fom
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12/4/12


a formal construction of Dedekind cuts
Posted:
Feb 21, 2013 8:21 PM


This is a formal construction.
As an initial context, any set theory that supports the use of braces, '{' and '}', to stand for representation of its collection finishing predicate will suffice.
The iterated enclosure of a symbol by braces shall be referred to as the Zermelo naming predicate,
x > {x} > {{x}} > {{{x}}} > {}{}{}
For each symbol 'x',
'{x} names x'
'{{x},{{x}},{{{x}}},...} describes x'
For each symbol 'x':
'x' stands as representative for a finished class if and only if the description for 'x' implies that 'x' has a representable name.
if 'x' stands as representative for a finished class, then 'x' is described by a Dedekind simply infinite class generated through successive iteration of the Zermelo naming predicate.
In order to formulate a representation for grounded von Neumann chains using only pairs of matched braces, the context must also admit an axiom of pairing over finished classes and an axiom of union across finished classes.
For each symbol 'x' that stands as representative for a finished class, the operation stipulated by
x :=> u{x,{{x}}}
shall be referred to as von Neumann succession.
For each symbol 'x' that stands as representative for a finished class:
'x is the von Neumann predecessor of u{x,{{x}}}'
'u{x,{{x}}} is the von Neumann successor of x'
'{u{x,{{x}}},{u{x,{{x}}},{{u{x,{{x}}}}}},...} is the von Neumann chain of x'
If 'x' is admissible as standing in representation for a finished class, then the von Neumann successor of 'x' is admissible as standing in reprsentation for a finished class.
If 'x' is admissible as standing in representation for a finished class, then the von Neumann chain of 'x' is admissible as standing in representation for a finished class and 'x' is said to be its ground.
If the concatenation
{}
is admissible as standing in representation for a finished class, and, if every proper initial segment of the concatenation
{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},...}
is admissible as standing in representation for a finished class, then
{{}},{{},{{}}},{{},{{}},{{},{{}}}},...}
is the von Frassen supervaluation chain grounded by
{}
The contingent finished class
{{},{{}},{{},{{}}},{{},{{}},{{},{{}}}},...}
shall be called the Wittgenstein enumeration.
This construction is intended to apply for for any set theory that admits every proper initial segment of the Wittgenstein enumeration and all of their proper parts as finished classes.
To ease the comprehensibility of the presented material, the following stipulations shall be made:
0:={} 1:={{}} 2:={{},{{}}} 3:={{},{{}},{{},{{}}}}
and so on,...
t:={{},{{}},{{},{{}}},...}
and so on,....
The colloquial term for the suggested sequence of marks
0,1,2,3,...
shall be used. The marks of this sequence shall be called the whole numbers and denoted by W.
The colloquial term for the suggested sequence of marks
1,2,3,...
shall be used. The marks of this sequence shall be called the natural numbers and denoted by N.
The cyclic group on 10 particular inscriptions,
0>1>2>3>4>5>6>7>8>9>0
shall constitute the letters of an alphabet to be called digits. An alphabet is a Dedekind system. It may be classified as a Dedekind closedchain system.
The digits shall be put into relation with the whole numbers according to the algorithm of long division terminating with remainders. Each remainder shall be a whole number.
Each whole number shall be put in relation with digits according to the formal relation
y=(a_n*x^n + a_(n1)*x^(n1) + ... + a_1*x^1 + a_0*x^0)
where the indeterminate terms are related to the described alphabet by the stipulations,
x^0:=0' /\ 0':={{0,1,2,3,4,5,6,7,8,9}}
x^1:=0'' /\ 0'':={{{0,1,2,3,4,5,6,7,8,9}}}
x^2:=0''' /\ 0''':={{{{0,1,2,3,4,5,6,7,8,9}}}}
and so on, as needed.
Thus, the alphabet is presumed to be a described finished set.
In any set theory that admits every proper initial segment of the Wittgenstein enumeration and all of their proper parts as finished classes, the alphabet may be taken as any initial segment of the Wittgenstein enumeration.
