fom
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a formal construction of Dedekind cuts
Posted:
Feb 21, 2013 8:21 PM
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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 closed-chain
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_(n-1)*x^(n-1) + ... + 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_(n-1)*x^(n-1) + ... + 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 order-isomorphic concatenation of
digits arranged sequentially according to
<a_n,a_(n-1),...,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_(n-1),...,x_1,x_0>:=<...<x_n,<x_(n-1),<...,<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_(n-1),...,a_1,a_0>
one may form a lossy numeral,
<b_m,b_(m-1),...,b_1,b_0>
using an order-preserving concatenation of digits
from
<a_n,a_(n-1),...,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 logically-constructed
system to be well-construed, 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_(n-1),...,a_1,a_0>
and its lossy numeral counterpart,
<b_m,b_(m-1),...,b_1,b_0>
the ordered pair,
{<a_n,a_(n-1),...,a_1,a_0>,{<a_n,a_(n-1),...,a_1,a_0>,<b_m,b_(m-1),...,b_1,b_0>}}
expressible as
<<a_n,a_(n-1),...,a_1,a_0>,<b_m,b_(m-1),...,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_(n-1),...,a_1,a_0>
is the foundational ground for the lossy numeral,
<b_m,b_(m-1),...,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_(n-1),...,a_1,a_0>
b:=<b_m,b_(m-1),...,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_(n-1),...,a_1,a_0>
is a lossless numeral.
Call any order-preserving 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 order-isomorphic with b.
Define a reduced numeral,
b:=<b_m,b_(m-1),...,b_1,b_0>
as a lossless numeral which:
1)
is derivative to some given lossless numeral;
2)
is obtained through order-isomorphism 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_(n-1),...,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),<(x-1),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),<(x-1),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_1^2-a_1^2)c_1*a_k
(b_k)'=2(b_1^2-a_1^2)c_1*b_k
(c_k)'=2(b_1^2-a_1^2)c_1*c_k
The single triangle having an odd length hypotenuse is
given by
(a_(n+1))'=(b_1^2-a_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 order-isomorphic 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 order-isomorphic 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/p-n/m and k/j-i/h form the differences
qpm-nmp and kjh-ihj
Then
(q/p-n/m)=(k/j-i/h) if and only if (qpm+ihj)=(kjh+nmp)
(q/p-n/m)<(k/j-i/h) if (qpm+ihj)<(kjh+nmp)
(q/p-n/m)>(k/j-i/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 well-ordering 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
one-to-one correspondence with the van Frassen supervaluation.
Typically, this would be called HOD for hereditarily ordinal-definable
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.
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