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.