****The Nature of Axioms, the Necessity of Infinite Regress as a Foundation and Non-Universality of (In)Completeness
Posted: Fri Sep 19, 2025 7:30 pm
For the following meditation the following symbols will be used to start:
The symbol ● is an axiom
The symbols ( ) are context by means of relation.
1. We know an axiom: ●
2. This axiom occurs relative to another axiom: ●●
3. This relationship of axioms is an axiom: (●●)●
4. We know an axiom that is but other axioms for the axiom is but self contained by other axioms as the axiom itself: (●)
5. The self contained axiom is but relative to other self contained axioms: (●)(●)
6. The relationship of self contained axioms is an axiom: ((●)(●))●
7. We know an axiom is but other self-contained axioms for the axiom is but self contained by other self contained axioms as the self contained axiom itself: ((●))
8. The self contained axioms as a self contain axiom is but relative to a self contained axiom as self-contained axioms: ((●))((●))
9. The relationship of self contained axioms is but an axiom:
(((●))((●)))●
10. We know an axiom is but other self-contained axioms for the axiom is but self contained by other self contained axioms as but the self contained axiom itself: (((●)))
11. The self-containment of an axiom is infinite, yet this self containment of the axiom is an axiom, an axiom relative to another axiom is but itself: (((●)))●
12. The axiom is a patterned process (Pp) by degree of its inherent relations that are not limited: △
13. The Pp is relative to another Pp: △△
14. The relationship of other patterns processes is but a patterned process (△△) △
15. The patterned process contains itself by other patterned processes: (△)
16. The self contained Pp is but relative to other self contained Pp: (△)(△)
17. The relationship of self contained Ppis a Pp:
((△)(△))△
18. We know a Pp is but other self-contained Pp for the Pp is but self contained by other self contained Pp as the self contained Pp itself: ((△))
19. The self contained Pp as a self contain Pp is but relative to a self contained Pp as self-contained Pp: ((△))((△))
20. The relationship of self contained Pp is but a Pp:
((△))((△))△
21. We know a Pp is but other self-contained Pp for the Pp is but self contained by other self contained Pp as but the self contained axiom itself: (((△)))
22. The self-containment of a Pp is infinite, yet this self containment of the Pp is a Pp, a Pp relative to another Pp is but itself: (((△)))△
23. The patterned process is an axiom by degree of its inherent relations that are not limited: ●
24. The axiom and patterned process are one and the same, isomorphic by nature as one results in the other as variable expressions of an infinite foundation by which both occur.
25. Axioms and patterned processes are self contained by there infinite nature, there completion is the foundation of infinity by which both occur and maintain.
26. The system is complete by degree of the containment of infinite recursion through infinite recursion making infinite recursion a self-evident foundation that justifies itself as complete.
27. Incompleteness is finiteness evidenced by regress.
28. Finiteness allows for infinite regress.
29. Infinite regress is a patterned process through said finiteness
30. A patterned process is finite infinity.
31. Finite infinite occurs recursively.
32. The recursion of infinity, by degree of finite infinities, justifies infinity by degree of a self-referential infinity.
33. Infinity is self contained, as self contained is complete by degree of the axiom and patterned process as one given that infinite recursion occurs recursively through the axiom/Pp thus maintaining its own foundation as infinite recursion. The recursion of infinite regress necessitates infinite regress as self-referencing and self contained thus such a system is complete by evidence of self-reference.
34. If a system is incomplete due to finiteness, and this finiteness contains infinite regress within it, than the system is complete by degree of containing infinity within the finite.
35. The containment of infinities, by degree of the finite, where the finite infinitely regresses, necessitates a recursion of infinities where said system is complete by nature of recursive infinities.
36. The recursion of infinities is but self referential completeness of a system where the infinite recursion of infinities is but the foundations of infinite recursion justifying themselves as complete through the nature of the infinite process where infinity is its own foundation expressed isomorphically thus containing itself.
