Date: Feb 21, 2013 8:21 PM Author: fom Subject: a formal construction of Dedekind cuts

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.