fom
Posts:
1,969
Registered:
12/4/12


natural numbers as pseudocircles
Posted:
Feb 4, 2014 9:17 PM


Some help to know what a pseudocircle is:
http://en.wikipedia.org/wiki/Pseudocircle
This post makes an explicit representation of Dana Scott's statement from
http://www.cs.nyu.edu/pipermail/fom/2003January/006087.html
concerning "enough Axiom of Foundation" to interpret
(All n,m)[n+1 = m+1 ==> n = m]
as "counting down".

This construction is motivated by the tree structure on the naturals recently described by Mr. Bacarisse in one of WM's threads. As can be seen,
"The infinite binary tree, B = (N, T) is a graph whose nodes are the natural numbers (1, 2, 3...) and whose edges are thepairs T = { (i, j)  j = 2i or j = 2i+1 }"
this is a graph theoretic description.
Now, a binary tree with distinctly labeled vertices can be interpreted as a system of tetrahedra sharing edges. So, another interpretation of the description above would be that of tetrahedra sharing edges. One would have edge sets of the form,
{ {1,2}, {2,4}, {2,5}, {1,4}, {1,5}, {4,5} }
{ {1,3}, {3,6}, {3,7}, {1,6}, {1,7}, {6,7} }
{ {2,4}, {4,8}, {4,9}, {2,8}, {2,9}, {8,9} }
{ {2,5}, {5,10,}, {5,11}, {2,10}, {2,11}, {10,11} }
{ {3,6}, {6,12}, {6,13}, {3,12}, {3,13}, {12,13} }
{ {3,7}, {7,14}, {7,15}, {3,14}, {3,15}, {14,15} }
{ {4,8}, {8,16}, {8,17}, {4,16}, {4,17}, {16,17} }
and so on
Now, the sets above correspond to the nodes labeled 2, 3, 4, 5, 6, 7, and 8. And, in the model being considered here, the domain is taken to be a pointed set where the distinguished point is the multiplicative identity. If we take differences between pairs for the last 6 edge sets, then we obtain the sets,
{ 2, 3, 4, 5, 6, 1 }
{ 2, 4, 5, 6, 7, 1 }
{ 3, 5, 6, 8, 9, 1 }
{ 3, 6, 7, 9, 10, 1 }
{ 4, 7, 8, 11, 12, 1 }
{ 4, 8, 9, 12, 13, 1 }
Each of these contains the original node label as well as the multiplicative identity. So, consider rewriting these as the pair of tuples,
{ < 1, 3 >, < 2, 4, 5, 6 > }
{ < 1, 4 >, < 2, 5, 6, 7 > }
{ < 1, 5 >, < 3, 6, 8, 9 > }
{ < 1, 6 >, < 3, 7, 9, 10 > }
{ < 1, 7 >, < 4, 8, 11, 12 > }
{ < 1, 8 >, < 4, 9, 12, 13 > }
Each of these has either the form,
{ < 1, m >, < k, m+1, n, n+1 > }
where
k < m+1 < n < n+1
If we consider the directed set structure of the naturals  that is, for any two given natural numbers there exist a natural number larger than both  in conjunction with the axiom,
m+1 = n+1 > m = n
then the form above expresses these facts together. That is, suppose one is given z with its successor,
z < z+1
Then, there is some relation of the form,
{ < 1, z >, < y, z+1, x, x+1 > }
relating z to the unit and asserting the existence of an x greater than both z and its successor which has a successor of its own.
Now, suppose the 4tuple
< k, m+1, n, n+1 >
is given a semigroup structure satisfying,
.......k...n+1...n...m+1.  ..k....k....k....n....n..  .n+1..n+1..n+1..m+1..m+1.  ..n....k....k....n....n..  .m+1..n+1..n+1..m+1..m+1. 