37. Completeness necessitates an element of finiteness by degree of containment. Incompleteness necessitates a finiteness by degree of perpetual regress. If regress, through recursion, is observed as the foundation and said regress occurs recursively (ie recursion of regresses) then all systems containing such foundations can be observed as simultaneously complete and incomplete.
38. A system that is simultaneously complete and incomplete ceases to be such things by degree of negation, thus a system must be observed as an axiomatic patterned process, patterned thus complete and processing thus incomplete.
39. Completeness and incompleteness synthesize as the axiomatic pattern process. Axioms are the patterns by which frameworks occur thus inevitably processes by degree of regress.
40. Infinity is its own foundational pattern evidenced as axiomatic by degree of the process within process for an infinite regress of axioms necessitates new axioms that each have an infinite regress of justifications. Infinite regress is nested within infinite regress thus making infinity finite, by degree of the form of regress, where furthermore infinity is nested within itself as its own foundation by degree of its finite dimension of expression.
41. Axiomatic patterned processes are self contained: □
42. Self contained processes exist across forms as a form: (□)□
43. All processes are fixed states by means of the continuity of the process thus the process is but a continuity of a specific form of change, this form of change is a form.
44. Forms are processes and by degree of continuity of said process and necessarily static. All processes are finite forms by degree of the form of the process.
45. While a process continues the expression of a variety of forms, the process is a form that underlies said variety of forms.
46. Form is the means by which change occurs, the form of infinite regress makes said form itself, linearistic branching, as infinite.
47. An infinite form necessitates infinity as finite while still necessitating an identity of the infinite for what it is. Axioms are stable processes by which interpretations exist.
48. "Completeness" and "incompleteness" are both axioms and follow said nature of axioms.
49. The axiom of completeness, by degree of the patterned process of axioms, cannot be completely proven as a complete axiom in one respect thus nullifies itself under its own terms as completeness is an incomplete axiom.
50. The axiom of incompleteness, by degree of the patterned process of axioms, can be proven as complete axiom in one respect thus nullifies itself under its own terms as incompleteness is a complete axiom.
51. Completeness and incompleteness exist only in grades, thus are relative. By degree of relativity they sometimes exist and sometimes do not, there universal validity is not stable. Systems derived from such axioms exhibit gradient behavior and cannot be considered universal axioms outside of specific contexts.
52. Axiomatic patterned processes are occurences, self-evidence is not self-evident by degree of its own standards.
53. Axioms are not universal outside of the occurence of form, only form is universally axiomatic by degree of occurence of said form.
54. Occurence is the universal axiom, evidenced by degree of axiomatic patterned processes being only occurences.
55. What is considered axiomatic is only the occurence of patterns for patterns give rise to evidence.
56. No single axiom can be observed as an axiom without corresponding patterns, there is no axiomatic interpretation of axioms outside of patterned processes of recognition.
57. Axioms are incomplete by progress, complete by self-referentiality. This dual nature of axiomatic patterned processes is an axiom within the nature of axioms.
58. "Incomplete" is an axiom.
59. "Complete" is an axiom.
60. "Axiom" is an axiom.
61. Axiom is thus relegated to assertion.
62. Assertions are pattern processes that exist simultaneously as forms.
63. Assertions are thus justified as occurences where the assertive nature of axioms is the occurence of patterns.
64. Truth values, derived from axiomatic patterned proccessive forms, are assertions by degree of occurence.
65. Frameworks of knowledge, derived from axioms, are asserted occurences. Evidence for such frameworks is subject to axiomatic process thus truth value is synonymous to pattern observation.
66. The truth value of an axiom is depth of percieved pattern.
67. Axiomatic pattern progressive forms, axioms within axioms as an axiom, contain potentially infinite axioms by degree of infinite regress within each axiom of an infinite regress.