In the formal relation given above,
y=(a_n*x^n + a_(n1)*x^(n1) + ... + a_1*x^1 + a_0*x^0)
each of the coefficients a_i are taken to be a letter of the alphabet  that is, a digit. A lossless numeral is defined as the orderisomorphic concatenation of digits arranged sequentially according to
<a_n,a_(n1),...,a_1,a_0>
where the delimiters, '<' and '>' stand for a particular use of the axiom of pairing by which ordered pairs may be represented as finished classes using the brace notation for the collection finishing predicate.
The schema
<x,y>:={{x},{x,y}}
stipulates an ordered pairing for any symbols 'x' and 'y'
The schemes
<x_1,x_0>:=<x_1,x_0>
<x_2,x_1,x_0>:=<x_2,<x_1,x_0>>
<x_3,x_2,x_1,x_0>:=<x_3,<x_2,<x_1,x_0>>>
and so on,...
<x_n,x_(n1),...,x_1,x_0>:=<...<x_n,<x_(n1),<...,<x_1,x_0>
and so on,....
convey the intended extended use of the base schema to arrange concatenations sequentially.
Given any lossless numeral,
<a_n,a_(n1),...,a_1,a_0>
one may form a lossy numeral,
<b_m,b_(m1),...,b_1,b_0>
using an orderpreserving concatenation of digits from
<a_n,a_(n1),...,a_1,a_0>
that are not syntactically equivalent with '0'.
Marks are syntactically equivalent if they can be placed in relation to one another using the sign of equality under bare quantification. For example,
2=2
expresses syntactic equivalence. Contingent to to prior stipulations, one might consider
2=1+1
to be syntactically equivalent in some system. But one might have a different system in which
0=1+1
expresses syntactic equivalence and '2' is in no alphabet across which the sign of equality expresses the relation of syntactic equivalence.
Note, however, that syntactic equivalence is distinct from inscriptional equivalence. The expression
2=2
conveys inscriptional equivalence, whereas
2=1+1
does not. In order for a logicallyconstructed system to be wellconstrued, every instance of inscriptional equivalence must correspond with syntactic equivalence. This is a pragmatic presupposition governing proper use for the sign of equality as a relation across an alphabet.
Since the alphabet of digits has been constructed using inscriptionally differentiated letters, a correspondence between lossless numerals and lossy numerals may be postulated. Given a lossless numeral,
<a_n,a_(n1),...,a_1,a_0>
and its lossy numeral counterpart,
<b_m,b_(m1),...,b_1,b_0>
the ordered pair,
{<a_n,a_(n1),...,a_1,a_0>,{<a_n,a_(n1),...,a_1,a_0>,<b_m,b_(m1),...,b_1,b_0>}}
expressible as
<<a_n,a_(n1),...,a_1,a_0>,<b_m,b_(m1),...,b_1,b_0>>
may be formed.
The grammar of logical construction dictates that the use of ordered pairs represented with
<x,y>:={{x},{x,y}}
convey an instance of a foundational ground as being prior to its relation with a derived counterpart. Thus, using the correspondence between lossless and lossy numerals as illustration, the lossless numeral,
<a_n,a_(n1),...,a_1,a_0>
is the foundational ground for the lossy numeral,
<b_m,b_(m1),...,b_1,b_0>.
To formulate this correspondence between lossless and lossy numerals as the ground for a described genus of numerals, it will be useful to simplify the presentation. Let the stipulations
a:=<a_n,a_(n1),...,a_1,a_0>
b:=<b_m,b_(m1),...,b_1,b_0>.
stand for representations of finished classes taken as lossless and lossy numerals, respectively.
Let
<a,b>={{a},{a,b}}
be a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
For any finished class, let its Russell ostension be defined as the pair composed of its description and its von Neumann chain.
For each symbol 'x' that stands as representative for a finished class:
Let ZD(x) be its description.
Let NC(x) be its von Neumann chain.
Then, 'this(x) indicates {ZD(x),NC(x)}'
and, the ordered pair
<ZD(x),<NC(x),{ZD(x),NC(x)}>>
is an instance of Russellian ostension.
Given an arbitrary collection of Russellian ostensions and any symbol 'x' that may stand as representative for a finished class:
'this(x) chooses <ZD(x),<NC(x),{ZD(x),NC(x)}>>'
is a substantiation predicate for the ordered pair,
<x,<ZD(x),<NC(x),{ZD(x),NC(x)}>>>
And, to say for any symbol 'x'
"'x' has a representable name by virtue of the description for 'x'" implies that 'this(x)' chooses well from a finished class of finished Russellian ostensions.
