On 7/24/2012 1:07 PM, Stephen P. King wrote:
On 7/22/2012 2:41 PM, Stephen P. King wrote:
Many (as implied by the word plural <http://www.google.com/#hl=en&gs_nf=1&pq=domain%20range%20map&cp=10&gs_id=l&xhr=t&q=plural+definition&pf=p&sclient=psy-ab&oq=plural+def&gs_l=&pbx=1&bav=on.2,or.r_gc.r_pw.r_qf.,cf.osb&fp=d0885a1bd80304c5&biw=1680&bih=894>) _is not just a number_. (It is at least a Gödel number.) A plurality of 1p is a mapping function from some domain to some co-domain (or range). So if there is no distinction between the domain and co-domain, what kind of map is it? Maybe it is an automorphism <http://www.google.com/#hl=en&sclient=psy-ab&q=automorphism+definition&oq=automorphism+definition&gs_l=serp.1.0.0j0i5i30.43772.43772.2.44960.,or.r_gc.r_pw.r_qf.,cf.osb&fp=d0885a1bd80304c5&biw=1680&bih=894>, but it is not something that allows us to extract a plurality over which variation can occur. You are talking as if the variation <http://www.google.com/#hl=en&gs_nf=1&gs_mss=automorphism%20definition&pq=automorphism%20definition&cp=10&gs_id=1d&xhr=t&q=variation+definition&pf=p&sclient=psy-ab&oq=variation+definition&gs_l=&pbx=1&bav=on.2,or.r_gc.r_pw.r_qf.,cf.osb&fp=d0885a1bd80304c5&biw=1680&bih=894> was present but not allowing the means for that variation to occur! The use of the word "plurality" is thus meaningless as you are using it: "first person plural view of physical reality". You must show first how it is that the plurality obtains without the use of a space if you are going to make claims that there is no space and yet plurality (of 1p) is possible. In the explanation that you give there is discussion of Moscow, Helsinki and Washington. These are locations that exists and have meaning in a wider context. At least there is assumed to be a set of possible locations and that the set is not a singleton (such as {0}) nor does it collapse into a singleton.

Dear Bruno and Friends,

I would like to add more to this portion of a previous post of mine (that I have revised and edited a bit).

Let us stipulate that contra my argument above that the "many" of a plurality is "just a number". What kind of number does it have to be? It cannot be any ordinary integer because it must be able to map some other pair of numbers to each other, ala a Gödel numbering scheme. But this presents a problem because it naturally partitions Gödel numbering schemes into separate languages, one for each Gödel numbering code that is chosen. This was pointed out in the Wiki article about

"Lack of uniqueness

A Gödel numbering is not unique, in that for any proof using Gödel numbers, there are infinitely many ways in which these numbers could be defined.

For example, supposing there are/K/basic symbols, an alternative Gödel numbering could be constructed by invertibly mapping this set of symbols (through, say, aninvertible function <http://en.wikipedia.org/wiki/Invertible_function>/h/) to the set of digits of abijective base-/K/numeral system <http://en.wikipedia.org/wiki/Bijective_numeration>. A formula consisting of a string of/n/symbolss_1 s_2 s_3 \dots s_nwould then be mapped to the number

   h(s_1) \times K^{(n-1)} + h(s_2) \times K^{(n-2)} + \cdots +
   h(s_{n-1}) \times K^1 + h(s_n) \times K^0 .

In other words, by placing the set of/K/basic symbols in some fixed order, such that the/i/^th symbol corresponds uniquely to the/i/^th digit of a bijective base-/K/numeral system,/each formula may serve just as the very numeral of its own Gödel number." "


This lack of uniqueness is a huge weakness! What it does is that it implies that ultimately any pair of sufficiently long strings of numbers will be equivalent to computations that are bisimilar and this isomorphic under functional equivalence. I do not know what kind of isomorphism this is or if it is already known.

So is a N -> NxN map identical to N? Did not Russell Standish make some comments that where proximate to this idea? What axioms are we assuming for this arithmetic?



"Nature, to be commanded, must be obeyed."
~ Francis Bacon

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