68. The nature of an axiom applies to all experiential reality by degree of occurence.
69. The definitive degree of an axiom is completeness, the pattern that occurs for what it is and is not.
70. The indefinitive degree of an axiom is incompleteness, the absence of pattern that occurs by necessitated regress.
71. Axioms, by nature of the axiom, are infinite.
72. There is no axiom for what is deemed axiomatic and what is not other than the occurence of the axiom by localization of infinite potential and possible axioms by the axiom itself.
73. Knowledge, derived from axioms, is purely occurence.
74. There is no axiomatic truth value outside of patterns, for the axiomatic notion of truth is derived from nested axioms thus necessitating truth corresponds to gradient patterns where the gradation is subject to the infinite regress of axioms.
75. Patterns occur, occurence is a pattern given the axiomatic nature of occurence is grounded in the recursion of occurence.
76. The occurence of patterns, by degree of axioms, is synonymous to assertion.
77. Truth value is assertion, to assert otherwise is a truth that is asserted.
78. Frameworks are recursive assertions by degree of infinite regress of axiomatic pattern processes as forms, axiomatic form is recursion.
79. No framework is complete.
80. No framework is incomplete.
81. Frameworks are purely occuring patterns where completion and incompletion are meta-patterns of their axiomatic natures.
82. The occurence of patterns is evident only relative to the occurence of other patterns for a pattern is derived from relation to other patterns, otherwise the quality of pattern ceases to be that of pattern and assymetry occurs.
83. Pattern is purely repetition of axioms, thus recursive.
84. Axioms require recursion or assymetry occurs. Assymetry occurs recursively across axioms, thus is a meta-pattern and isomorphic to symmetry.
85. Given the nature of axiomatic patterned process forms there is no axiomatic derived system which is axiomatocally irrational, at the minimum degree it is assymetric.
86. Axiom derived frameworks are justified by occurence, contradiction is an axiom by virtue of occurence.
87. The axiomatic nature of contradiction necessitates contradiction can be observed as a framework.
88. All frameworks derived from axioms are limited by the axioms thus the creation of a framework is to create an incomplete set of axioms by degree of the limits of the framework by finite axioms.
89. Axiomization results in an absence of axioms by degree of the limits of the axioms for the localization of finite axioms from infinite potential axioms excludes potential axioms that where not localized.
90. There is no truly axiomatic framework outside of patterns. Without patterns axioms cease and yet patterns are deemed axiomatic thus necessitating a system requires multiple axioms that are isomorphisms of eachother, ie at the meta-level "axiom" and "pattern" are isomorphisms.
91. Isomorphic nature of axioms necessitates a framework is dualistic in foundation where any framework can be derived from any dualism.
92. Dualism is the process by which frameworks measure reality, thus necessitating axioms as fixed processes within a framework.
93. The foundation of any axiom derived frameworks is the process by which the axioms occur as a pattern by which further axioms are derived and justified. Infinite regress of axiom, and the infinite regress of each relative axiom, is the foundation of a framework itself as infinite regress sustains both itself, through self reference, and there are infinite regresses of infinite regress.
93. Infinite regress of an axiom is an axiom that justifies itself by further infinite regress thus making infinite regress a foundational axiom that justifies itself.
94. Axioms are self justifying by degree of infinite regress necessitating further axioms, any system has an infinite number of foundational axioms by degree of one axiom.
95. Axiomatic patterned processes necessitate a framework's foundations as dynamic. What is an axiomatic foundation at one time is different in another.
96. Axioms are not fixed but are processes, as processes of infinite regress, one axiom effectively can sustain a whole framework by degree of the axioms it contains within regression. To observe one axiom it to observe infinite.
97. Given axioms justifying axioms there is no set limit to the number of axioms within a system given the proof of an axiom is not only an axiom but observes the multidimensional nature of an axiom by degree of an isomorphic variation where proof is just the mirroring of the axiom.
98. There is no proof outside of an axiom that in itself is not an axiom.
99. Proof is isomorphic axioms, variations of the core axiom or axioms. The variation of axiom is an axiom thus necessitating nested axiomatic patterned processes within axiomatic patterned processes where the nesting is an axiomatic patterned process.
