Date: Dec 17, 2013 2:43 AM
Author: fom
Subject: Thoughts on unique factorization interpretation


When I had been thinking about arithmetic
because of Dan Christensen's threads, I
did a great deal of analysis only to have
arrived at a seemingly trivial observation.
Although motivated by the unique factorization
theorem, the introduction of succession seemed
only to dictate that the multiplicative identity
be included with incrementally larger exponents.

This post provides some additional analysis.

Recalling that there is no 0, that only the
multiplicative identity and prime numbers are
primitive notions as aliquot parts, and that
exponentiation is taken as metamathematical
accounting practice, the sequence of numbers
would be something like:


1 = 1^1

2 = 1^2 * 2^1

3 = 1^3 * 3^1

4 = 1^4 * 2^2

5 = 1^5 * 5^1

6 = 1^6 * 2^1 * 3^1

7 = 1^7 * 7^1

8 = 1^8 * 2^3

9 = 1^9 * 3^2


In this way, the "monadic divisors" would all be
participating in the uniqueness of the factorization.
No mathematician would, at first thought, think to
arrange elements in this fashion as canonical
representatives. Treating the multiplicative identity
in this way is counterintuitive.

The notion of "monadic divisor" and "monadic division"
comes from trying to use the axioms,


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 ) ) )


Where 'mdiv' is specifically intended to admit ground
forms such as


1 mdiv 1

2 mdiv 2

3 mdiv 3

5 mdiv 5


Succession is given by the formula,


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) ) ) ) )


It may be better to refer to this as posteriority
since it does not express immediate succession.
But, "immediate successor" is a selection from
among "successors". So, "succession" seems adequate.

The identity criteria (different from grammatical
equality) are given by the axioms,


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) ) ) ) )


The expression "Az( x mdiv z )" from the first axiom --
and its counterpart, "Av( u mdiv v )", from the second
axiom -- do nothing more than choose a basepoint from
a collection. Hence, the ground here is a "pointed set"
or a "pointed space". There is no assertion based upon
"knowing" what actually constitutes a base of an
arithmetical system.

Now, the additional observations I would like
to make are as follows.

Consider the sequence of tuples as if they are elements
in real analysis,


( 1 )

( 2, 1 )

( 3, 0, 1 )

( 4, 2, 0, 0 )

( 5, 0, 0, 0, 1 )

( 6, 1, 1, 0, 0, 0 )

( 7, 0, 0, 0, 0, 0, 1 )

( 8, 3, 0, 0, 0, 0, 0, 0 )

( 9, 0, 2, 0, 0, 0, 0, 0, 0 )

( 10, 1, 0, 0, 1, 0, 0, 0, 0, 0 )

( 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 )


Now, where composite numbers expand the
dimension of the space, they do so ambiguously.

Each prime number, however, is associated
with a substantive orthogonal extension of
dimension.

Put another way, the null space of a decomposition
is ambigous. With each prime, the space is
extended. So, from 1 to 2 and from 2 to 3 the
extension is a 1-space. From 3 to 5 and from
5 to 7 the extension is a 2-space. From 7 to 11
the extension is a 4-space.

So, although Euclid's axiom is true and algebraic
extension occurs with the successor, the number
in Euclid's axiom does not have a definite position
in the additional algebraic dimension.

In this construction, consider a vector with
unit value in all dimensions. Its length increments
according to the sequence,


surd(1), surd(2), surd(3), surd(4), ...


It's angular coordinate from each dimension -- that
is, the sequence of direction cosines -- decrements
according to the sequence,


surd(1)/1, surd(2)/2, surd(3)/3, ...


Under the assumption of a completed infinity, this
ray becomes an orthogonal dimension. That is,
the direction cosine would correspond with


cos(pi/2) = 0.


Setting these observations aside, there is one of
more interest.

What the successor does here is to order the
primes so that a unique factorization is possible.
So, there is a sense by which prime factorization
is based upon a projection into the "first" dimension.
With that in mind, consider the sequences of exponents
for the prime divisors without concern for the base
of the exponents.

1 :=> <>

2 :=> < 1 > ( -> 'p':'p' )

3 :=> < 1 > ( -> 'p':'p' )

4 :=> < 2 > ( -> 'p^2':'2p' )

5 :=> < 1 > ( -> 'p':'p' )

6 :=> < 1, 1, > ( -> 'p*q':'p+q' )

7 :=> < 1 > ( -> 'p':'p' )

8 :=> < 3 > ( -> 'p^3':'3p' )

9 :=> < 2 > ( -> 'p^2':'2p' )

10 :=> < 1, 1 > ( -> 'p*q':'p+q' )

11 :=> < 1 > ( -> 'p':'p' )

12 :=> < 2, 1 > ( -> 'p^2*q':'2p+q' )

:
:
:

30 :=> < 1, 1, 1 > ( -> 'p*q*r':'p+q+r' )


The first element of the list is the
empty string because 1 has no prime
divisor.

The numbers comprising the sequences in
each line are bounded above by the succession
of exponents for the multiplicative identity.

Each ordered sequence corresponds to an
initial segment of natural numbers in the
well-founded tree of finite sequences which
grounds the Polish space of such sequences.

Each ordered sequence is a type. That is,
there is a collection of the exponents
of the multiplicative identity which are
correlated by these sequences. For example,
the prime numbers are in the correlation,


< 1 > :=> { 2, 3, 5, ... }


Note that the definition of this mapping
from the canonical syntax naturally excludes
the unit from the partition


{ 2, 3, 5, ... }


that distinguishes the primes.


The first "independent triple" is given by


30 :=> < 1, 1, 1 > ( -> 'p*q*r':'p+q+r' )


So, its class would be,



{ 30 (= 2*3*5 ), 42 (= 2*3*7 ), 66 (= 2*3*11 ), 105 (= 3*5*7 ), ... }



But, the notation also yields a second class
since exponentiation is a metamathematical
count. Unique factorization governs the canonical
representation, but the general interpretation
of this metamathematical usage in group theory
is ambigous. Thus, there is also a class of
numbers


{ 10 (= 2+3+5 ), 12 (= 2+3+7 ), 16 (= 2+3+11 ), 15 (= 3+5+7 ), ... }


The point here is that the priority of this construction
is to represent relations between objects without regard
for "rules" between "known" objects. With a uniquely
given syntax, multiple interpretations present themselves.
In the case of natural numbers, three well-ordered collections
are associated with each signature.

In addition, each signature is associated with a finite
initial segment for sequences in the Baire space. Since
the configuration is fixed and since the first element
can take any natural number, let the second element
correspond with the Baire space configuration. Hence,



30 :=> < 1, 1, 1 > :=> [ 1; 1, 1 ]


Where


[ a_1; a_2, a_3, ..., a_k ]


yields the continued fraction,


a_1 + ( 1 / a_2 + ( 1 / a_3 + ( 1 / ... + 1 / a_k ) ) ) )


In this way, one canonical representation of the natural
numbers supports a general notion of number based upon
the part relation rather than a syntactic definition.

This is, by no means, intended as a replacement for the
notion of number in "primitive recursive arithmetic".
Rather, this is an investigation of how one obtains
foundational theories uniformly. The first axiom above
can be interpreted set-theoretically as "proper subset",
mereologically as "proper part", and possibly as "wf-part"
with respect to recursively-generated languages.