Obections to Cantor's Theory (Wikipedia article)
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From: Jesse F. Hughes on 31 Jul 2005 17:27 "Jiri Lebl" <jirka(a)5z.com> writes: > BTW from the formulation of your answer I assume you're a programmer > and not a mathematician. Vaughan Pratt is a mathematician as well as a computer scientist (and logician). I don't know if one should call him a programmer too. Google is your friend. -- Jesse F. Hughes "Love songs suck and losing you ain't worth a damn." -- The poetry of Bad Livers
From: george on 1 Aug 2005 19:41 sradhakr wrote: > Thanks for acknowledging > that NAFL exists. Now can you take the next > step and understand why I > am objecting to Cantor, and in fact, > classical/intuitionistic logics? No, not yet. Infinity is problematic but that's no reason to wax all phobic about it. > In particular, take a look at math.LO/0506475 > and let me know what you think. I can't expand that reference. What I can reply to is the abstract of the original article. > In the proposed non-Aristotelian finitary logic (NAFL), > truths for formal propositions can exist only with respect > to axiomatic theories, This is already on the verge of a category-mistake, by classical measures. Classically, truths for formal propositions exist only with respect to INTERPRETATIONS (or candidate MODELS) of theories, NOT of theories directly. Theories have theorems, not truths. > essentially as temporary axiomatic declarations in the > human mind. An undecidable proposition $P$ in a consistent > NAFL theory T is Is basically a contradiction in terms, because in NAFL theories, unlike in classical ones, nothing is undecidable; rather, things that WOULD be, classically, undecidable, are decided to have some 3rd superposed truth-value. > true/false with respect to T if and only if it has been > axiomatically declared as true/false by virtue of its > provability/refutability So far, so good, except that if the axioms have in fact declared the statement to be true or false, then they have DECIDED it, so it is NOT undecidable! > in an interpretation T* of T. NO, you CAN'T have provability/refutability in ONE interpretation! RATHER, things have TRUTH or FALSITY in ONE interpretation of the theory, and they AS A RESULT OF THAT wind up having/lacking provability/refutability IN THE THEORY, ITSELF, and NOT in any particular interpretation specifically! Because the whole point, classically, is that TO BE provable/refutable/decided is to have THE SAME truth-value in ALL models of the theory. If all models of the theory make the thing true, then the thing is provable in, is decided by, is a theorem of, the theory. If all of them make it false then it is again decided (and this time refuted), by the theory, and the thing's denial is a theorem of the theory. > In the absence of any such axiomatic declarations, $P$ is > in a superposed state of "neither true nor false" In every MODEL, everything gets decided. But if you are going to just look at the uninterpreted T, then P is simply undecided by it, because different models of T decide it different ways. To say about a MODEL that it leaves P undecided is to EXPAND THE CLASSICAL DEFINITION of "model" to include partial/incomplete interpretations. This is arguably perfectly reasonable if the models are non- recursive; it is not like anybody ever finished specifying one anyway. Still, the fact that these interpretations have to decide "everything, totally" in principle, is a constraint on them and you could get a very different model theory if you expanded the definition. > and consistency of T requires the existence of a > non-classical model for T in which $P \& \neg P$ > is the case. Classically, if P is a statement undecided by an incomplete theory T, then every model of T decides P anyway. Every classical model of T "forgets" that T failed to decide P. You, by contrast wind up seeking a "model" that remembers more, and assigns true to "P&~P" while assigning some 3rd value to P itself (and to ~P itself -- do they get the same 3rd value or does ~P get a 4th?). > Here T* is an axiomatic NAFL theory that, like T, > resides in the human mind and acts as the "truth-maker" > for (a model of) T. Again, the same old category-mistake, flaunted this time by ()'s. IT MATTERS whether you mean T *XOR* A MODEL OF T! T is a theory! A model of T is a model! THESE ARE FUNDAMENTALLY *DIFFERENT* breeds of cat! Models live in the realm of interpretations. Theories live in the realm of sets-of-sentences. I'm not saying that NEVER the twain shall meet but I *am* saying that the classical treatment is at least NOT CONFUSED about the difference between the two, whereas YOU ARE! Classically, INTERPRETATIONS are "truth-makers" for theories. If they in fact succeed in making all the theorems true, then they get admitted into a special relationship with the theory: they become MODELS of it. But interpretations and theories are still FUNDAMENTALLY different KINDS of things! You cannot just gloss over that! You do not necessarily have to have a different definition of "logic" or of "theory". But you definitely need a different notion of what an interpretation is. > Quantum superposition is justified by identifying >"axiomatic declarations" for the truth/falsity of $P$ > (by virtue of its provability/refutability in T*) > with "measurement" in the real world. I did this better classically. Every interpretation is a measurement. To the extent that your interpretation has not yet been completely specified, you can allege that it has some superposed state for sentences that the partial specification has not yet decided. The interesting cases are going to be interpretations that are non-recursive and that therefore in some sense CANNOT be completely specified, so you might be making a contribution in saying something about how we should talk about the places in the interpretation that haven't been "filled in" yet. > NAFL also explains and de-mystifies the phenomenon > of entanglement. Sorry, never heard of it. > NAFL severely restricts classical infinitary reasoning, > but possibly provides sufficient machinery for a > consistent axiomatization of quantum mechanics. How are we supposed to be doing quantum mechanics if we can't even do arithmetic? Isn't one of these supposed to be HARDER than the other??
