details re Z-R |- PA consistent
in [Math]
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From: Transfer Principle on 20 Jul 2010 00:31 On Jul 16, 8:07 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 14, 3:39 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > And so I'm going to continue to have discussions about alternatives > > to ZFC, no matter how dogmatic the posters discussing it become. > Hey! George Greene made six posts commenting on the Ed Nelson pager > that you refer to so much. Why haven't you responded and discussed it? > (After all, you don't find many people more dogmatic than George). With so many posts in so many threads, it's easy for me to lose track of them all. Once again, I'd rather post here than make a fortnight bump. Yes, Greene certainly ripped apart Nelson's paper. Among Greene's objections, he asked about Nelson's claim about the difficulty in calculating "superexponentiation" (tetration), which is the cornerstone of Nelson's proof attempt. In particular, Greene asked for a paper, written by someone other than Nelson, which discusses this. I attempted to make a Google Scholar search, but since I don't have university library access, I can't retrieve any paper to tell whether it's what Greene is looking for or not. But while searching, I noticed that there is a distinction between two types of functions, "elementary primitive" vs. "primitive recursive." For there exist functions which are "primitive recursive" but not "elementary recursive" -- and as it turns out, the "superexponentiation" (tetration) function is the foremost example of a function which is primitive recursive but not elementary recursive. And so I wonder whether elementary recursion is the key to making Nelson's proof work. Numbers such as Nelson's: 2^^2^^2^^2^^2^^2^^2^^2^^2^^2^^2^^2^^2^^2^^2^^2 (aka 2^^^16, called "2 pentated to the 16") contain superexponentiations which require primitive recursion to define, but perhaps only functions defined via elementary recursion can be proved to be "counting numbers" in Nelson's sense. Another objection Greene had was that Nelson adds symbol "^^" to the language of PA, then goes back to state that instances of the induction containing that symbol are forbidden. I see what Greene is getting at here, since this problem came up for me in the TO thread as well. In particular, I wanted to add a new primitive symbol "tav" to the language of ZF, then add axioms mentioning "tav." But as it turns out, one can use an instance of ZF's Separation Schema to derive a contradiction involving "tav" (such as tav is both finite and infinite). I couldn't go back and retroactively declare that Separation doesn't apply to instances including the new symbol "tav," just as Greene wrote that Nelson can't go back and declare that induction doesn't apply to instances including the new symbol "^^." Both Induction and Separation apply to all formulas of their respective theories' languages, even if we extend them to add a new symbol.
From: MoeBlee on 20 Jul 2010 10:56 On Jul 19, 10:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > are there explicit axioms that we can write, similar to the > axioms of PA, that can represent WM's ideas? Keep in mind that WM has said more than once that he rejects formal axiomatization. MoeBlee
From: Transfer Principle on 20 Jul 2010 22:45 On Jul 20, 7:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Jul 19, 10:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > are there explicit axioms that we can write, similar to the > > axioms of PA, that can represent WM's ideas? > Keep in mind that WM has said more than once that he rejects formal > axiomatization. Notice that I was responding to Jeffries, who had said: "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." Now by "non-Peanoan system," did Jeffries intend a formal axiomation of this system? If so, then how can we connect this to WM, who opposes formal axioms? In other words, how can we have something that satisfies both Jeffries and WM? I suppose we can come up with the following: declare the set N of (WM-)naturals to equal N_n for some natural number n, and then if anyone asks how to prove this, I respond that the posters whose ideas I'm trying to represent with this N_n, namely WM, is opposed to axiomatization. Thus, I can't give axioms to describe this N_n, as giving an axiomatization isn't true to the desiderata of WM.
From: FredJeffries on 24 Jul 2010 06:17 On Jul 20, 7:45 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jul 20, 7:56 am, MoeBlee <jazzm...(a)hotmail.com> wrote: > > > On Jul 19, 10:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > > are there explicit axioms that we can write, similar to the > > > axioms of PA, that can represent WM's ideas? > > Keep in mind that WM has said more than once that he rejects formal > > axiomatization. > > Notice that I was responding to Jeffries, who had said: > > "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." > > Now by "non-Peanoan system," did Jeffries intend a formal > axiomation of this system? If so, then how can we connect > this to WM, who opposes formal axioms? In other words, how > can we have something that satisfies both Jeffries and WM? > I was merely referring to a couple of features of the modern world that seem to imply that the notion that you "get" the set of natural numbers by starting at 0 and repeatedly incrementing just doesn't describe the numbers that we actually use. As you point out, we are using RSA cryptography but those numbers are absurd according to Yessenin-Volpin (and Nelson?). Also, when we take into account the resources (register size, swap space available to hold intermediate results, ...) needed to do computations, we can calculate 10^500 + 10^500 rather easily but many calculations involving numbers quite smaller than 10^500 give us overflow problems. > I suppose we can come up with the following: declare the set > N of (WM-)naturals to equal N_n for some natural number n, Isn't this a circular definition? You are using a natural number n to define the set of natural numbers? But this is a problem that comes up often: someone may say that 10^500^500^500 does not exist, but there must be some sense in which it does exist in order to be able to point to it and say that it doesn't exist...
From: FredJeffries on 24 Jul 2010 06:49
On Jul 19, 8:46 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > 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))} But what are you using for your indexes? It seems to me that there are (at least) two systems of "natural numbers" being referred to here: Counting numbers used for your indexes and keeping track of steps (and as a measure of complexity? Or is that a separate type?) and computation numbers, the elements of your N_n. You've also got the binary operations addition and multiplication but what is the number 2 which is used in "binary"? These different systems are also indicated by the quarrel "Do the natural numbers start at 0 or at 1?" Well, counting starts at 1 (most of the time) but computation is better served by having a bit pattern of all 0's as the base. |