Given any lossless numeral a having a lossy counterpart b, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of numerals. Such a finished class is called a genus of numerals.
In particular, the numerals whose digits constitute the alphabet
{0,1,2,3,4,5,6,7,8,9}
are a genus. Each numeral is either a lossless numeral or a lossy numeral. Thus, in relation to this alphabet  that is, the digits of whole numbers  there is a species of lossless numerals and a species of lossy numerals.
Suppose now that
a:=<a_n,a_(n1),...,a_1,a_0>
is a lossless numeral.
Call any orderpreserving concatenation of digits from the numeral a that does not include the digit a_n a representative trailing segment of a. Say that a representative trailing segment is full if there exists a lossless numeral b such that the representative trailing segment of a is orderisomorphic with b.
Define a reduced numeral,
b:=<b_m,b_(m1),...,b_1,b_0>
as a lossless numeral which:
1) is derivative to some given lossless numeral;
2) is obtained through orderisomorphism with a full representative trailing segment of the given lossless numeral;
3) corresponds with that full representative trailing segment determined according to the condition that every representative trailing segment of which it is a proper part has an initial segment of concatenated 0's but its has no initial segment beginning with 0.
Given a lossless numeral, say a, and any reduced numeral corresponding to a, say b, let
<a,b>={{a},{a,b}}
be a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any lossless numeral and any lossless numeral that satisfies the definition of a reduced counterpart b, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of lossless numerals. Such a finished class is called a genus of lossless numerals.
In particular, the lossless numerals whose digits constitute the alphabet
{0,1,2,3,4,5,6,7,8,9}
are a genus. Each lossless numeral is either the reduced counterpart of a lossless numeral or it is not. Thus, in relation to this alphabet  that is, the digits of whole numbers  there is a species of reduced numerals and a species of raw numerals.
Suppose now that
a:=<a_n,a_(n1),...,a_1,a_0>
is a lossless numeral.
Suppose that a_n is the digit 0.
Define any lossless numeral whose leading term is 0 to be an expansion numeral.
Define any lossless numeral whose leading term is inscriptionally differentiated from 0 a floor numeral.
Given any expansion numeral, form
<0,a>={{0},{0,a}}
and let this be a representation of its grounded construction.
Then
{<0,a>} names <0,a>.
and
{{<0,a>},{{<0,a>}},{{{<0,a>}}},...} describes <0,a>
Moreover,
u{<0,a>,{{<0,a>}}} is the von Neumann successor of <0,a>
and
{u{<0,a>,{{<0,a>}}},{u{<0,a>,{{<0,a>}}},{{u{<0,a>,{{<0,a>}}}}}},...} is the von Neumann chain of <0,a>
Given any expansion numeral, it is always assumed that
'this(<0,<ZD(<0,a>),<NC(<0,a>),{ZD(<0,a>),NC(<0,a>)})'
chooses well from a finished class of lossless numerals. Such a finished class is called a genus of lossless numerals.
In particular, the lossless numerals whose digits constitute the alphabet
{0,1,2,3,4,5,6,7,8,9}
are a genus. Each lossless numeral is either an expansion numeral or it is a floor numeral. Thus, in relation to this alphabet  that is, the digits of whole numbers  there is a species of expansion numerals and a species of floor numerals.
By construction, every reduced numeral is a floor numeral.
By construction, every floor numeral is a reduced numeral.
The system of whole numbers is taken to be the species of floor numerals over the alphabet
{0,1,2,3,4,5,6,7,8,9}
related to one another systematically by the algorithm of long division terminating with remainders such that each remainder shall be a whole number.
With each instance of long division is associated a task. The ground for the task is called a 'dividend'. The associate to the ground is called a 'divisor'.
With regard to tasks, the whole number whose floor numeral is 0 may not be a divisor. Consequently, the whole numbers whose digits constitute the alphabet
{0,1,2,3,4,5,6,7,8,9}
is a genus. Each whole number may either be a divisor relative to the systematic relations imposed by the algorithm of long division terminating with whole number remainders or it is the number 0. Thus, in relation to this alphabet  that is, the digits of whole numbers  there is a species of divisors and a species consisting only of 0.