100. The infinite regress of axioms is multidimensional by degree of nesting.
101. Infinite regress is a constant axiom, an absolute foundation of the axiom as an axiom within it.
102. Nothing is complete outside an infinite regress, nothing is incomplete outside an infinite regress.
103. Completeness and incompleteness are unnecessary axioms for a framework as an axiom, by degree of necessary relations and regress, is empty in itself thus effectively is always a potential axiom.
104. Axioms are potential other axioms this are axiomatic as potential axioms: ( )●
105. Axiom as potentiality necessitates potentiality underlies patterned processes: (( )●)△
106. Axiomatic potential by means of a patterned process necessitates axiomatic forms are the potential for further axioms: (( )△) □
107. All Axioms are simultaneously actual, by degree of form, and potential, by degree of further form.
108. Axioms as both actual and potential necessitates the axiom as simultaneous meta-axioms, as the axiom of actuality and the axiom of potentiality, where actuality and potentiality are axiomatic as distinctions.
109. An axiom is axioms, axioms are an axiom.
110. Axioms are the degree of measurement according to a unity and by the application of unity, be it a localization of a phenomena or a set of phenomena, an axiom is.
111. Axiomatic expression of unity necessitates an axiom is a process of measurement.
112. The unity of one axiom relative to the unity of another is a unity by degree of a set of axioms as an axiom.
113. The unity of an axiom is a dualism of a localization and set.
114. The relationship of localized axioms is a unified set of axioms.
115. The unified set of axioms is a localized axiom.
116. Localization and set are expressions of how distinctions are actualized, they are the root of an axiom within the axiom as axioms.
117. Context determines the application of how these forms of unity occur, thus axioms are contextualization of experiential reality.
118. By context an axiom is, by axioms there is context, axioms are context.
119. Completeness and incompleteness are axioms thus contexts.
120. Incompleteness as a context requires a context beyond it to make it axiomatic, by said nature of definition incompleteness self-negates and is only valid by self-reference.
121. Completeness follows this same pattern process as incompleteness.
123. Incompleteness ceases by nature of having a complete definition by self-reference and incomplete by degree of regress. Completeness only occurs by degree of absence of incompletion thus necessitating self-referential recursion.
124. There is only completeness by self referential recursion, the term completeness lacks distinction from incompleteness thus nullifying the term leaving only recursion.
125. Completeness and incompleteness are only axioms by degree of assertion thus assumed. A framework strictly is a axiomatic patterned process of form, there is no completion or imcompletion that does not negate itself thus leaving only patterns of form and patterns of function.
126. Some axioms are just paradoxes that mirror foundational processes that reach a logical end.
127. Paradox is the means of completion of a framework and the beginning of a new, paradoxes are potential frameworks.
128. Axioms are paradoxes by degree of being both form and function, actual and potential, one and many.
129. Axioms are non-axiomatic by degree of their limits necessitating potentiality by which they are justified.
130. There is no fully axiomatic framework outside of patterns where patterns are only axiomatic by degree of potentiality.
131. Axioms defined by potentiality define the actual axiom by what it is not, ie the potential. Non-axioms are required.
132. The requirement of non-axioms necessitates a non-axiomatic framework to occur if the non-axiom is to be observed.
133. Frameworks are derived by what they are absent of thus a framework requires its negation by the non-axiom.
134. The negation of a non-axiom is an axiom.
135. Negation is the foundational distinction by which axiomatic patterned processes of forms occur.
136. Negation is a foundational axiom by degree of the non-axiom.
137. Non-axioms are universal by what they are not, axioms are not universal by degree of what they are.
138. Non-axioms are the foundation to axioms as the means by which the axiom is.
++++++
X. An axiom is a set of axioms by degree of the relations that define an axiom thus necessitating an axiom by degree as a pattern through the relations and a process by degree of necessary regress. This patterned process necessitates that axiom as a form by which reality is localizes conceptually amidst all potential and possible axioms.