From: sradhakr on 2 Aug 2005 11:05 george wrote: > sradhakr wrote: [...] > > > In particular, take a look at math.LO/0506475 > > and let me know what you think. > > I can't expand that reference. > What I can reply to is the abstract of the original article. Here is the link to the article: http://arxiv.org/abs/math.LO/0506475 This paper explains how one can do practically useful real analysis in NAFL while getting rid of the paradoxes (Zeno, Banach-Tarski, etc.), despite the non-existence of infinite sets and despite the NAFL ban on quantification over proper classes. > > > In the proposed non-Aristotelian finitary logic (NAFL), > > truths for formal propositions can exist only with respect > > to axiomatic theories, > > This is already on the verge of a category-mistake, > by classical measures. Classically, truths for formal > propositions exist only with respect to INTERPRETATIONS > (or candidate MODELS) of theories, NOT of theories > directly. Theories have theorems, not truths. NAFL truth is with respect to theories, for the following reason. The NAFL model of a theory T is a different beast from the classical model, which is "pre-existing", independent of T. Whereas the NAFL interpretation T* of a theory T instantaneously *generates* the model of T. Here is where you have to make the leap in order to understand NAFL. See below for further explanations. > > > essentially as temporary axiomatic declarations in the > > human mind. An undecidable proposition $P$ in a consistent > > NAFL theory T is > > Is basically a contradiction in terms, because in NAFL > theories, unlike in classical ones, nothing is undecidable; > rather, things that WOULD be, classically, undecidable, > are decided to have some 3rd superposed truth-value. What is meant here is that P is undecidable in T, just as in classical logic, i.e., neither P nor ~P is provable in T. The human mind then specifies the interpretation T* of T, say by adding axioms to T. T* generates the NAFL model of T; thus if P is provable/refutable in T*, then P is true/false w.r. to T, i.e., it is true/false in the classical model of T that gets generated by T*. If P is undecidable in T* as well (i.e., if P is neither provable nor refutable in T*), then P is *classically* "neither true nor false" w.r.to T and P&~P is the case in the non-classical model for T that gets generated. In this non-classical model, `P' expresses that "~P is not provable in T*" and '~P' expresses that "P is not provable in T*". So P&~P, and hence, both P and ~P, are indeed *non-classically* true in this sense. To see why such a non-classical model is required to exist (for consistency of T), I recommend that you read the above arXiv reference, in particular, Proposition 1 and its proof in the Appendix. > > > true/false with respect to T if and only if it has been > > axiomatically declared as true/false by virtue of its > > provability/refutability > > So far, so good, except that if the axioms have in fact > declared the statement to be true or false, then they have > DECIDED it, so it is NOT undecidable! > In NAFL it makes no sense to talk of absolute decidability. Here T*, the interpretation of T, has (possibly) decided P, but nevertheless, P is still undecidable in T. > > in an interpretation T* of T. > > NO, you CAN'T have provability/refutability in ONE > interpretation! RATHER, things have TRUTH or FALSITY > in ONE interpretation of the theory, and they AS A RESULT > OF THAT wind up having/lacking provability/refutability IN > THE THEORY, ITSELF, and NOT in any particular interpretation > specifically! Because the whole point, classically, is that > TO BE provable/refutable/decided is > to have THE SAME truth-value in ALL models of the theory. > If all models of the theory make the thing true, then the > thing is provable in, is decided by, is a theorem of, the > theory. If all of them make it false then it is again > decided (and this time refuted), by the theory, and the > thing's denial is a theorem of the theory. Note that the interpretation T* which, like T, resides in the human mind and can vary in time according to the free will of the individual interpreting T. Further different humans can have different interpretations of T in mind (and so can generate different models of T). Here the proposition P is (possibly) decided in T*. So if the observer takes T* as T+P, then certainly P is true in every model of T*. But here we are talking about models of T, not T*. The observer can switch to T*=T+~P at any given time, or even T*=T, and so can generate different models of T. Again, note that T* generates the NAFL models of T, which are NOT pre-existing. T* can vary in time in the observer's mind, while T is fixed. > > > In the absence of any such axiomatic declarations, $P$ is > > in a superposed state of "neither true nor false" > > In every MODEL, everything gets decided. > But if you are going to just look at the uninterpreted > T, then P is simply undecided by it, because different > models of T decide it