As it is isolated by a nested sequence of genera and species, the whole number whose floor numeral is 0 is a paradigmatic instance of Aquinian individuation. This is used to endow ontological import to the singularity of terms as follows.
Given any symbol 'x' such that 'x' stands as representative for a finished class, form
<0,x>={{0},{0,x}}
and let this be a representation of its grounded construction.
Then
{<0,x>} names <0,x>.
and
{{<0,x>},{{<0,x>}},{{{<0,x>}}},...} describes <0,x>
Moreover,
u{<0,x>,{{<0,x>}}} is the von Neumann successor of <0,x>
and
{u{<0,x>,{{<0,x>}}},{u{<0,x>,{{<0,x>}}},{{u{<0,x>,{{<0,x>}}}}}},...} is the von Neumann chain of <0,x>
Given any symbol 'x' such that 'x' stands as representative for a finished class, it is always assumed that
'this(<0,<ZD(<0,x>),<NC(<0,x>),{ZD(<0,x>),NC(<0,x>)})'
chooses well from a finished class of individuals. Such a finished class is called a genus of individuals.
With each instance of long division is associated a completion. The ground for the completion is called a 'quotient'. The associate to the ground is called a 'remainder'.
For some instance of long division, let:
a be the remainder
b be the dividend
c be the divisor
d be the quotient
and form the nesting of ordered pairs
<a,<<b,c>,<d,a>>>
called a base relation.
Given any base relation, say <z,<<w,x>,<y,z>>>, it has the form of the ordered pair
<a,b>={{a},{a,b}}
where
a:=z
b:=<<w,x>,<y,z>>
So, take
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any base relation and its species within the genus of base relations, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of completed tasks. Such a finished class is called a systemic relation of whole number bases.
For each whole number z, the collection of base relations corresponding to
<z,<<w,x>,<y,z>>>
is taken to be a finished collection. As a describable finished collection within a describable finished collection, it is a species in relation to its genus.
For each base relation <z,<<w,x>,<y,z>>>, denote its species within the genus of base relations as
[<z,<<w,x>,<y,z>>>]
Let
a:=<z,<<w,x>,<y,z>>>
b:=[<z,<<w,x>,<y,z>>>]
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any base relation and its species within the genus of base relations, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of base relation quotient maps. Such a finished class is called a congruence identification of base relations.
For each whole number different from 0, form the nesting of ordered pairs
<x,<(x+1),<(x1),x>>>
This nesting of ordered pairs shall be called the Peano relation corresponding to the natural number x.
Let
{{1},{2},{3},...}
be the finished class of Zermelo names for natural numbers
For each natural number x, let
a:=<x,<(x+1),<(x1),x>>>
b:={x}
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any Peano relation of whole numbers and the Zermelo name for its corresponding natural number, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of Peano relation quotient maps. Such a finished class is called a Peano correspondence map.
For each natural number x there is a collection of Pythagorean triples of cardinality x. This collection is determined by a recurrence relation such that each of the triangles for a given x have the same area. Among the collection of triangles for each natural number under this recurrence relation, precisely 1 has a hypotenuse with odd length. In the following description, that set of triples for each x is designated as (a_1,b_1,c_1).
The recurrence relation is described as:
Given n>=1, let there be n given Pythagorean triples (a_k,b_k,c_k) such that a_k<b_k<c_k and k=1,...,n
The n triangles having an even length hypotenuse are given by
(a_k)'=2(b_1^2a_1^2)c_1*a_k
(b_k)'=2(b_1^2a_1^2)c_1*b_k
(c_k)'=2(b_1^2a_1^2)c_1*c_k
The single triangle having an odd length hypotenuse is given by
(a_(n+1))'=(b_1^2a_1^2)^2
(b_(n+1))'=4(a_1*b_1*c_1^2)
(c_(n+1))'=4(a_1^2*b_1^2) + c_1^4
For each natural number x, let
<x,<<0,1,2>,{<a_*,b_*,c_*>}>>
be taken as a correspondence between each natural number, the grounding Peano relation, and the finished collection of Pythagorean triples of whole numbers under the recurrence relation just discussed.
