details re Z-R |- PA consistent
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From: Transfer Principle on 12 Aug 2010 00:16 On Aug 9, 8:05 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Aug 4, 9:45 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > ZF Axiom of Infinity: > > Ex (0ex & Ay (yex -> yu{y}ex)) > > The dual of 0 is V, but then one has to wonder what the > > dual of yu{y} is supposed to be. For that matter, we > > wonder what the dual of {0} is. Since {0} is a set such > > that the only element of {0} is 0, its dual ought to be > > a set such that it is a structure of only V. > Perhaps the non-cantorian sets -- there is more than one, but they > might fit into the proper gap? Let me think about this for a moment. In this theory, V is the simplest object. It is a universal set in that every set, including V itself, is a structure of it. So what would the second simplest object be? It would be a set of which every set _except_one_ is a structure of it. So we take V and remove one of its structures, but which one? We know that every set is a structure of V, but only one of those structures has a name so far -- V itself! Thus what we want is an object such that every set other than V is a structure of it: Ex (Ay (ysx <-> ~y=V)) Such an object x cannot equal V because VsV yet ~Vsx, so x and V are distinct via Extensionality. For lack of a better name, let's call this object V_1. Every object other than V is a structure of V_1 -- and notice that this includes V_1 itself. V_1 is also a structure of V as well, since every set is a structure of V. So this raises the question, can there exist an object other than V and V_1 such that V_1 is a structure of it? If we had Trichotomy, it would be immediate that there isn't. But we don't. At this point I am stuck. Let me think about this for a while.
From: Transfer Principle on 12 Aug 2010 00:37 On Aug 7, 7:49 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Aug 6, 9:14 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > I've been thinking about how to come up with an axiom that > > states the existence of _TO's_ infinitesimals. > > And we need a theory other than ZF, and an axiom other > > than the ultrafilter axiom, to obtain TO's infinitesimals. > So you are looking for the Magic Bullet? That entity so small that any > finite number of them strung together is infinitesimal but infinitely > many together have a finite size? The solution to "since the sum from > 1 to n of 1/n is 1, take the limit as n goes to infinity"? A uniform > distribution for the natural numbers? > I don't know how to give you a formalization, but I have thought of a > real world application: > Let's call our magic bullet M and we know that an omega sequence of M > stuck together has size 1 (sum for i = 1 to infinity of M yields 1) . In thinking about this, I just realized something here. What does it mean for infinitesimals like M to be "stuck together"? Well, Jeffries gave us a Join Axiom, so perhaps we can Join all these infinitesimals together. To recall, Jeffries's Join Axiom is: > Axy (Ez (xsz & ysz)) But here's the problem: z is supposed to be the Join of x and y, but notice that since every set is a structure of the universal set V, z=V satisfies this formula for every set x and y! Thus, what is there stopping us from declaring that the Join of any two sets x and y is V, even if x and y are both infinitesimals like M? Of course, what we really want to say is that the join of x and y is the "smallest" set z such that xsz and ysz, but unless we know what "smallest" means, what is there stopping us from declaring that V is the "smallest"? What this does show, unfortunately, is that we have a long way to go in developing this theory before we can even consider infinitesimals like M, as much as I hate to admit it. Notice that with, say, Pairing in ZFC, {a,b} isn't merely a set containing the elements a and b, but it's the _smallest_ such set in that it contains _only_ a and b. I mention earlier that the Pairing Axiom is usually stated to give this set directly, or by applying Separation Schema to a set containing a and b. > Since M is positive, an omega+1 sequence of M stuck together has size > (infinitesimally) larger than 1, but if we take the bullet at omega+1 > and move it around to the beginning, we get an omega sequence which > has size 1. Further, if we take an omega+omega (which has size 2) and > interweave them we get an omega sequence (size 1). > This one-dimensional Banach-Tarski-like behavior seems to me to > resemble Wall Street economics where options and derivatives trading > can create or destroy fortunes from a small amount of actual capital. But TO, the poster whose infinitesimals we are trying to represent, is opposed to this B-T-like behavior. If there is a sequence of Joined infinitesimals with size 1, then TO wants them to remain of unit size after reordering. > Maybe your theory can somehow ban non-initial infinite ordinals? This appears to be necessary, since TO opposes to them for the reasons given in this post.