different ways. To say about > a MODEL that it leaves P undecided is to EXPAND > THE CLASSICAL DEFINITION of "model" to include > partial/incomplete interpretations. This is arguably > perfectly reasonable if the models are non- > recursive; it is not like anybody ever finished specifying > one anyway. Still, the fact that these interpretations have > to decide "everything, totally" in principle, is a constraint > on them and you could get a very different model theory if > you expanded the definition. Yes, the NAFL model theory is very different, as I have explained above. In the NAFL model of T (generated by T*), only the decidable propositions of T* get decided; the remaining propositions (undecidable in T*) are in a superposed state of "neither true nor false". > > > and consistency of T requires the existence of a > > non-classical model for T in which $P \& \neg P$ > > is the case. > > Classically, if P is a statement undecided by an incomplete > theory T, then every model of T decides P anyway. Every > classical model of T "forgets" that T failed to decide P. You, by > contrast wind up seeking a "model" that remembers > more, and assigns true to "P&~P" while assigning some 3rd > value to P itself (and to ~P itself -- do they get the same > 3rd value or does ~P get a 4th?). Here note the crucial difference in the NAFL model theory noted above. If P is undecidable in T and further, if, say, the observer takes T*=T, then a non-classical model of T gets generated in which P&~P is the case. The *non-classical* interpretation of P&~P was explained above. In other words, P, ~P and P&~P are *non-classically* true. If you look at P in the *classical* sense in this non-classical model, then it is "neither true nor false". This seems a bit confusing, so consider the following Schrodinger cat example. Say T is the theory of quantum mechanics, and say, `P' denotes that `the cat is alive'. Here neither P nor ~P is provable in T. So upon opening the box and finding the cat to be alive, say, the observer could take T*=T+P (i.e., when he `measures' the cat to be alive in the real world, he also simultaneously *declares* it to be alive). But prior to opening the box, the observer does not know the status of the cat and so, takes T*=T, say. Then prior to opening the box, P&~P is the case in a non-classical model for T. Here `P' denotes that ~P is not provable in T*, which means that observer has not declared (measured) the cat to be dead. Similarly `~P' in the non-classical model denotes that the observer has not declared (measured) the cat to be alive. So P is *classically* neither true nor false in the non-classical model, i.e., the cat is not in either of the classical 'alive' or 'dead' states. > > > Here T* is an axiomatic NAFL theory that, like T, > > resides in the human mind and acts as the "truth-maker" > > for (a model of) T. > > Again, the same old category-mistake, flaunted this time > by ()'s. IT MATTERS whether you mean T *XOR* A MODEL OF > T! T is a theory! A model of T is a model! THESE ARE > FUNDAMENTALLY *DIFFERENT* breeds of cat! Models live > in the realm of interpretations. Theories live in the > realm of sets-of-sentences. I'm not saying that NEVER > the twain shall meet but I *am* saying that the classical > treatment is at least NOT CONFUSED about the difference > between the two, whereas YOU ARE! I do hope my explanations above have clarified matters for you. > > Classically, INTERPRETATIONS are "truth-makers" for theories. > If they in fact succeed in making all the theorems true, > then they get admitted into a special > relationship with the theory: they become MODELS of it. > But interpretations and theories are still FUNDAMENTALLY > different KINDS of things! You cannot just gloss over > that! You do not necessarily have to have a different > definition of "logic" or of "theory". But you definitely > need a different notion of what an interpretation is. > > > Quantum superposition is justified by identifying > >"axiomatic declarations" for the truth/falsity of $P$ > > (by virtue of its provability/refutability in T*) > > with "measurement" in the real world. > > I did this better classically. Every interpretation is > a measurement. To the extent that your interpretation > has not yet been completely specified, you can allege > that it has some superposed state for sentences that the > partial specification has not yet decided. The interesting > cases are going to be interpretations that are non-recursive > and that therefore in some sense CANNOT be completely > specified, so you might be making a contribution in saying > something about how we should talk about the places in > the interpretation that haven't been "filled in" yet. > Precisely. The human mind may not be in a position to specify truth values for infinitely many propositions, in which case those that are not specified have to be in a superposed state. This requirement completely changes the notion of "consistency" and "recursive" itself, as explained in my