The symbol ● is an axiom
The symbols ( ) are context by means of relation.
1. We know an axiom: ●
2. This axiom occurs relative to another axiom: ●●
3. This relationship of axioms is an axiom: (●●)●
4. We know an axiom that is but other axioms for the axiom is but self contained by other axioms as the axiom itself: (●)
5. The self contained axiom is but relative to other self contained axioms: (●)(●)
6. The relationship of self contained axioms is an axiom: ((●)(●))●
7. We know an axiom is but other self-contained axioms for the axiom is but self contained by other self contained axioms as the self contained axiom itself: ((●))
8. The self contained axioms as a self contain axiom is but relative to a self contained axiom as self-contained axioms: ((●))((●))
9. The relationship of self contained axioms is but an axiom:
(((●))((●)))●
10. We know an axiom is but other self-contained axioms for the axiom is but self contained by other self contained axioms as but the self contained axiom itself: (((●)))
11. The self-containment of an axiom is infinite, yet this self containment of the axiom is an axiom, an axiom relative to another axiom is but itself: (((●)))●
12. The axiom is a patterned process (Pp) by degree of its inherent relations that are not limited: △
13. The Pp is relative to another Pp: △△
14. The relationship of other patterns processes is but a patterned process (△△) △
15. The patterned process contains itself by other patterned processes: (△)
16. The self contained Pp is but relative to other self contained Pp: (△)(△)
17. The relationship of self contained Ppis a Pp:
((△)(△))△
18. We know a Pp is but other self-contained Pp for the Pp is but self contained by other self contained Pp as the self contained Pp itself: ((△))
19. The self contained Pp as a self contain Pp is but relative to a self contained Pp as self-contained Pp: ((△))((△))
20. The relationship of self contained Pp is but a Pp:
((△))((△))△
21. We know a Pp is but other self-contained Pp for the Pp is but self contained by other self contained Pp as but the self contained axiom itself: (((△)))
22. The self-containment of a Pp is infinite, yet this self containment of the Pp is a Pp, a Pp relative to another Pp is but itself: (((△)))△
23. The patterned process is an axiom by degree of its inherent relations that are not limited: ●
24. The axiom and patterned process are one and the same, isomorphic by nature as one results in the other as variable expressions of an infinite foundation by which both occur.
25. Axioms and patterned processes are self contained by there infinite nature, there completion is the foundation of infinity by which both occur and maintain.
26. The system is complete by degree of the containment of infinite recursion through infinite recursion making infinite recursion a self-evident foundation that justifies itself as complete.
27. Incompleteness is finiteness evidenced by regress.
28. Finiteness allows for infinite regress.
29. Infinite regress is a patterned process through said finiteness
30. A patterned process is finite infinity.
31. Finite infinite occurs recursively.
32. The recursion of infinity, by degree of finite infinities, justifies infinity by degree of a self-referential infinity.
33. Infinity is self contained, as self contained is complete by degree of the axiom and patterned process as one given that infinite recursion occurs recursively through the axiom/Pp thus maintaining its own foundation as infinite recursion. The recursion of infinite regress necessitates infinite regress as self-referencing and self contained thus such a system is complete by evidence of self-reference.
34. If a system is incomplete due to finiteness, and this finiteness contains infinite regress within it, than the system is complete by degree of containing infinity within the finite.
35. The containment of infinities, by degree of the finite, where the finite infinitely regresses, necessitates a recursion of infinities where said system is complete by nature of recursive infinities.
36. The recursion of infinities is but self referential completeness of a system where the infinite recursion of infinities is but the foundations of infinite recursion justifying themselves as complete through the nature of the infinite process where infinity is its own foundation expressed isomorphically thus containing itself.
