details re Z-R |- PA consistent
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From: Jesse F. Hughes on 16 Jul 2010 12:23 FredJeffries <fredjeffries(a)gmail.com> writes: >> idea that proper subsets can have different "sizes" (Bigulosities)? > > I addressed this issue a couple weeks ago to which you have made no > reply. Perhaps you didn't see it: > http://groups.google.com/group/sci.math/msg/c39c4f58b2b37d91 Perhaps he prefers to talk about how people like you try to suppress certain discussions rather than actually contribute to such discussions with an evil suppressor like you. -- "The Hammer is not force. It is absolute power. The Hammer is from Idea Space. That's the real world. Here is the magical realm. You are creatures in that realm, who do not quite understand. But it doesn't matter. There is a story to be told..." James S. Harris, poet.
From: Transfer Principle on 19 Jul 2010 22:45 On Jul 16, 7:01 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 15, 7:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > With Jeffries, Nguyen, and MoeBlee all questioning this axiom, > > it's best that I drop it. > I have nothing against the axiom. It's unfortunate that you are > dropping it because it's the only interesting thing that you've posted > this month. The problem is that any time that I post a theory, there are two often competing notions that I must consider. 1) Is the theory that I'm posting in any sense interesting? 2) Is the theory true to the ideas of the "crank," which it was intended to represent? And so, out of fear that someone would criticize my bringing up NFU as being too far removed from the ideas of the one whose ideas I was trying to make rigorous (in this case Herc), I was going to drop the axiom. But since Jeffries is still asking about it, I'll continue to discuss it. > http://en.wikipedia.org/wiki/New_Foundations#Models_of_NFU describes > "a fairly simple method for producing models of NFU" (which I don't > claim to understand on first reading) which concludes "If alpha is a > natural number n, we get a model of NFU which claims that the universe > is finite (it is externally infinite, of course)", which seems similar > to another possibility I should have asked you about: even if the > existence of a natural number H which is equal to {} implies that all > "greater" natural numbers are also equal to {} (and hence equal to > each other?) couldn't it be the case that H is some kind of non- > standard number, so that there are infinitely many natural numbers > none of which are equal to {}? Ironically, I started bringing up nonstandard numbers in the context of the ideas of another "crank," Tony Orlow. And I was criticized for mentioning them, since they were irrelevant to TO's ideas. (IIRC someone told me that the explanation of something being infinite inside a model and finite outside a model flew right over TO's head). In fact, this incident had discouraged me from mentioning nonstandard naturals in the context of Herc. But now that Jeffries has mentioned them, it's now OK for me to consider nonstandard numbers with Herc. I've heard of nonstandard naturals before, but only in the context of theories like IST, not NFU. Jeffries cites an interesting Wikipedia page describing nonstandard naturals in NFU. I am reluctant to consider Wikipedia pages because previous posters have criticized me for my over-reliance on Wikipedia, but now that Jeffries has cited the page, it's now OK for me to consider it. Now, to answer Jeffries's question, is this possible? Based on what the Wikipedia page says, I'll say that yes, it is possible. In that case, can such a model still describe Herc's ideas? Can Herc's I equal Jeffries's H-1, which is nonstandard? I point out that Herc never did state exactly how large his I is, so it might be remotely possible that Herc's I could be nonstandard. Then the number I would be greater than any physical counterexample that Burns or anyone else can throw at him, yet internally, the set I is still finite. On the other hand, if Herc insists that his I be a standard natural, then we can continue to look to Srinivasan for finitist tricks, similar to his D=0, in order to make the universe finite (though I'm not sure yet whether D=0 works in NFU). > Another intriguing sentence (which I also don't understand yet) I've > come across inhttp://stanford.library.usyd.edu.au/entries/settheory-alternative/#Cr... > is that "the ordinals are not well-ordered in any set model of NFU". Holmes explains this on his own webpage: "The Burali-Forti paradox of the largest ordinal: Ordinals are defined in NF as equivalence classes of well-orderings. There is a set of all ordinals, and it has the usual natural order, and this order has an order type. So far, so good. But the order type of the set of ordinals less than an ordinal a does not need to be equal to a (this is proved in the usual set theory by transfinite induction on an unstratified condition!); the order type Omega of all the ordinals is an ordinal, but it is less than the largest ordinals. The order type Omega_2 of the ordinals less than Omega is less than Omega. There is an apparent descending sequence of ordinals Omega_i suggested here, but it is not a set because (you guessed it) its definition is not stratified (fortunately!)."