paper. > > NAFL also explains and de-mystifies the phenomenon > > of entanglement. > > Sorry, never heard of it. It is explained in the references [1] and [4] of the above arXiv paper (these refs. are also available on the web). > > > NAFL severely restricts classical infinitary reasoning, > > but possibly provides sufficient machinery for a > > consistent axiomatization of quantum mechanics. > > How are we supposed to be doing quantum mechanics if we > can't even do arithmetic? Isn't one of these supposed > to be HARDER than the other?? Actually we CAN do Peano Arithmetic (PA) in NAFL; one doesn't need infinite sets to justify PA. The question is, can we manage to axiomatize quantum mechanics with the NAFL version of real analysis outlined in the above arXiv paper. I believe it can be done, but it's going to be very hard. Regards, R. Srinivasan
From: malbrain on 2 Aug 2005 13:11 sradhakr wrote: > george wrote: > > sradhakr wrote: > [...] > > > > > In particular, take a look at math.LO/0506475 > > > and let me know what you think. > > > > I can't expand that reference. > > What I can reply to is the abstract of the original article. > > Here is the link to the article: > http://arxiv.org/abs/math.LO/0506475 > This paper explains how one can do practically useful real analysis in > NAFL while getting rid of the paradoxes (Zeno, Banach-Tarski, etc.), > despite the non-existence of infinite sets and despite the NAFL ban on > quantification over proper classes. After reading your paper, I concur that it is an excellent PROOF of the LAW OF TRANSFORMATION: The item under consideration, similar to "if a tree falls in the forest...", or a "sound-check" is a paradox from philosophy and is subject to the laws of transformation: a) unity and struggle of opposites; b) negation of the negation; and c) quantitative change leads to qualitative change. That's my VOTE. karl m
From: malbrain on 2 Aug 2005 15:07
sradhakr wrote: > george wrote: > > sradhakr wrote: > [...] > > > > > In particular, take a look at math.LO/0506475 > > > and let me know what you think. > > > > I can't expand that reference. > > What I can reply to is the abstract of the original article. > > Here is the link to the article: > http://arxiv.org/abs/math.LO/0506475 > This paper explains how one can do practically useful real analysis in > NAFL while getting rid of the paradoxes (Zeno, Banach-Tarski, etc.), > despite the non-existence of infinite sets and despite the NAFL ban on > quantification over proper classes. > > > > > > In the proposed non-Aristotelian finitary logic (NAFL), > > > truths for formal propositions can exist only with respect > > > to axiomatic theories, > > > > This is already on the verge of a category-mistake, > > by classical measures. Classically, truths for formal > > propositions exist only with respect to INTERPRETATIONS > > (or candidate MODELS) of theories, NOT of theories > > directly. Theories have theorems, not truths. > > NAFL truth is with respect to theories, for the following reason. The > NAFL model of a theory T is a different beast from the classical model, > which is "pre-existing", independent of T. Whereas the NAFL > interpretation T* of a theory T instantaneously *generates* the model > of T. Here is where you have to make the leap in order to understand > NAFL. See below for further explanations. > > > > > > essentially as temporary axiomatic declarations in the > > > human mind. An undecidable proposition $P$ in a consistent > > > NAFL theory T is > > > > Is basically a contradiction in terms, because in NAFL > > theories, unlike in classical ones, nothing is undecidable; > > rather, things that WOULD be, classically, undecidable, > > are decided to have some 3rd superposed truth-value. > > What is meant here is that P is undecidable in T, just as in classical > logic, i.e., neither P nor ~P is provable in T. The unity in struggle of opposites. > The human mind then > specifies the interpretation T* of T, say by adding axioms to T. T* Quantitative change leads to qualitative change. > generates the NAFL model of T; thus if P is provable/refutable in T*, > then P is true/false w.r. to T, i.e., it is true/false in the classical > model of T that gets generated by T*. If P is undecidable in T* as well > (i.e., if P is neither provable nor refutable in T*), then P is > *classically* "neither true nor false" w.r.to T and P&~P is the case in > the non-classical model for T that gets generated. The negation of the negation. > In this > non-classical model, `P' expresses that "~P is not provable in T*" and > '~P' expresses that "P is not provable in T*". So P&~P, and hence, both > P and ~P, are indeed *non-classically* true in this sense. To see why > such a non-classical model is required to exist (for consistency of T), > I recommend that you read the above arXiv reference, in particular, > Proposition 1 and its proof in the Appendix. Your work is exactly what I was looking for in M. Blum/D. Wagner's concrete mathematics course at Cal. Thanks, karl m |