Let
a:=x
b:=<<0,1,2>,{<a_*,b_*,c_*>}>
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any natural number x, the grounding Peano relation, and the collection of Pythagorean triples of whole numbers uniformly described under recurrence, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of Peano relation quotient maps. Such a finished class is called a Pythagorean correspondence map.
Let
<1,<(2),<0,1>>>
be the grounding Peano relation.
Let
{u{<1,<(2),<0,1>>>,{{<1,<(2),<0,1>>>}}},...}
be the von Neumann chain of <1,<(2),<0,1>>>
Each natural number is to be successively put into correspondence with each successively grounded von Neumann chain. The natural number 1 shall be put in correspondence with the von Neumann chain of <1,<(2),<0,1>>>. The natural number 2 shall be put into correspondence with the von Neumann chain of
u{<1,<(2),<0,1>>>,{{<1,<(2),<0,1>>>}}}
and so on....
So, for the natural number 1, one has
a:={u{<1,<(2),<0,1>>>,{{<1,<(2),<0,1>>>}}},...}
b:=1
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given the grounded von Neumann chain for <1,<(2),<0,1>>> and the natural number 1, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of Aquinian individuals. Such a finished class is called a Cantorian unit base.
Let a and b be any two natural numbers and consider the fraction b/a. The denominator be specifies the nature of the units to be enumerated. The numerator a specifies the cardinality of the enumeration. The denominator is the ground of the relationship.
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any two natural numbers a and b, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of Aquinian individuals. Such a finished class is called a rationalization base.
By virtue of arithmetical operations with fractions, there is an arithmetical equivalence relation such that for fractions,
b/a=d/c
whenever
b*c=a*d
as natural numbers in relation to the congruence identification class corresponding with the whole number 0.
Each such equivalence class is taken to be a describable finished collection within the ratio base. As the rationalization base is taken to be a describable finished collection, each equivalence class is a species in relation to its genus.
Within each equivalence class, there is a fraction whose denominator and numerator are a coprime pair. This is a unique fraction within the class that shall be called the accepted ratio.
Let b/a be the fraction corresponding to the accepted ratio for some equivalence class of the rationalization base.
For each fraction y/x from the given equivalence class, let <<a,b>,<x,y>> denote its grounded relation to the accepted ratio. Then, to construct an indentification map, form the nested sequence of pairs
<<x,y>,<<a,b>,<x,y>>>
Let
a:=<x,y>
b:=<<a,b>,<x,y>>
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any fraction and its grounded relation to an accepted fration, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of rationalization base quotient maps. Such a finished class is called a accepted ratio identification for the rationalization base.
The finished class corresponding to the equivalence relations over the rationalization base is presumed to be a describable class in lexicographic order with the grounded denominate number lexicographically prior to the numerator.
The finished class corresponding to the equivalence relations over the rationalization base is presumed to be orderisomorphic with an initial segment of the Wittgenstein enumeration.
Let
{[b_0/a_0],[b_1/a_1],[b_2/a_2],...}
represent the finished collection of identified equivalence classes over the rationalization base.
By construction, the accepted ratio is the canonical representative for each equivalence class.
Each pair of distinct equivalence classes is differentiated by virtue of the underlying order relation of the natural numbers. Given accepted ratios,
q/p and n/m,
if (m*q)<(p*n), then (q/p)<(n/m)
if (m*q)>(p*n), then (q/p)>(n/m)
Thus,
(q/p)=(n/m) if and only if (m*q)=(p*n)
in the natural numbers.
For any pair of accepted ratios, say q/p and n/m, form the grounded pairing
<<p,q>,<m,n>>
The accepted ratio in the grounding position shall be the minuend and the derivative accepted ratio shall be the subtrahend. The finished class of such pairs shall be the rational difference base.
With these difference pairs, the arithmetical equivalence classes can be taken to be in correspondence with the rational numbers, Q.
When (q/p)=(n/m) does not hold, the statements
'<<p,q>,<m,n>> is the negative of <<m,n>,<p,q>>'
'<<m,n>,<p,q>> is the negative of <<p,q>,<m,n>>'
are satisfied.
