details re Z-R |- PA consistent
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From: FredJeffries on 7 Aug 2010 10:49 On Aug 6, 9:14 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Yes, I'm already aware of the construction of Robinson's > hyperreals from ultrafilters using AC. > > But the infinitesimals as described by RF and TO can't > so be constructed. TO's infinitesimals directly contradict > ZFC, since, for example, to TO the set size of the set N > of natural numbers is invertible as an infinitesimal, but > one can't invert the cardinalities of ZFC to obtain any > sort of infinitesimal hyperreals. Furthermore, TO has > _expressly_ rejected hyperreals (and surreals) as > equivalent to his infinitesimals. > > So let me correct what I 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) . 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. By the way, what happens if you arrange the bullets in the order type of the rational numbers? Maybe your theory can somehow ban non-initial infinite ordinals? Or maybe instead of defining an infinite sum as the limit of the partial sums you could have a different definition for an infinite sum? Or perhaps the bullets could have a quantum-fluctuating size so that you don't really know where one stops and the next one begins and the fluctuations of infinitely many all working together is what gives a finite size? Your system seems neither to be Archimedean (since there are infinitesimals) nor strictly non-Archimedean (since the sum of infinitely many of an infinitesimal yields a finite value). Perhaps "anti-Archimedean" is not yet taken.
From: FredJeffries on 9 Aug 2010 11:05 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?
From: Ross A. Finlayson on 9 Aug 2010 19:41 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: > > > > > Yes, I'm already aware of the construction of Robinson's > > hyperreals from ultrafilters using AC. > > > But the infinitesimals as described by RF and TO can't > > so be constructed. TO's infinitesimals directly contradict > > ZFC, since, for example, to TO the set size of the set N > > of natural numbers is invertible as an infinitesimal, but > > one can't invert the cardinalities of ZFC to obtain any > > sort of infinitesimal hyperreals. Furthermore, TO has > > _expressly_ rejected hyperreals (and surreals) as > > equivalent to his infinitesimals. > > > So let me correct what I 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? > EF is its CDF. > I don't know how to give you a formalization, but I have thought of a > real world application: > You can just formalize it as a non-real function modeled by real functions (for example n/d for d->oo with n-d) just like some other non-real functions like the impulse or unit step functions are modeled, preserving their analytical properties. > 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) . > > 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). > It's more like the unit is divided into omega (or what other ordinal is used as its prototype) many partitions, and scalar relations between those would be preserved. > 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. > > By the way, what happens if you arrange the bullets in the order type > of the rational numbers? > Good question. > Maybe your theory can somehow ban non-initial infinite ordinals? > No, not necessary, where that they would follow under the behavior of their initial ordinals (in asymptotics) would follow from induction through the non-initial ordinals. > Or maybe instead of defining an infinite sum as the limit of the > partial sums you could have a different definition for an infinite > sum? > Geometrically, the sum equals the limit. The limit is the sum. > Or perhaps the bullets could have a quantum-fluctuating size so that > you don't really know where one stops and the next one begins and the > fluctuations of infinitely many all working together is what gives a > finite size? > That sounds reasonable. > Your system seems neither to be Archimedean (since there are > infinitesimals) nor strictly non-Archimedean (since the sum of > infinitely many of an infinitesimal yields a finite value). Perhaps > "anti-Archimedean" is not yet taken. There are multiple meanings of Archimedean, vis-a-vis existence of complete infinities, here this function could be modeled by real functions which can be founded in real analysis which the standard defers to as founded in the finite (in methods of exhaustion). Warm regards, Ross Finlayson
From: Ross A. Finlayson on 9 Aug 2010 20:12 On Aug 6, 9:14 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Aug 5, 9:54 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > > I've been thinking about how to come up with an axiom that > > > states the existence of infinitesimals. > > An axiom added to what other axioms? > > We can prove the existence of a number system with infinitesimals from > > ZF by adding "every filter can be extended to an ultrafilter", which > > is entailed by the axiom of choice but is weaker than the axiom of > > choice. > > It's not clear to me that you understand that non-standard analysis > > can be done in ZFC and that, indeed, the non-standard analysis > > introduced by Robinson was within ZFC. > > Yes, I'm already aware of the construction of Robinson's > hyperreals from ultrafilters using AC. > > But the infinitesimals as described by RF and TO can't > so be constructed. TO's infinitesimals directly contradict > ZFC, since, for example, to TO the set size of the set N > of natural numbers is invertible as an infinitesimal, but > one can't invert the cardinalities of ZFC to obtain any > sort of infinitesimal hyperreals. Furthermore, TO has > _expressly_ rejected hyperreals (and surreals) as > equivalent to his infinitesimals. > > So let me correct what I 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. > > One thing that I was wondering, though, is how to deal with > zero and negative values. According to RF, we want to > divide a real number into smaller and smaller parts (or > structures) corresponding to smaller infinitesimals, but at > how do we obtain zero or negative values? > > 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?) Hi, Thanks that is an interesting direction for exposition, about what definitions of infinitesimals would match our expectations that a partition of the unit interval into them would have them being a la Vitali's (and Banach-Tarski's) constant 'c', that summing those would yield a finite and fixed value (where that this is impossible is why it is claimed there are non-measurable sets.) 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. (The fluents and fluxions of Newton are instead recursively analytical as defining differential elements.) With the idea that the infinitesimals from zero through one, these iota-values, have basically a constant real value in difference, gets into that then these dots drawn to the line are the same numbers as all the real numbers from zero through one as the complete ordered field, for example where Hardy has real numbers as interchangeably geometric. The properties of the complete ordered field have that between any two there are infinitely many, that doesn't seem to allow that they progress in some order (their natural well-order for that matter on well-ordering segments of reals), unless the topological properties of that sequence maintain the topological properties of the elements of the complete ordered field. Then, it seems more a matter for the conscientious mathematician as to why they are than why they aren't. Now, besides things like properties of infinitesimals posited within the real number continuum, there are properties of the real number continuum above yet still within the structure, for example the projectively extended real numbers with the "points" at infinity, these properties could be seen as representative of features of symmetry about the missing middle of the line between zero and infinity, conveniently as well a natural compactification with the infinite integers having (at least) an infinite integer. Then, to talk about these infinite values as scalars is just in process, for example along the lines of Sergeyev's, or in ordinals about use of infinite ordinals and comparing them and those of the same polynomial degree as ratios (between zero and one), where ordinals are polynomials (in omega, though ordinals). In the standard way, the real numbers are the complete ordered field (of course not counting zero, except modern methods with for example meromorphic projection remove that singularity, or rather that is so modeled by real functions). Then as to what the course is along them, that leads to a consideration of "how" the real numbers are a complete ordered field, compared to their perhaps "standard" construction as Dedekind/Cauchy/Eudoxus, or continued fractions, how they are in a line. It is like defining a line as all of (or two of) the points on it, compared to defining a point as all the lines through it. (I think geometry's primitives are points and spaces instead of points and lines with a space-filling spiral continuum from the origin that via several considerations is minimizes need for expression, is expressive.) 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). In terms of axioms generally I promote an axiomless system of natural deduction. I'm a Platonist I think the objects have their properties. Again Larry your interesting article covers various ideas, most of which you describe are simply enough accommodated in the standard. Warm regards, Ross Finlayson
From: MoeBlee on 10 Aug 2010 11:52
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." One way to put a damper on certain people being called 'a crank' is to get those people to stop being cranks. Though, that's seems quite difficult to do. 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. MoeBlee |