Cantor's Diagonal?
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From: Transfer Principle on 9 Feb 2010 23:52 On Feb 8, 8:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Transfer Principle wrote: > > I. Standard theorists think of mathematics in terms of the > > objects that they want to _exist_ in their own conception > > of the mathematical universe. > > "Cranks," on the other hand, think of mathematics in terms > > of the objects that they _don't_ want to exist. > What about those posters who subscribe to none of those "thinking"? > Are they "crank", or "standard"? The two options that I gave were not intended to be either mutually exclusive or mutually exhaustive. They are intended merely as generalizations of the two sides of the issue. To illustrate what I mean, let's look at an old thread from last year, one in which Nam Nguyen himself was a participant (via Google archive): http://groups.google.com/group/sci.logic/browse_thread/thread/7634dadf33044697/dba25c60398020dd?lnk=gst&q=cantor+crank# In this thread, the standard theorist MoeBlee wrote: "Why would the point of such exercises merely be to fiddle with axioms irrespective of what those axioms PROVE, especially when we're talking about comparisons with a foundational theory such as Z that axiomatizes calculus?" Typical of the standard theorists, MoeBlee desires a theory in which calculus can be axiomatized. It matters little to such standard theorists what "extra" can be axiomatized in theory, however "useless" such objects may be (objects such as the set P(P(P(R)))). As long as calculus is axiomatized, that's what mainly matters in a foundational theory that's adequate for standard theorists. An axiomatization for the calculus is MoeBlee's priority in this post. On the other hand, the following WM quote appears in the thread to which I linked above: > "Axiom Of Potential Infinity: For every natural number there is a set > that contains this number together with all smaller natural numbers, > and for every set of natural numbers there is a natural number that > is larger than every number of the set." In this post, WM has expressed that he wants potentially infinite sets to exist, but for no _actually_ infinite sets to exist. It matters little to "cranks" like WM that without actually infinite sets, calculus is most likely impossible. Avoiding actually infinite sets is WM's priority in this post. The priorities of MoeBlee and WM in these quotes are typical of the priorities of the standard theorists and the "cranks," respectively. The standard theorist MoeBlee says to include calculus in the foundational theory (no matter what else gets included). The "crank" WM says to exclude actually infinite sets in the foundational theory (no matter what what else gets excluded). Standard theorists think in terms of what to _in_clude. So-called "cranks" think in terms of what to _ex_clude. Hence my generalization above. > What about those posters who subscribe to none of those "thinking"? > Are they "crank", or "standard"? It's possible (indeed probable) that Nguyen intends to include himself as one of "those posters who subscribe to none of those" generalizations. In this case, I would use a different criterion in order to determine whether Nam Nguyen is "crank" or standard. Consider the thread in the link above. In this thread, Nguyen used mostly symbolic language, which according to criterion III above, suggests that he is standard. But then again, the argument that most of the standard theorists was making is that Nguyen actually was thinking in terms of _excluding_ something from standard logic, in this case, excluding the proof of "Axy (x+y=0)" from a theory with axiom "Axy (x=y)" yet no axioms containing the symbol "+" from the language of the theory, was enough to consider Nguyen to be a "crank" -- hence the existence of that long discussion with him in the link. Of course, Nguyen hasn't argued in a long time that "Axy (x+y=0)" isn't derivable from the lone axiom "Axy (x=y)" in a theory with "+" in the language. So I have no idea whether the standard theorists with whom he argued in that thread still consider him to be a "crank" anymore or not. Remember that theose standard theorists are the ones who came up with the "crank" label in the first place, so they are the ultimate judges as to whether Nguyen is a "crank" or not.