37. Completeness necessitates an element of finiteness by degree of containment. Incompleteness necessitates a finiteness by degree of perpetual regress. If regress, through recursion, is observed as the foundation and said regress occurs recursively (ie recursion of regresses) then all systems containing such foundations can be observed as simultaneously complete and incomplete.
38. A system that is simultaneously complete and incomplete ceases to be such things by degree of negation, thus a system must be observed as an axiomatic patterned process, patterned thus complete and processing thus incomplete.
39. Completeness and incompleteness synthesize as the axiomatic pattern process. Axioms are the patterns by which frameworks occur thus inevitably processes by degree of regress.
40. Infinity is its own foundational pattern evidenced as axiomatic by degree of the process within process for an infinite regress of axioms necessitates new axioms that each have an infinite regress of justifications. Infinite regress is nested within infinite regress thus making infinity finite, by degree of the form of regress, where furthermore infinity is nested within itself as its own foundation by degree of its finite dimension of expression.
41. Axiomatic patterned processes are self contained: □
42. Self contained processes exist across forms as a form: (□)□
43. All processes are fixed states by means of the continuity of the process thus the process is but a continuity of a specific form of change, this form of change is a form.
44. Forms are processes and by degree of continuity of said process and necessarily static. All processes are finite forms by degree of the form of the process.
45. While a process continues the expression of a variety of forms, the process is a form that underlies said variety of forms.
46. Form is the means by which change occurs, the form of infinite regress makes said form itself, linearistic branching, as infinite.
47. An infinite form necessitates infinity as finite while still necessitating an identity of the infinite for what it is. Axioms are stable processes by which interpretations exist.
48. "Completeness" and "incompleteness" are both axioms and follow said nature of axioms.
49. The axiom of completeness, by degree of the patterned process of axioms, cannot be completely proven as a complete axiom in one respect thus nullifies itself under its own terms as completeness is an incomplete axiom.
50. The axiom of incompleteness, by degree of the patterned process of axioms, can be proven as complete axiom in one respect thus nullifies itself under its own terms as incompleteness is a complete axiom.
51. Completeness and incompleteness exist only in grades, thus are relative. By degree of relativity they sometimes exist and sometimes do not, there universal validity is not stable. Systems derived from such axioms exhibit gradient behavior and cannot be considered universal axioms outside of specific contexts.
52. Axiomatic patterned processes are occurences, self-evidence is not self-evident by degree of its own standards.
53. Axioms are not universal outside of the occurence of form, only form is universally axiomatic by degree of occurence of said form.
54. Occurence is the universal axiom, evidenced by degree of axiomatic patterned processes being only occurences.
55. What is considered axiomatic is only the occurence of patterns for patterns give rise to evidence.
56. No single axiom can be observed as an axiom without corresponding patterns, there is no axiomatic interpretation of axioms outside of patterned processes of recognition.
57. Axioms are incomplete by progress, complete by self-referentiality. This dual nature of axiomatic patterned processes is an axiom within the nature of axioms.
58. "Incomplete" is an axiom.
59. "Complete" is an axiom.
60. "Axiom" is an axiom.
61. Axiom is thus relegated to assertion.
62. Assertions are pattern processes that exist simultaneously as forms.
63. Assertions are thus justified as occurences where the assertive nature of axioms is the occurence of patterns.
64. Truth values, derived from axiomatic patterned proccessive forms, are assertions by degree of occurence.
65. Frameworks of knowledge, derived from axioms, are asserted occurences. Evidence for such frameworks is subject to axiomatic process thus truth value is synonymous to pattern observation.
66. The truth value of an axiom is depth of percieved pattern.
67. Axiomatic pattern progressive forms, axioms within axioms as an axiom, contain potentially infinite axioms by degree of infinite regress within each axiom of an infinite regress.
68. The nature of an axiom applies to all experiential reality by degree of occurence.
69. The definitive degree of an axiom is completeness, the pattern that occurs for what it is and is not.
70. The indefinitive degree of an axiom is incompleteness, the absence of pattern that occurs by necessitated regress.
71. Axioms, by nature of the axiom, are infinite.
72. There is no axiom for what is deemed axiomatic and what is not other than the occurence of the axiom by localization of infinite potential and possible axioms by the axiom itself.
