details re Z-R |- PA consistent
in [Math]
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From: MoeBlee on 12 Aug 2010 13:46 On Aug 11, 11:55 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Aug 10, 8:52 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > 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. No, the quote marks around the thrid occurrence of 'cranks' is incorrect. I don't say that the word 'crank' applies to certain people because they are 'cranks'. That would be ridiculous, since posters are not WORDS. Rather I say that the word 'crank' applies to certain people because they are indeed cranks. That is not circular. No more than saying that the word 'snow' applies to the stuff in my hand because the stuff in my hand is indeed snow. I call them 'cranks' for behaving in certain ways that I've several times listed (I've posted such lists several times; I'm not gonna bother to do it again). So I've said just what kinds of posting behaviors I take as being crank behavior. Then I say that certain people are cranks based on the fact that they behave in some number of ways as mentioned. That is, the word 'crank' applies to them because they post in certain ways, which is to say, the word 'crank' applies to them because they fit the definition of the word 'crank'. Not circular. You're being intellectually dishonest by pretending that I haven't listed such behavior and as you avoid that point to instead claim falsely that I merely say people are cranks in some question begging way. > The mathematician Y-V has been frequently mentioned as a > counterexample to my claim that finitists are called "cranks." Not by me, just for the record. I've not posted any opinion about him. > 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. So what? His definition is not automatically what some crank has in mind. > 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." Okay. But just be clear that this is not a formal definition. > Then this will become a non-"crank" > definition since it's given in terms of the non-"crank" Y-V. That doesn't even follow. Just because a person is not a crank doesn't entail that everything that person says is correct, acceptable, coherent, or even un-crank-like. It can be the case that a person is not so hopeless as to be a crank but still display certain crankish behaviors. (For the record, I'm not claiming anything about Yessin- Volpin specifically.) > 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. And what would be the point of that? What's the point of giving a quite NON-mathematical definition like that? Surely, you don't think "x is a blah blah iff so and so considers x to be a blah blah" is any kind of mathematical definition? And, as to 'infinitesimal', there are quite adequate formal and MATHEMATICAL definitions of 'infinitesimal' even in the langauge of Z set theory, and then proof that there exist systems with infinitesimals even in ZC. > The trick is for me not to complain that certain posters are > treated differently, but rather to take full advantage of this > inequity. Did you witness some horrible crime when you were a very young child that has caused you this bizarre preoccupation you have? MoeBlee
From: MoeBlee on 12 Aug 2010 13:48 On Aug 12, 12:01 am, Transfer Principle <lwal...(a)lausd.net> wrote: > Hyperreals are basically a > "non-operation," I don't know what you mean by that. > so MoeBlee can't use hyperreals to satisfy > the infinitesimals as described by RF. I have no intention of using anything in the world to satisfy anything at all described by Ross Finlayson. MoeBlee
From: FredJeffries on 12 Aug 2010 16:00 On Aug 11, 9:37 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > 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"? Silly of me -- I am completely wrong here. The magic bullets do not form an omega sequence. In the first place, an anti-cantorian would not use transfinite ordinals which were invented/discovered by Cantor. (One of the few things that is somewhat clear from anti-cantorian writings is the disdain for the cardinal/ordinal distinction.) But also, from the "sum from 1 to n of 1/n is 1, take the limit as n goes to infinity" example it is clear that omega is not an appropriate order type because we need a last bullet and a next to last and a next to next to last and ... (Oh no! I used an ellipsis which implies order type omega but that wrong here!) Anyhow, we are not just adding terms onto (the end of) a sequence. We are holistically altering the sequence at each iteration. So we need a new order type altogether which a) has infinitely many elements b) each element except the first has an immediate successor and c) each element except the last has an immediate predecessor I apologize for the misunderstanding. This new order type would probably take care of my re-ordering problem also.
From: Ross A. Finlayson on 12 Aug 2010 18:59 On Aug 12, 10:48 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Aug 12, 12:01 am, Transfer Principle <lwal...(a)lausd.net> wrote: > > > Hyperreals are basically a > > "non-operation," > > I don't know what you mean by that. > > > so MoeBlee can't use hyperreals to satisfy > > the infinitesimals as described by RF. > > I have no intention of using anything in the world to satisfy anything > at all described by Ross Finlayson. > > MoeBlee What it means is that there aren't results in nonstandard analysis with hyperreals unavailable to standard analysis with reals, and vice versa, positing hyperreals adds no analytical structure, it's a non- event or no-op. These infinitesimals as so described or as some analysts might know them better the raw/bare/naked differentials, yeah we can point to Newton or Leibniz or further back towards the originators of general methods of exhaustion for pretty much the same description, the goal of modern mathematics being to make their use rigorous, formally. So, that kind of stuff, basically historical with the context of the development of the integral calculus in mathematics, yeah Moe maybe you'd care to avoid that kind of conversation, where I say so too. I just pick up the arguments that the partition of the unit interval into infinitely many partitions each with an extent no different than the other is a mathematical object, that it would have various properties as deduced from its structure, and that then when these elements are mapped to the naturals in their natural order this resulting function is a bijection between the naturals and unit interval of reals, in the context of each of the four or five different arguments for the uncountability of the reals. This satisfies for those results, for example the anti-diagonal argument, like no other function, that is, in a way without the same conclusion. They're countable. This function also has interesting and what might seem surprising results with regards to features of uniform probability distributions over infinite spaces. So, in terms of modern mathematics as a foundation for analysis, thanks Moe they didn't need your help to figure that out, what were called infinitesimals with the usual meaning for the hundreds of years between Newton and Weierstrass are modeled by the standard. Of course, in terms of what constructions I describe that aren't necessarily consistent with the set theory underpinning measure theory underpinning real analysis, with real infinitesimals, and how that as the image of a function from the natural integers these real infinitesimals (or iota-values) illustrate the countability of the unit interval of reals, then, those are mine. Also the function is standardly modeled by real functions, so, nope, don't need your help. Thanks for nothing, though! It's appreciated. Warm regards, Ross Finlayson
From: FredJeffries on 13 Aug 2010 04:55
On Aug 11, 9:37 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Well, Jeffries gave us a Join Axiom I merely proposed a possible path of investigation as an alternative to the narrow trichotomy property > 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"? You want it to satisfy a universal property http://en.wikipedia.org/wiki/Universal_property thusly: (xsz and ysz) and (if (xsw and ysw) then zsw) Exercise: Draw your commutative diagrams. |