From: Ross A. Finlayson on 9 Feb 2010 23:57 On Feb 3, 9:04 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <c3c98452-4379-412c-8eda-26b5e5546...(a)k2g2000pro.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > Yes it is well known that the binary case requires refinement and > > there is one anti-diagonal, of the matrix expansion of the elements. > > If one goes by the "infinite matrix" model of a proof, it is quite easy > to show that there are at least as many anti-diagonals as there are rows > in that infinite matrix, one for each natural number (including zero) > offset before starting to anti-diagonalize. The equivalency function, though, that works in general models of the reals, and variously as I show in the proof machinery of results about uncountability of the reals not similarly to any other function you describe, by proof, still sees that you are selecting a different by the symmetry of the diagonal, with respect to EF's about the other. Regards, Ross Finlayson
From: Transfer Principle on 10 Feb 2010 00:03 On Feb 8, 1:33 am, "Peter Webb" <webbfam...(a)DIESPAMDIEoptusnet.com.au> wrote: > I don't think zuhair is a "crank" in the more general Usenet sense. He does > appear to eventually agree he was wrong, which cranks don't - they either > change the topic, make an ad-hominem attack, or do both. Standard theorists make ad-hominem attacks all the time. But I do concede that zuhair does admit that he's wrong -- indeed he did so in this very thread. > And of course your views on crankdom are very specific to logic. It's hard for me to generalize beyond mathematics and logic what traits make a "crank." Of course, Webb does mention how "cranks" rarely admit their errors, which does generalize to fields outside mathematics and logic. > One thing I disagreed on: > > II. Standard theorists believe that physics favors their > > side of the debate. Physical processes can be modeled by > > differential equations, which require the existence of a > > complete ordered field to solve. Thus, they believe that > > standard analysis is necessary for the sciences, and so > > science is impossible in the "crank" theories. > All physical science can be done using only the countable (but uncomputable) > set of computable Reals. Calculus also works fine over this subset of R. > There is no hard evidence to me that uncountable sets need exist in nature > at all. I grant that it would be somewhat arbitrary for magnetic field > strengths to only take only computable values, but for all we know they may > do so. You might call this UltraCountabilism in comparison to UltraFinitism > Indeed, if you accept that you can make precise measurements of anything at > all in nature, and you believe in QM, then in this Universe I can well order > the Reals. I do so as follows. I place an electron inside an energy well > such that its location between 0 and 1 is randomly determined by QM. The > location at which I precisely measure the electron is the lowest (first) > Real. Then I continue doing so for the second and subsequent measurements, > making sure each time I haven't listed that Real already. In this manner, I > can select a random Real from any subset of Reals, which in ZFC requires a > limitted form of choice. The random outcomes in QM provides a mechanism not > directly available inside ZF for choosing elements, and may provide a choice > function in this Universe. Interesting idea! > The point of all this being it is very dangerous to try and use the physical > world to validate set theory. I agree with Webb here -- in the end, it's a wash as to which side physics really favors, so perhaps it's best just to avoid physics in any discussion of set theory. The point that I was making is that members of both sides _do_ make physical-based arguments, not that such arguments are actually valid.
From: Virgil on 10 Feb 2010 01:34 In article <a8554fc6-0b17-44e5-88dc-4543a9338365(a)s33g2000prm.googlegroups.com>, "Ross A. Finlayson" <ross.finlayson(a)gmail.com> wrote: > On Feb 3, 9:04�pm, Virgil <Vir...(a)home.esc> wrote: > > In article > > <c3c98452-4379-412c-8eda-26b5e5546...(a)k2g2000pro.googlegroups.com>, > > �"Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > Yes it is well known that the binary case requires refinement and > > > there is one anti-diagonal, of the matrix expansion of the elements. > > > > If one goes by the "infinite matrix" model of a proof, it is quite easy > > to show that there are at least as many anti-diagonals as there are rows > > in that infinite matrix, one for each natural number (including zero) > > offset before starting to anti-diagonalize. > > The equivalency function, though, that works in general models of the > reals, and variously as I show in the proof machinery of results about > uncountability of the reals not similarly to any other function you > describe, by proof, still sees that you are selecting a different by > the symmetry of the diagonal, with respect to EF's about the other. Would you translate that into comprehensible English please.
From: Ross A. Finlayson on 10 Feb 2010 01:43
On Feb 9, 10:34 pm, Virgil <Vir...(a)home.esc> wrote: > In article > <a8554fc6-0b17-44e5-88dc-4543a9338...(a)s33g2000prm.googlegroups.com>, > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > On Feb 3, 9:04 pm, Virgil <Vir...(a)home.esc> wrote: > > > In article > > > <c3c98452-4379-412c-8eda-26b5e5546...(a)k2g2000pro.googlegroups.com>, > > > "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > > > > > Yes it is well known that the binary case requires refinement and > > > > there is one anti-diagonal, of the matrix expansion of the elements.. > > > > If one goes by the "infinite matrix" model of a proof, it is quite easy > > > to show that there are at least as many anti-diagonals as there are rows > > > in that infinite matrix, one for each natural number (including zero) > > > offset before starting to anti-diagonalize. > > > The equivalency function, though, that works in general models of the > > reals, and variously as I show in the proof machinery of results about > > uncountability of the reals not similarly to any other function you > > describe, by proof, still sees that you are selecting a different by > > the symmetry of the diagonal, with respect to EF's about the other. > > Would you translate that into comprehensible English please. The symmetry about the diagonal has the notion that there is the opposite diagonal in the matrix. Now, there is no end of the rows and columns of the infinite matrix with a corner. Yet, there is a natural representation in finite aproximation. Then, where Cauchy sequences underdefine reals for their natural properties, where it is proven that Cauchy sequences are complete in the complete ordered field, has by simple extention of projection of admittance of finite approximations that exhaust, and the resulting calculus on them that could be as well defined by the unit measure. Of course, that is generally in my polydimensional model of space, with the space-filling. Polydimensional: one. It's simple. It's because theyre perpendicular, the diagonal and opposite diagonal. Those are the axes of symmetry in the matrix. If EF went from zero to one, the entire row begins with all zeros for each initial segment of its expansion, unless you define them in terms of each standard point. All simple. Thanks, Ross Finlayson |