The accepted difference from each equivalence relation is that difference from among differences formed only between the accepted ratios from the underlying equivalence classes over the rationalization base whose minuend appears first in the lexicographic ordering of the rationalization base.
Let <<p,q>,<m,n>> be the accepted difference for some equivalence class of the rational difference base.
For each difference <<y,x>,<z,w>> from the given equivalence class, let <<<p,q>,<m,n>>,<<y,x>,<z,w>>> denote its grounded relation to the accepted difference. Then, to construct an indentification map, form the nested sequence of pairs
<<<y,x>,<z,w>>,<<<p,q>,<m,n>>,<<y,x>,<z,w>>>>
Let
a:=<<y,x>,<z,w>>
b:=<<<p,q>,<m,n>>,<<y,x>,<z,w>>>
Next, form the ordered pair
<a,b>={{a},{a,b}}
as a representation of its grounded construction.
Then
{<a,b>} names <a,b>.
and
{{<a,b>},{{<a,b>}},{{{<a,b>}}},...} describes <a,b>
Moreover,
u{<a,b>,{{<a,b>}}} is the von Neumann successor of <a,b>
and
{u{<a,b>,{{<a,b>}}},{u{<a,b>,{{<a,b>}}},{{u{<a,b>,{{<a,b>}}}}}},...} is the von Neumann chain of <a,b>
Given any difference and its grounded relation to an accepted difference, it is always assumed that
'this(<a,<ZD(<a,b>),<NC(<a,b>),{ZD(<a,b>),NC(<a,b>)})'
chooses well from a finished class of rational difference base quotient maps. Such a finished class is called a accepted difference identification for the rational difference base.
The finished class corresponding to the equivalence relations over the rational difference base is presumed to be a describable class in lexicographic order with the grounded minuend lexicographically prior to the subtrahend and the global ordering inherited from the lexicographic order of the rationalization base.
The finished class corresponding to the equivalence relations over the rational difference base is presumed to be orderisomorphic with an initial segment of the Wittgenstein enumeration.
Let
{[<b_0_m/a_0_m,b_0_s/a_0_s>],[<b_1_m/a_1_m,b_1_s/a_1_s>],...}
represent the finished collection of identified equivalence classes over the rational difference base.
By construction, the accepted difference is the canonical representative for each equivalence class.
Each pair of distinct equivalence classes is differentiated by virtue of the underlying order relation of the natural numbers inherited through the construction of the rationalization base. Given accepted ratios,
For q/pn/m and k/ji/h form the differences
qpmnmp and kjhihj
Then
(q/pn/m)=(k/ji/h) if and only if (qpm+ihj)=(kjh+nmp)
(q/pn/m)<(k/ji/h) if (qpm+ihj)<(kjh+nmp)
(q/pn/m)>(k/ji/h) if (qpm+ihj)>(kjh+nmp)
One may now consider the question of Dedekind cuts.
Since the numerals for the whole numbers are floor numerals, the orientation of the Dedekind cuts will be to accept cuts corresponding to greatest lower bounds.
Under interpretation relative to the construction at hand, Dedekind presumed that for every binary partition of the rational numbers, one part of each partition could be interpreted as a species in relation to a genus relative to a uniform choice of cuts. That is, a sequence of cuts such that one of every pair is a proper part of the other could first, be taken individually as species to genus with the full collection of rationals, and, second, be taken pairwise as species to genus in terms of antecedent cuts and succeedent cuts.
Clearly, that is not possible to formulate using descriptions.
However, if one accepts the real numbers as a set, then they are present in the theory, as individuals and as a system whose identity criterion is based upon the inherited order from the natural numbers by the construction just completed. The final step, of course, is to obeserve that the rationals are dense in the reals. Therefore, the fact that the construction has yielded an identity criterion for the rationals, means that it has yielded an identity criterion for the Dedekind cuts as the real number system.
Given this, a wellordering for the reals is only possible if one stipulates that the elements different from {} of whatever model in which one chooses to perform this construction can be placed in onetoone correspondence with the van Frassen supervaluation.
Typically, this would be called HOD for hereditarily ordinaldefinable since the intended relation between the Wittgenstein enumeration and the van Frassen supervaluation is that the van Frassen supervaluation is taken to be lossless by comparison.