Then, each element can be fixed in relation to arbitrary pairings of the remaining elements. For example,
(m+1)(k)(n)(m+1) = (m+1)(n)(k)(m+1) = m+1
(m+1)(k)(n+1)(m+1) = (m+1)(n+1)(k)(m+1) = m+1
(m+1)(n+1)(n)(m+1) = (m+1)(n)(n+1)(m+1) = m+1
The reason for this is that succession follows from the neighborhood structure of the unit given by the neighborhoods,
{ 1 }
{ 1, 2 }
{ 1, 2, 3 }
{ 1, 2, 3, 4 }
and so on
In contrast, what is expressed in the construction,
{ < 1, 3 >, < 2, 4, 5, 6 > }
{ < 1, 4 >, < 2, 5, 6, 7 > }
{ < 1, 5 >, < 3, 6, 8, 9 > }
{ < 1, 6 >, < 3, 7, 9, 10 > }
{ < 1, 7 >, < 4, 8, 11, 12 > }
{ < 1, 8 >, < 4, 9, 12, 13 > }
is the fact that the multiplicative identity is a divisor of every number. The relation to succession can be seen in the pairs of triples,
{ { 3, 4, 5 }, { 4, 5, 6 } }
{ { 4, 5, 6 }, { 5, 6, 7 } }
{ { 5, 6, 7 }, { 6, 7, 8 } }
But, the domain is coordinatized in relation to succession:
0 exists.
If x exists, Sx exists.
SSx = SSy > Sx = Sy
I wrote it this way to emphasize that the Peano axioms express the successor relations as triples,
< x, Sx, SSx >
The third statement is usually interpreted as an assertion that the successor function is welldefined. In this analysis, no function has been asserted. So, the overlapping triples correspond to what is expressed by the axioms.
Note that this system of overlapping triples closely resembles the structure for the domain of a free Heyting algebra on one generator,
http://en.wikipedia.org/wiki/File:RiegerNishimura.svg
The diagram gives the logical relations, but, the definition in Balbes and Dwinger is
<begin quote>
The free Heyting algebra on one free generator is the sublattice
L = [ ( union_[q = 1..oo] { (q+1,p) : q <= p <= q+3 }) union ( { (1,1), (1,2), (1,3) } ) ] + ONE
<end quote>
The expression '... + ONE' corresponds to a a point at ifinity in the sense that any element of the set of ordered pairs is less than ONE. So,
(1,3) <= ONE
for instance.
And, there are a number of rules for using the ordered pairs that give the lattice its usual operations. But, it is a system of quadruples grounded by an initial triple and related to one another by 3's.
It is true that what I have done above does not correspond to the definition as written. In fact, only
{ < 1, 3 >, < 2, 4, 5, 6 > }
could be used. That will have to be kept in mind moving forward. However, the system described has yielded the basic structure upon which Heyting algebras are represented without asserting it.
For the next step, give each quadruple the topology,
{ { k, m+1, n, n+1 }, { k, n, n+1 }, { m+1, n, n+1 }, { k, m+1 }, { k }, { m+1 }, null }
This is the pseudosphere topology. Notice that neither n nor n+1 are presented as individuals. Once again, this speaks to the axiom
p+1 = q+1 > p = q
in relation to directed sets. There is some number greater than k and m+1 by the directed set structure. That number has a successor. Those numbers need not exist as "individuals"; but, they serve to individuate k and m+1 by the pseudocircle topology.
We come now to the hard cases. For the first vertex labled by 2 the edge set is
{ {1,2}, {2,4}, {2,5}, {1,4}, {1,5}, {4,5} }
and its differences yield
{ 1, 2, 3, 4}
This is not disturbing in and of itself since the axioms motivating this construction introduce 1 and 2 simultaneously,
ExEy( Az( x mdiv z ) /\ S(x,x,y) )
AxAy( x = y <> Au( Av( u mdiv v ) > Ev( S(u,x,v) <> S(u,y,v) ) ) ) )
where
AxAy( x pdiv y <> ( Az( y pdiv z > x pdiv z ) /\ Ez( x pdiv z /\ ~( y pdiv z ) ) ) )
AxAy( x mdiv y <> ( Az( y pdiv z > x mdiv z ) /\ Az( z mdiv x > z mdiv z ) ) )
AxAyAz( S(x,y,z) <> ( ( x mdiv x /\ ( x mdiv y /\ ( x mdiv z /\ x pdiv z ) ) ) /\ ( y mdiv x > ( S(y,x,z) <> S(x,y,z) ) ) ) )
So, the pseudocircle topology,
{ { 1, 2, 3, 4 }, { 1, 3, 4 }, { 2, 3, 4 }, { 1, 2 }, { 2 }, { 1 }, null }
and the associated semigroup product can be associated with the quadruple.
And, this is enough for the unit, 1. It will have the neighborhood structure of succession.
All of this, however, has been relating structure to composite numbers. Moreover, 4 is a composite number.
