, Podolsky, and Rosen were trying to make. > Their point was that if we take it for granted that nature is local, > then the EPR experiment shows that the wave function can't be a complete > description of nature. This is because the wave function at the two > separated detectors is the same, so if the detectors are far enough > apart that a measurement at one can't influence the other, then you > can't explain the observed correlations. > I think your analysis is essentially correct. But if I understand correctly, this relies on a particular definition of "element of reality"; which in the Copenhagen interpretation is the probability distribution, not some real number. Again, it's been a while since I've looked at this stuff, but considering the comments found at: http://www.mth.kcl.ac.uk/~streater/EPR.html http://www.mth.kcl.ac.uk/~streater/lostcauses.html#I and http://www.mth.kcl.ac.uk/~streater/lostcauses.html#XI it seems that while EPR + Bell inequalities implies there are no local hidden variable descriptions; it doesn't imply that therefore QM need be a non-local theory. If I read the comments above correctly, it remains local in spite of Bell, because it identifies the "elements of reality" as probability distributions (self adjoint operators). Despite his obviously unsympathetic view of Bohm mechanics as a "Lost Cause", his arguments seem appealing as at least a logical account of EPR and Bell in the Copenhagen interpretation. I would be interested in hearing your thoughts, particularly regarding his statements regarding classical probability theory versus quantum probability theory (which are frankly a bit over my head). > I think this also explains why Einstein never embraced Bohmian > Mechanics. It was always clear that Bohmian Mechanics was non-local. > Einstein was looking for a local theory, so he wouldn't be interested in > Bohmian Mechanics. This was quite reasonable on Einstein's part, since > until John Bell came along and proved his inequality, there was no > reason to think that you couldn't come up with a local theory that > agreed with all the experiments. Once you realize you have to give up > locality, then Bohmian Mechanics looks very attractive. > Again, in light of Streater's statements, I'm not convinced we have to give up locality (and with it, causality). Whether the trade off (a non-classical definition of what the "elements of reality" are) is worth it is a more philosophic |