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Topic: natural numbers as pseudocircles
Replies: 1   Last Post: Feb 4, 2014 9:42 PM

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fom

Posts: 1,969
Registered: 12/4/12
natural numbers as pseudocircles
Posted: Feb 4, 2014 9:17 PM
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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/2003-January/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 4-tuple


< 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 well-defined. 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:Rieger-Nishimura.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".







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