Prev: geometry template for infinite-line-ray to precision define infinite-number in Peano Axioms #627 Correcting Math
Next: The Necessary vs Contingent Distinction
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.
First  |  Prev  |  Next  |  Last
Pages: 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Prev: geometry template for infinite-line-ray to precision define infinite-number in Peano Axioms #627 Correcting Math
Next: The Necessary vs Contingent Distinction