73. Knowledge, derived from axioms, is purely occurence.
74. There is no axiomatic truth value outside of patterns, for the axiomatic notion of truth is derived from nested axioms thus necessitating truth corresponds to gradient patterns where the gradation is subject to the infinite regress of axioms.
75. Patterns occur, occurence is a pattern given the axiomatic nature of occurence is grounded in the recursion of occurence.
76. The occurence of patterns, by degree of axioms, is synonymous to assertion.
77. Truth value is assertion, to assert otherwise is a truth that is asserted.
78. Frameworks are recursive assertions by degree of infinite regress of axiomatic pattern processes as forms, axiomatic form is recursion.
79. No framework is complete.
80. No framework is incomplete.
81. Frameworks are purely occuring patterns where completion and incompletion are meta-patterns of their axiomatic natures.
82. The occurence of patterns is evident only relative to the occurence of other patterns for a pattern is derived from relation to other patterns, otherwise the quality of pattern ceases to be that of pattern and assymetry occurs.
83. Pattern is purely repetition of axioms, thus recursive.
84. Axioms require recursion or assymetry occurs. Assymetry occurs recursively across axioms, thus is a meta-pattern and isomorphic to symmetry.
85. Given the nature of axiomatic patterned process forms there is no axiomatic derived system which is axiomatocally irrational, at the minimum degree it is assymetric.
86. Axiom derived frameworks are justified by occurence, contradiction is an axiom by virtue of occurence.
87. The axiomatic nature of contradiction necessitates contradiction can be observed as a framework.
88. All frameworks derived from axioms are limited by the axioms thus the creation of a framework is to create an incomplete set of axioms by degree of the limits of the framework by finite axioms.
89. Axiomization results in an absence of axioms by degree of the limits of the axioms for the localization of finite axioms from infinite potential axioms excludes potential axioms that where not localized.
90. There is no truly axiomatic framework outside of patterns. Without patterns axioms cease and yet patterns are deemed axiomatic thus necessitating a system requires multiple axioms that are isomorphisms of eachother, ie at the meta-level "axiom" and "pattern" are isomorphisms.
91. Isomorphic nature of axioms necessitates a framework is dualistic in foundation where any framework can be derived from any dualism.
92. Dualism is the process by which frameworks measure reality, thus necessitating axioms as fixed processes within a framework.
93. The foundation of any axiom derived frameworks is the process by which the axioms occur as a pattern by which further axioms are derived and justified. Infinite regress of axiom, and the infinite regress of each relative axiom, is the foundation of a framework itself as infinite regress sustains both itself, through self reference, and there are infinite regresses of infinite regress.
93. Infinite regress of an axiom is an axiom that justifies itself by further infinite regress thus making infinite regress a foundational axiom that justifies itself.
94. Axioms are self justifying by degree of infinite regress necessitating further axioms, any system has an infinite number of foundational axioms by degree of one axiom.
95. Axiomatic patterned processes necessitate a framework's foundations as dynamic. What is an axiomatic foundation at one time is different in another.
96. Axioms are not fixed but are processes, as processes of infinite regress, one axiom effectively can sustain a whole framework by degree of the axioms it contains within regression. To observe one axiom it to observe infinite.
97. Given axioms justifying axioms there is no set limit to the number of axioms within a system given the proof of an axiom is not only an axiom but observes the multidimensional nature of an axiom by degree of an isomorphic variation where proof is just the mirroring of the axiom.
98. There is no proof outside of an axiom that in itself is not an axiom.
99. Proof is isomorphic axioms, variations of the core axiom or axioms. The variation of axiom is an axiom thus necessitating nested axiomatic patterned processes within axiomatic patterned processes where the nesting is an axiomatic patterned process.