From: Transfer Principle on 19 Jul 2010 22:57 On Jul 16, 7:13 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 15, 8:19 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > To repeat, Holmes uses all these definitions in NFU. In ZFC, > > none of these objects (except {{}}) are sets. Holmes's > > definitions can be found at his website: > >http://math.boisestate.edu/~holmes/holmes/nf.html > I had to dig a bit, but I eventually found Holmes's textbook > "Elementary Set Theory with a Universal Set" athttp://math.boisestate.edu/~holmes/holmes/head.pdf > where the natural numbers are defined in chapter 12. Sorry about that. Yes, I did get Holmes's definition of natural number from the second link. So I should've just posted the link directly to that textbook.
From: Transfer Principle on 19 Jul 2010 23:46 On Jul 16, 7:26 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 14, 2:42 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > the number googolplex exists, but not all the > > classical naturals less than googolplex exist. > I do not find it inconceivable that some non-peano-an system could be > of some value, perhaps drawing on (or contributing to) the notion of > Kolmogorov complexity -- as you have pointed out, numbers used in RSA > cryptography are not arrived at by starting at 0 and adding 1 > repeatedly. Interesting. As I've said before, this is WM's desiderata. So now we ask, are there explicit axioms that we can write, similar to the axioms of PA, that can represent WM's ideas? Let's start out by writing the following axioms, where "is a number" is a primitive notion: 1) 1 is a number. 2) If m and n are numbers, then m+n is a number. 3) If m and n are numbers, then mn is a number. 4) If m and n are numbers, then m^n is a number. But then this reproduces all of the standard naturals, since there's no limit as to how many times we can apply the rules. Since Jeffries mentions a measure of "complexity," perhaps we can assign complexities to numbers and then state that there is an upper bound, say C, on the complexity of a natural number. With "of complexity" primitive: 1) 1 is a number of complexity 1. 2) If m and n are numbers of complexity a and b respectively, then m+n, mn, m^n are numbers of complexity max(a,b)+1. But then we have the following dilemma: 1 is of complexity 1. 2, being 1+1, is of complexity max(1,1)+1 = 2. 3, being 2+1, is of complexity max(2,1)+1 = 3. 4, being 2+2, is of complexity max(2,2)+1 = 3. 4, being 3+1, is of complexity max(3,1)+1 = 4. So 4 would have two complexities, 3 and 4? Of course, we want the complexity of a number to represent the _shortest_ path from 1 to that number, so that the complexity of 4 ought to be 3. So instead we write: 2) If m and n are numbers of complexity a and b respectively, then m+n, mn, m^n are numbers of complexity at most max(a,b)+1. But then what's to stop us from declaring every number to have complexity 1, so that every natural exists? Sure, the axioms as written so far don't prove that every natural has complexity 1, but neither do they refute it. And I admit that this is where I'm stuck. I think back to the surreal numbers of Conway, where he can define {L|R} to be the "simplest" surreal between those in L and R, and the surreals all have a complexity called "birthday." Yet somehow, "simplest" surreal is welldefined. So there might be a trick similar to Conway for defining the WM-naturals. A few other notes of interest: I'm considering including subtraction, since WM likely wouldn't mind googol-1 to have complexity only one more than that of googol. (Otherwise, we'd have to do something like (((...(((9)10+9)10+9)10+9)...))) which might be much more complex than googol itself. Also, I might exclude exponentiation, since including it might make the numbers increase well more rapidly than WM intends them to. A few interesting sets (using addition and multiplication only): N_1 = naturals of complexity 1 = {1} N_2 = naturals of complexity at most 2 = {1,2} N_3 = naturals of complexity at most 3 = {1,2,3,4} N_4 = naturals of complexity at most 4 = {1,2,3,4,5,6,7,8,9,12,16} N_5 = naturals of complexity at most 5 = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21, 22,23,24,25,27,28,30,32,35,36,40,42,45,48,49,54,56 60,63,64,72,80,81,84,96,108,112,128,144,192,256} i.e., we define N_n recursively as: N_1 = {1} N_(n+1) = {meN | Eab (aeN_n & beN_n & (a+b=m v ab=m))}