The axioms above take the multiplicative identity and the primes as fundamental objects. So, the sense by which these cases are handled must involve primes.
For the prime integer topology, subbasic neigborhoods are of the form,
{ x : x = b + np }
for any b and any prime p.
However, this system does not have an additive identity. Neighborhoods of that form converge to { 2p } and { b + p } for b different from p.
So, each prime p depends on its relations as described above for its individuation. That is, 3, 5, and 7 are singletons in the system because of the relations,
{ < 1, 3 >, < 2, 4, 5, 6 > }
{ < 1, 5 >, < 3, 6, 8, 9 > }
{ < 1, 7 >, < 4, 8, 11, 12 > }
And, so it is for every prime.
Thus, 2 remains a problem.
In a recent post, a neighborhood system for 2 had been devised by using a combination of methods from analytical number theory. The first two iterations of the method produce a 4 tuple. The neighborhoods are the "residuals from the parititioning algorithm:
interval_1 :=> { ..., 33 } interval_2 :=> { 31, ..., 17 } [5 primes] [length: 14] [gap: 2] interval_3 :=> { 13, 11 } [2 primes] [length: 2] [gap: 4] interval_4 :=> {} [0 primes] [length: null] [gap: null] interval_5 :=> { 7 } [1 prime] [length: 0] [gap: 4] interval_6 :=> {} [0 primes] [length: null] [gap: null] interval_7 :=> { 5 } [1 prime] [length: 0] [gap: 2] interval_8 :=> {} [0 primes] [length: null] [gap: null]
residual :=> { 2 } [1 prime] [length: 0] [gap: 3] omitted :=> { 3 } [1 prime] [length: 0] [gap: 0]
The second sequence is
0 < 1/128 < 2/128 < 3/128 < 4/128 < 5/128 < 1/18 < 1/16 < 1/9
Its partition will be,
interval_1 :=> { ..., 131 } interval_2 :=> { 127, ..., 67 } [13 primes] [length: 60] [gap: 4] interval_3 :=> { 61, ..., 43 } [5 primes] [length: 18] [gap: 6] interval_4 :=> { 41, 37 } [2 primes] [length: 4] [gap: 2] interval_5 :=> { 29 } [1 prime] [length: 0] [gap: 8] interval_6 :=> { 23, 19 } [2 primes] [length: 4] [gap: 6] interval_7 :=> { 17 } [1 prime] [length: 0] [gap: 2] interval_8 :=> { 13, 11 } [2 primes] [length: 2] [gap: 4]
residual :=> { 7, 5, 3, 2 } [4 primes] [length: 5] [gap: 4] omitted :=> { 31 } [1 prime] [length: 0] [gap: 0]
So, for
{ 7, 5, 3, 2 }
form the topology,
{ { 2, 3, 5, 7 }, { 2, 5, 7 }, { 3, 5, 7 }, { 2, 3 }, { 3 }, { 2 }, null }
and provide it with the same semigroup structure discussed above.
With this, then, each element and its successor is individuated,
1 has { 2 }:
{ { 1, 2, 3, 4 }, { 1, 3, 4 }, { 2, 3, 4 }, { 1, 2 }, { 2 }, { 1 }, null }
2 has { 3 }:
{ { 2, 3, 5, 7 }, { 2, 5, 7 }, { 3, 5, 7 }, { 2, 3 }, { 3 }, { 2 }, null }
3 has { 4 }:
{ { 2, 4, 5, 6 }, { 2, 5, 6 }, { 4, 5, 6 }, { 2, 4 }, { 4 }, { 2 }, null }
4 has { 5 }:
{ { 2, 5, 6, 7 }, { 2, 6, 7 }, { 5, 6, 7 }, { 2, 5 }, { 5 }, { 2 }, null }
5 has { 6 }:
{ { 3, 6, 8, 9 }, { 3, 8, 9 }, { 6, 8, 9 }, { 3, 6 }, { 6 }, { 3 }, null }
6 has { 7 }
{ { 3, 7, 9, 10 }, { 3, 9, 10 }, { 7, 9, 10 }, { 3, 7 }, { 7 }, { 3 }, null }
and so on
Notice the smallest value in each topology. For all numbers x >= 3, the tree structure means that they will occur exactly 2 times. The unit has a multiplicity of 1 and its successor, namely 2, has a multiplicity of 3.
It is because 2 has to be "the smallest prime" in much the same way that 1 has to be "the smallest number".

