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

When I had been thinking about arithmeticbecause of Dan Christensen's threads, Idid a great deal of analysis only to havearrived at a seemingly trivial observation.Although motivated by the unique factorizationtheorem, the introduction of succession seemedonly to dictate that the multiplicative identitybe included with incrementally larger exponents.This post provides some additional analysis.Recalling that there is no 0, that only themultiplicative identity and prime numbers areprimitive notions as aliquot parts, and thatexponentiation is taken as metamathematicalaccounting practice, the sequence of numberswould be something like:1 = 1^12 = 1^2 * 2^13 = 1^3 * 3^14 = 1^4 * 2^25 = 1^5 * 5^16 = 1^6 * 2^1 * 3^17 = 1^7 * 7^18 = 1^8 * 2^39 = 1^9 * 3^2In this way, the "monadic divisors" would all beparticipating in the uniqueness of the factorization.No mathematician would, at first thought, think toarrange elements in this fashion as canonicalrepresentatives.  Treating the multiplicative identityin 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 groundforms such as1 mdiv 12 mdiv 23 mdiv 35 mdiv 5Succession 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 posterioritysince it does not express immediate succession.But, "immediate successor" is a selection fromamong "successors".  So, "succession" seems adequate.The identity criteria (different from grammaticalequality) 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 secondaxiom -- do nothing more than choose a basepoint froma 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 anarithmetical system.Now, the additional observations I would liketo make are as follows.Consider the sequence of tuples as if they are elementsin 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 thedimension of the space, they do so ambiguously.Each prime number, however, is associatedwith a substantive orthogonal extension ofdimension.Put another way, the null space of a decompositionis ambigous.  With each prime, the space isextended.  So, from 1 to 2 and from 2 to 3 theextension is a 1-space.  From 3 to 5 and from5 to 7 the extension is a 2-space.  From 7 to 11the extension is a 4-space.So, although Euclid's axiom is true and algebraicextension occurs with the successor, the numberin Euclid's axiom does not have a definite positionin the additional algebraic dimension.In this construction, consider a vector withunit value in all dimensions.  Its length incrementsaccording to the sequence,surd(1), surd(2), surd(3), surd(4), ...It's angular coordinate from each dimension -- thatis, the sequence of direction cosines -- decrementsaccording to the sequence,surd(1)/1, surd(2)/2, surd(3)/3, ...Under the assumption of a completed infinity, thisray becomes an orthogonal dimension.  That is,the direction cosine would correspond withcos(pi/2) = 0.Setting these observations aside, there is one ofmore interest.What the successor does here is to order theprimes so that a unique factorization is possible.So, there is a sense by which prime factorizationis based upon a projection into the "first" dimension.With that in mind, consider the sequences of exponentsfor the prime divisors without concern for the baseof 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 theempty string because 1 has no primedivisor.The numbers comprising the sequences ineach line are bounded above by the successionof exponents for the multiplicative identity.Each ordered sequence corresponds to aninitial segment of natural numbers in thewell-founded tree of finite sequences whichgrounds the Polish space of such sequences.Each ordered sequence is a type.  That is,there is a collection of the exponentsof the multiplicative identity which arecorrelated by these sequences.  For example,the prime numbers are in the correlation,< 1 > :=> { 2, 3, 5, ... }Note that the definition of this mappingfrom the canonical syntax naturally excludesthe unit from the partition{ 2, 3, 5, ... }that distinguishes the primes.The first "independent triple" is given by30 :=> < 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 classsince exponentiation is a metamathematicalcount.  Unique factorization governs the canonicalrepresentation, but the general interpretationof this metamathematical usage in group theoryis ambigous.  Thus, there is also a class ofnumbers{ 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 constructionis to represent relations between objects without regardfor "rules" between "known" objects.  With a uniquelygiven syntax, multiple interpretations present themselves.In the case of natural numbers, three well-ordered collectionsare associated with each signature.In addition, each signature is associated with a finiteinitial segment for sequences in the Baire space.  Sincethe configuration is fixed and since the first elementcan take any natural number, let the second elementcorrespond 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 naturalnumbers supports a general notion of number based uponthe part relation rather than a syntactic definition.This is, by no means, intended as a replacement for thenotion of number in "primitive recursive arithmetic".Rather, this is an investigation of how one obtainsfoundational theories uniformly.  The first axiom abovecan be interpreted set-theoretically as "proper subset",mereologically as "proper part", and possibly as "wf-part"with respect to recursively-generated languages.
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