From: Transfer Principle on 12 Aug 2010 00:55 On Aug 10, 8:52 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Aug 6, 10:51 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > By "raise their reputations," I mean convince the mainstream > > to stop using five-letter insults against them. > > For example, right here in this thread, MoeBlee refers to RF > > and TO as "cranks." Many sci.math finitists, especially > > Herc, HdB, and WM, are regularly so insulted. I want to > > make it so that these finitists are [not] insulted as much, and the > > only way to do so is to "raise their reputations." > By the way, what you just wrote is tautological anyway. You say that > by "raising reputations" you mean for them not to be insulted (as > much). Then you say that only way for them not to be insulted (as > much) is to raise their reputations. Yeah, duh, in other words, the > only way for them to be not insulted as much is for them to be not > insulted as much. > Also, just to be yet again clear, people aren't ordinarily called > 'cranks' merely for being finitists or ultra-finitists. Yes, this circularity is inevitable. Posters like MoeBlee say that they don't call Herc, HdB, and WM "cranks" because they're finitists -- they call them "cranks" because they're "cranks." And that turns into another dead-end argument. The mathematician Y-V has been frequently mentioned as a counterexample to my claim that finitists are called "cranks." So, rather than ask why Herc, HdB, WM are called "cranks" and not Y-V (since no satisfactory answer can ever be given), I decide that I will take _advantage_ of this fact that Y-V is a finitist who isn't called a "crank." To take advantage of this, when a sci.math finitist is asked to give a coherent definition (as the inability to give such a definition is one often-cited reason for assigning a "crank" label), I can jump in and give a definition in terms of Y-V. Say a sci.math states that a finite natural number is one which has a real-world physical application. I can then recast this definition as, "n is a finite natural number iff when asked 'Is n a natural number?' at his birth, Y-V will answer 'yes' sometime prior to his death." Then this will become a non-"crank" definition since it's given in terms of the non-"crank" Y-V. Similarly, one can define infinitesimal as, "x is an infinitesimal iff Robinson (or some other known non-"crank" infinitesimalist) considers x to be infinitesimal." And so on. The trick is for me not to complain that certain posters are treated differently, but rather to take full advantage of this inequity.
From: Transfer Principle on 12 Aug 2010 01:01 On Aug 9, 5:12 pm, "Ross A. Finlayson" <ross.finlay...(a)gmail.com> wrote: > On Aug 6, 9:14 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Interestingly enough, in another thread, there is discussion > > of a system of "infinitesimals" where the symbol "z" is an > > infinitesimal, but there is no standard 0. Perhaps the > > infinitesimals of TO will turn out the same way. (For that > > matter, does TO ever discuss negative values?) > Now, these infinitesimals surely are exactly the same as differential > patches' extent in the dimension of analysis, where the hyperreals as > being infinitesimals surrounding a real number in a dense point cloud > have no analytical character extra the finite, modeling real > analysis. Now, we all grew up with the instruction of the use of > limits in methods of exhaustion like the integral calculus (also known > as infinitesimal analysis, real analysis, continuum analysis), and > they reasonably explain and meet expectations in the results of > analysis matching those of areal geometry, about maintaining ratio in > the limit. Claiming hyperreals in real analysis is basically a non- > operation. MoeBlee should read that last line. Hyperreals are basically a "non-operation," so MoeBlee can't use hyperreals to satisfy the infinitesimals as described by RF. > For the conscientious mathematician, to work from these > principles of the existence of these iota-values to their meaning as > elements of the complete ordered field (and not necessarily without > modification to as elements of Argand C), then it may well be that > there are indeed true features of these numbers, that are obviously > outside the standard which speaks not to it, in the polydimensional > (where the point is zero-dimensional but in all dimensions at once). Here RF mentions "iota-values." Notice that RF's iota is incompatible with TO's infinitesimals. This is because for RF iota is the smallest positive infinitesimal (since it is supposed to be the successor to 0 where < is a wellordering), but TO has no smallest infinitesimal. I might have more to say on RF's post later.
From: Jesse F. Hughes on 12 Aug 2010 01:26
Transfer Principle <lwalke3(a)lausd.net> writes: > Yes, this circularity is inevitable. Posters like MoeBlee say that > they don't call Herc, HdB, and WM "cranks" because they're > finitists -- they call them "cranks" because they're "cranks." And > that turns into another dead-end argument. No, others have explained the features that make them cranks. As it happens, belief in finitism is not one of these features. An inability to understand (or at least convey) basic mathematical reasoning, on the other hand, while still claiming great mathematical insight necessary for a theoretical revolution *is* pretty good indication that one is a crank. Herc, WM and (often?) Han exhibit this symptom. There is no circularity there. -- Jesse F. Hughes "You shouldn't hate Mother Mathematics." -- James S. Harris |