100. The infinite regress of axioms is multidimensional by degree of nesting.
101. Infinite regress is a constant axiom, an absolute foundation of the axiom as an axiom within it.
102. Nothing is complete outside an infinite regress, nothing is incomplete outside an infinite regress.
103. Completeness and incompleteness are unnecessary axioms for a framework as an axiom, by degree of necessary relations and regress, is empty in itself thus effectively is always a potential axiom.
104. Axioms are potential other axioms this are axiomatic as potential axioms: ( )●
105. Axiom as potentiality necessitates potentiality underlies patterned processes: (( )●)△
106. Axiomatic potential by means of a patterned process necessitates axiomatic forms are the potential for further axioms: (( )△) □
107. All Axioms are simultaneously actual, by degree of form, and potential, by degree of further form.
108. Axioms as both actual and potential necessitates the axiom as simultaneous meta-axioms, as the axiom of actuality and the axiom of potentiality, where actuality and potentiality are axiomatic as distinctions.
109. An axiom is axioms, axioms are an axiom.
110. Axioms are the degree of measurement according to a unity and by the application of unity, be it a localization of a phenomena or a set of phenomena, an axiom is.
111. Axiomatic expression of unity necessitates an axiom is a process of measurement.
112. The unity of one axiom relative to the unity of another is a unity by degree of a set of axioms as an axiom.
113. The unity of an axiom is a dualism of a localization and set.
114. The relationship of localized axioms is a unified set of axioms.
115. The unified set of axioms is a localized axiom.
116. Localization and set are expressions of how distinctions are actualized, they are the root of an axiom within the axiom as axioms.
117. Context determines the application of how these forms of unity occur, thus axioms are contextualization of experiential reality.
118. By context an axiom is, by axioms there is context, axioms are context.
119. Completeness and incompleteness are axioms thus contexts.
120. Incompleteness as a context requires a context beyond it to make it axiomatic, by said nature of definition incompleteness self-negates and is only valid by self-reference.
121. Completeness follows this same pattern process as incompleteness.
123. Incompleteness ceases by nature of having a complete definition by self-reference and incomplete by degree of regress. Completeness only occurs by degree of absence of incompletion thus necessitating self-referential recursion.
124. There is only completeness by self referential recursion, the term completeness lacks distinction from incompleteness thus nullifying the term leaving only recursion.
125. Completeness and incompleteness are only axioms by degree of assertion thus assumed. A framework strictly is a axiomatic patterned process of form, there is no completion or imcompletion that does not negate itself thus leaving only patterns of form and patterns of function.
126. Some axioms are just paradoxes that mirror foundational processes that reach a logical end.
127. Paradox is the means of completion of a framework and the beginning of a new, paradoxes are potential frameworks.
128. Axioms are paradoxes by degree of being both form and function, actual and potential, one and many.
129. Axioms are non-axiomatic by degree of their limits necessitating potentiality by which they are justified.
130. There is no fully axiomatic framework outside of patterns where patterns are only axiomatic by degree of potentiality.
131. Axioms defined by potentiality define the actual axiom by what it is not, ie the potential. Non-axioms are required.
132. The requirement of non-axioms necessitates a non-axiomatic framework to occur if the non-axiom is to be observed.
133. Frameworks are derived by what they are absent of thus a framework requires its negation by the non-axiom.
134. The negation of a non-axiom is an axiom.
135. Negation is the foundational distinction by which axiomatic patterned processes of forms occur.
136. Negation is a foundational axiom by degree of the non-axiom.
137. Non-axioms are universal by what they are not, axioms are not universal by degree of what they are.
138. Non-axioms are the foundation to axioms as the means by which the axiom is.
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X. An axiom is a set of axioms by degree of the relations that define an axiom thus necessitating an axiom by degree as a pattern through the relations and a process by degree of necessary regress. This patterned process necessitates that axiom as a form by which reality is localizes conceptually amidst all potential and possible axioms.