On 3/23/2012 3:44 PM, Joseph Knight wrote:

On Thu, Mar 22, 2012 at 6:40 AM, Stephen P. King<stephe...@charter.net <mailto:stephe...@charter.net>> wrote:On 3/21/2012 8:16 PM, Joseph Knight wrote:On Tue, Mar 20, 2012 at 10:25 AM, Stephen P. King <stephe...@charter.net <mailto:stephe...@charter.net>> wrote: Dear Joseph, How do numbers implement that necessary capacity to define each other and themselves? What kind of relational structure is necessary? From what I can tell, it looks like a "net of Indra" where every jewel, here a number, reflects all others. This is a non-well founded structure. You'll have to be more explicit than this if I am to make any sense of it.Dear Joseph, I first must say that I appreciate very much this exchange as it forces me to better refine my wordings and explanations. In the passage above I was trying to get at something that I see in the implied structure of numbers, given Bruno's amazing ideas. Remember, I "think in pictures", so the relations between numbers - with their Goedelizations and Loeb references - is to me a network where any one entity - here an integer - is defined by and related to all others. It looks like the structure of an infinite Webster Dictionary! What I also see is that the "links" are not of a constant length - some connections between numbers are tiny - like the link between prime pairs - while others are infinitely long. What I am trying to point out is that this structure, is very much_unlike_ the structure that we think of when we just consider the "number line" where such a line is made up only of integers - 0, 1, 2, 3, ...This is all nice, but I can't understand it unless you give make thismore formal/precise.

`Do you only think in words? I'm just curious... I will try harder`

`to sketch the idea in words for you.`

`Think of how Goedelizations and Goedel numbers work as a visual`

`picture, perhaps as a poitrait by Matisse or Dali. We have a string of`

`numbers that "represents" another set of numbers *and* some arithmetic`

`operation on those numbers. Any such Goedel number is thus the`

`equivalent to a "handle" on the "space" of numbers (which is, by`

`definition, a one dimensional manifold`

`<http://en.wikipedia.org/wiki/Curve#Topology>, also see 1`

`<http://en.wikipedia.org/wiki/Evenly_spaced_integer_topology> and 2`

`<http://en.wikipedia.org/wiki/Open_sets#Topological_spaces>), therefore`

`if it is possible to have an infinity of goedel numbers in the integers`

`then the resulting manifold would an infinity of handles (disjoint`

`manifolds) on it. How many unique paths would exist on such a manifold?`

`What is the "average" length of a path? (Please recall the fact that a`

`handle can have any size iff it is simply connected and analytic) There`

`is no such an average for the only faithful sample of the set of`

`possible lengths of paths is the set itself (infinite sets are`

`isomorphic to any of their proper subsets).`

`Remember that we can also have goedel numbers operating on (mapping`

`into) dovetailed strings of goedel numbers and goedel numbers can have`

`arbitrarily long number string lengths...... This makes the dimension of`

`this manifold to be infinite because of the disjointness of the`

`"handles" that are induced by the Goedelizing, thus making it (modulo`

`the requirements of spaces to exist) an infinite space. It is only if`

`the requirements of a space`

`<http://en.wikipedia.org/wiki/Space_%28mathematics%29> not being met`

`that this would not occur. Given that a geodelization introduces`

`arithmetic into the set of numbers then is automatically qualifies a`

`goedelized number line to be the dual of a space (via the Stone`

`representation theorem`

`<http://en.wikipedia.org/wiki/Stone_representation_theorem>).`

QED.

`The visual mode and the symbol mode of languages seem to have a`

`strange conjugacy....`

Numbers as Bruno is considering them, I contend, has a structure that mathematicians denote as "non-well founded" in the sense that there is no "basic" building block out of which this structure is constructed unless we force it into a very tight straight jacket. One example of just a constraint occurs when we think of numbers as von Neumann numerals <http://bmanolov.free.fr/von-neumann-integer.php> or something like: s, s(), s(()), s((())), ... - where s is the null set which we can define in terms of Spencer-Brown's laws of form as the Double Cross (see http://upload.wikimedia.org/wikipedia/commons/f/ff/Laws_of_Form_-_double_cross.gif), my point being that we only obtain a 'well-founded' version when we impose a constraint of the "natural' structure.Well, of course. To talk about well-foundedness you need a class and arelation, not just a class. It doesn't make sense to say "numbers arenon-wellfounded".

`They are non-well founded if they are identified with`

`Goedelizations or computations as I have shown above. I would be happy`

`to see a counterexample, for this seems to be a proof of the`

`non-existence of Bruno's global measure. :-(`

Before I divagate off in a ADHD haze, let me state my conclusion: this structure that we find as defined by COMP is very much unlike the nice and well behaved collection of Integers and should not be mistaken for it. How? Where is the justification for this?

See my proof above.

It is not uniform nor regular nor well-founded and I dare say that it is not computable in the usual sense of recursively enumerable mappings from N to N.I don't even know what structure you are talking about, because youhaven't defined it. If you had defined it, we could learn about it.

`Its all about the Goedel numbering and the self-referencing that it`

`induces. Loedbian operations would be even more vicious as they allow`

`plenums (a form of continua for discrete spaces, its similar to a`

`spectrum`

`<http://en.wikipedia.org/wiki/Spectrum_%28functional_analysis%29>) of`

`self-loops, which look like higher dimensional versions of totally`

`disconnected discrete spaces, when we consider the topological view of`

`numbers. All of this idea simply follows from the Stone duality; Logics`

`(the 'stuff" of minds) are the ontological dual to topological Spaces`

`(the "stuff" of physics). This is all flowing from my Prattian dualist`

`alternative to material or ideal monist ontological theories. I see`

`Bruno's work as being the equivalent for logic as what Newton did for`

`physics. I mean this with all sincerity!`

snip Onward! Stephen -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.