Equality as Contextual Derivation Recursive Formalism; Potential Transcendence of Incompleteness

[eodnhoj7]

***Updated

Emergent equivalence operator identity by degree of self-embedding recursion.

Equality and inequality are purely contextual identities by degree of having identities that effectively are relation. Where standard operators of equality and inequality are subject to a fixed status within standard formalisms the following formalistic proof will observe such fixed statuses as fundamentally relative identities by nature where what is constant is context itself.

Context is subject to a dualism of presence and absence, or rather negation of negation and negation thus effectively is a binary state, or dualism, that effectively self-references to form higher level binary/dualistic distinctions

The following distinctions can be taken both syntactically and semantically. They are not axioms in the traditional sense, they are not assumed, but rather emergent patterns by degree of recursion thus effectively are complete by self-embedding:

“=” equality

“=/=” inequality

“( )” context/set/container

“{ }” context/set/container different by degree

“-” negation, absence, negative

“–” negation of negation, presense, positive

(A = A)

(A = (=))

(=) = (=)

(=) =/= (=/=)

(=/=) = (=/=)

((=) = (=)) =/= ((=/=) = (=/=))

((=){=}(=)) =/= ((=/=){=}(=/=))

{=} =/= {=}

(=/=){=}(=/=)

(=/=) =/= (=/=)

(=) = {(=),(=/=)}

(=/=) = {(=),(=/=)}

( ) = {( ),( )}

( )

( )( )

{( )( )}( )

{( )( )}( )( )

{( )( )}{( )( )}

…

{ }

{ }{ }

({ }{ }){ }

…

=

== (=, -=(=/=))

(==)= (-=(=/=),–=(=))

…

=/=

=/==/= (=/=,-=/=(=))

(=/==/=)=/= (-=/=(=),–=/=(=))

…

-

-- (-, --)

(–)- (–, —)

(–)-- (—, ----)

…

****

The nature of the operators: =, =/=, -, – effectively reduce to binary contexts that are effectively self-embedding in the degree that recursion results in an identity of pattern, a meaningful tautology akin to the identity law of A=A, which dually inverts into an isomorphic expression that contains the sequence as a new variable. Identity is thus context dualism by degree of repetition which results in a contrast:

( ), { } are both single contexts that each differ by degree with said degree being the dimension of recursion.

In one respect:

{( )( )} observes ( ) as a sequence contained as a differing dimensional context as { }

In an inverse respect:

({ }{ }) observes { } as a sequence contained as a differing dimensional context as ( ).

Regardless of the dimensionality of both ( ) and { } the degree of one relative to another is relatively higher or lower relative to scale; in one respect { } represents a higher dimensional context while in another ( ) is a higher dimensional context. The nature of a higher dimension is one of general inclusivity:

{( )( )} observes { } as higher dimensional.

({ }{ }) observes ( ) as higher dimensional.

In these respects a higher dimension can be view as a context of containment.

All variables can be reduced to contexts;

A=A → 1=1, 2=2, 3=3… Cat=Cat, Dog=Dog, …

and all operators can be reduced to variables as what they operate one defines the operator themselves by degree of relationality:

(A=A, B=B, C=C, … X=X…) →

(=) ↔ (A,B,C,…X)

In these respects all operators are effectively contexts that effectively are contained by there degree of:

Recursion:

(A=A, B=B, C=C, … X=X…) ↔

(=, =, =, …=…)

and Inversion:

(=) ↔ (A,B,C,…X)

In these respect both operator, F, and operand, f, are biconditional when both operator and operand are seen as variables: F ↔ f

This biconditionality is an operator as well and in these respects is a self contained variable in one respect:

(B ↔ B) = ((<->)<->(<->))

while dually observing that its recursive latice results in its negation by inversion into its opposite. Where biconditionality, <->, inverts into is opposite then occurs “therefore, transitions to, becoming, etc.”, ->. Any form of recursion inherently contains sequences of double which results in a self contained identity, akin to A=A, while dually a self contrast occurs by its opposite, -A:

↔

<-><-> (<->,- <->(->))

(<-><->)<-> (- <->(->), – <->(<->))

…

->

->-> (->, - ->(<->))

(->->)-> (- ->(<->), – ->(->)

…

and in these respect recursion results in gradient dualisms.

This dualism of biconditionality and transitioning results in the further degree of operator emergence under ∨∧;

∧ observes “both/and” that is a reflection of biconditionality and transitioning in the respect that ∧ reflects ↔ as connection of context and → as dependent contrast. Mathematically ∧ is addition and multiplication as compounding.

∨ observes “either/or” that is a reflection of biconditionality and transitioning in the respect that ∨ reflects ↔ as distinct identities and → as identity divergence. Mathematically ∨ is subtraction and division as reduction.

[huphuphup123]

hello. i am huphuphup123 from philosophy now but the site went down, i dont have my posts with you from there. but i had a question, i remember you had the chain of presence absence, but my question is wouldn’t it jus go on for infinite? shouldn’t there be a stop to it

[eodnhoj7]

What’s up dude, I remember you, hope all is well.

To answer your question I will give the visual of a line segment:

There is a line segment A

It is composed of infinite line segments.

This infinity is bound as the finite line segment A.

Another example, there is a line now, not a line segment but rather a line.

This line goes on infinitely.

Now this infinitey implies an absence of limits under base appearance. But that is not the case, the line is bound as 1 dimension, so while it is infinite it is bound finitely as the dimension through which it exists. This infinitey is a continuum but this continuum is always expressed by the dimension it occurs within.

Here is another example, the basic number line. There is 1, 2, 3, 4, etc….infinitely.

Now given the number line is infinite it gives the impression of there being no boundaries, but that is only an impression. All numbers on th number line are compounded 1. 1 and 1 for 2, 1 and 1 and 1 and 1, for 4, etc. So any number you pick on the number line has a holographic structure for 1. You can pick X and X will always being X times 1, doesn’t matter the number. Now what I say is obvious, you are well aware of all this, but I used the number line as an example.

Now rather than 1 compounding upon itself there is a limit to replace it. This limit is presence. Where the limit exists requires a relative absence of another limit. So the presense of one limit is the absence of another, and this occurs repeated, the repetition of one limit is the repetition of absences of others.

Hopefully that answered your question, if not then fire a few more questions and I will see what I can do.

So yes, it does go on infinitely but all infinites occur through finite dimensions thus are bounded. A visual would be a snow flake, there may be infinite snow flakes but this infinitey is bounded in the form of a snow flake. Infinite and finiteness are context, they both exist simultaeously.