From: Transfer Principle on 20 Jul 2010 00:15
On Jul 16, 8:01 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 14, 3:01 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Thus, how can I tell who really are the open-minded posters? > You can't. Get over it. > > Instead, I want to consider those who shut out their opponents ideas > > to be closed-minded. > You want to put people in boxes? Fine. What I want is to be able to defend certain posters, but if I defend everyone, then I'm really defending no one. So, I want to have some criteria for determining which posters to defend. Is open-mindedness is a poor criterion for deciding which posters I should defend? Then fine. But there has be _some_ way for me to decide which posters to defend. I'd love to do this without "putting people in boxes." I'd love it if I could defend certain posters and have it be seen not as "putting people in boxes," but as adhering to perfectly reasonable criteria for choosing which posters to defend, since -- as I repeat -- it is neither possible nor desirable to for me defend _everyone_. > > idea that proper subsets can have different "sizes" (Bigulosities)? > I addressed this issue a couple weeks ago to which you have made no > reply. Perhaps you didn't see it: > http://groups.google.com/group/sci.math/msg/c39c4f58b2b37d91 Believe it or not, I actually was going to give a lengthy response to this post, but then I clicked on the wrong button and deleted it instead of posted it. (The problem stems from the fact that I post on Google, and if I click "reply" and it takes me more than an hour to type my response, the message "Your session has expired, please try again later" appears instead.) But of course, all of this sounds as plausible as "my dog ate it." So instead of continuing to post excuses, let me respond to the key points in that post. I respond here rather than bump a thread that has been inactive for over two weeks. > Suppose the universe can be well ordered. We must keep TO's desiderata in mind. So the question is, does TO want a wellorderable universe? On one hand, TO is opposed to AC, and wellordering the universe certainly sounds like global Choice. On the other hand, he isn't opposed to the set R of reals being wellordered, since his H-riffics were an attempt to wellorder R. > Of course this order may have other objectionable features, For > instance, if 1 is the first element of the well order then any > infinite set containing 1 will be larger than any infinite set not > containing 1. And I assume that TO would object as well, since he'd want: X = {1,4,7,10,13,16,19,...} to have Bigulosity tav/3 Y = {2,4,6,8,10,12,14,...} to have Bigulosity tav/2 But at this point, I wonder whether there might be a trick to using a wellorder of the universe to define "<=" anyway, in a more complicated than Jeffries has proposed, to give a definition that satisfies TO's desiderata. At this point, we might go back and try to define a "strong" ("Tconsistent") bijection in terms of the wellorder, rather than "<=" directly. I tried to come up with some axioms earlier, but failed, since this will not be simple to accomplish (and had my Google session "expire" in the process). Going back to the wellorder issue, there might be some other internal-vs.-external trick where outside the model, we can see that the universe is wellordered, but inside the model, we can't see the wellordering directly, so that TO's ~AC isn't violated. But then again, depending on how we ultimately define "<=" it might be possible to recover the wellorder of the universe anyhow. (P.S., in the process of posting this, my session "expired" again, but hopefully this posted correctly this time.) |