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List of axioms
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| ARTICLES RELATED TO List of axioms | | | |  List of axioms: Encyclopedia II - Laws of Form - The primary algebraGiven any valid primary arithmetic expression, insert into one or more locations any number of Latin letters, with or without numerical subscripts; the result is a pa formula. Variable is the conventional name for a letter of this sort. A pa variable signifies that location where one can write either primitive value. Multiple instances of the same variable stand for multiple locations where the same primitive value must be written.
The sign '=' denotes that what appears to the left and right of = are logically equivalent ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The primary algebra |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - Applying the form to Boolean algebra and logic The marked and unmarked state can be read as the Boolean values 1 and 0, or as True and False. The first reading transforms the pa into a notation for 2; the second into a notation for sentential logic. Since the Cross also denotes crossing the boundary of a distinction, it may be read as Not. This may seem odd at first blush; however True is equivalent to Not False and both ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - Applying the form to Boolean algebra and logic |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - Resonances in religion philosophy and science The mathematical and logical content of LoF is wholly consistent with a secular point of view. Nevertheless, LoF's "first distinction", and the Notes to its chapter 12, bring to mind the following landmarks in religious belief, and in philosophical and scientific reasoning, presented in rough historical order:
Vedic, Hindu and Buddhist: Related ideas can be noted in the ancient Vedic Upanishads, which form the monastic foundations of Hinduism and later Buddhism. As stated in the Aitareya Upanishad ("The Micr ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - Resonances in religion philosophy and science |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - Related work Charles Peirce (1839-1914) anticipated the pa in three veins of work:
Two papers he wrote in 1886 proposed a logical algebra employing "one single symbol", the streamer-cross that is almost identical to the Cross of LoF. An excerpt from one of these papers was published in 19761, but they were not published in full until 19932,3
A closely related notation appears in an encyclopedia article he published in 1902, reprinted in vol. 4 of his Collected Papers, paragraphs 378- ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Footnotes Read more here: » Laws of Form: Encyclopedia II - Laws of Form - Related work |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - Resonances in religion philosophy and science The mathematical and logical content of LoF is wholly consistent with a secular point of view. Nevertheless, LoF's "first distinction", and the Notes to its chapter 12, bring to mind the following landmarks in religious belief, and in philosophical and scientific reasoning, presented in rough historical order:
Vedic, Hindu and Buddhist: Related ideas can be noted in the ancient Vedic Upanishads, which form the monastic foundations of Hinduism and later Buddhism. As stated in the Aitareya Upanishad ("The Micr ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - Resonances in religion philosophy and science |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - Related work Charles Peirce (1839-1914) anticipated the pa in three veins of work:
Two papers he wrote in 1886 proposed a logical algebra employing "one single symbol", the streamer-cross that is almost identical to the Cross of LoF. An excerpt from one of these papers was published in 19761, but they were not published in full until 19932,3
A closely related notation appears in an encyclopedia article he published in 1902, reprinted in vol. 4 of his Collected Papers, paragraphs 378- ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - Related work |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The book There are several editions of LoF, the first in 1969, the most recent (a German translation) in 1997. The mathematics fills only about 55pp and is not difficult. But LoF's mystical and declamatory prose style, and its love of paradox, make it a challenging read for mathematicians and non-mathematicians alike. In this and other respects, Spencer-Brown was much influenced by Wittgenstein and R. D. Laing. At the same time, LoF also echoes a number of themes from the work of Charles Peirce, Bert ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The book |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The Form The symbol:
also called the Mark or Cross, is the essence of the Laws of Form.
In Spencer-Brown's initimable and enigmatic fashion, the Mark symbolizes the root of cognition, i.e., the dualistic Mark indicates the capability of differentiating a "this" from a "that."
In LoF, a Cross denotes the drawing of a "distinction", and can be thought of as signifying the following, all at once:
The act of drawing a boundary around something, thus separa ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The Form |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The primary arithmetic and its axioms Begin with the void, the only "atomic" expression. Then posit two inductive rules:
Given any expression, a Cross can be written over it;
Any two expressions can be concatenated.
Thus the syntax of the primary arithmetic, a Dyck language of order 1 with a null alphabet, and the simplest instance of a context-free language in the Chomsky hierarchy. LoF often uses the phrase calculus of indications in place of "primary arithmetic".
The primary arithmetic and algebra begin with a definitio ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The primary arithmetic and its axioms |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - The primary algebra Given any valid primary arithmetic expression, insert into one or more locations any number of Latin letters, with or without numerical subscripts; the result is a pa formula. Variable is the conventional name for a letter of this sort. A pa variable signifies that location where one can write either primitive value. Multiple instances of the same variable stand for multiple locations where the same primitive value must be written.
The sign '=' denotes that what appears to the left and right of = are logically equivalent ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - The primary algebra |
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| | | | | List of axioms: Encyclopedia II - Laws of Form - Applying the form to Boolean algebra and logic The marked and unmarked state can be read as the Boolean values 1 and 0, or as True and False. The first reading transforms the pa into a notation for 2; the second into a notation for sentential logic. Since the Cross also denotes crossing the boundary of a distinction, it may be read as Not. This may seem odd at first blush; however True is equivalent to Not False and both True and Not False are represented the same way — with a Cross.
= ...
See also:Laws of Form, Laws of Form - The book, Laws of Form - The Form, Laws of Form - The primary arithmetic and its axioms, Laws of Form - The notion of 'canon', Laws of Form - The primary algebra, Laws of Form - Applying the form to Boolean algebra and logic, Laws of Form - An example calculation, Laws of Form - A technical digression, Laws of Form - Resonances in religion philosophy and science, Laws of Form - Related work, Laws of Form - Bibliography Read more here: » Laws of Form: Encyclopedia II - Laws of Form - Applying the form to Boolean algebra and logic |
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| | | | | List of axioms: Encyclopedia II - List of basic philosophical topics - Basic philosophical concepts A priori -- A posteriori -- abduction -- absolute -- Aesthetics -- Age of Enlightenment -- Agnosticism -- Altruism -- Ambiguity -- American Philosophical Association -- analytic philosophy -- analogy -- Aristotle -- atheism -- awareness -- axiom --
being -- belief -- Buddhist philosophy --
causality -- Cogito, ergo sum -- consciousness -- cosmogony -- cosmology -- continental philosophy -- creation --
deconstruction -- deduction -- determinism -- dialectics -- dualism --
Eastern philosophy -- emergent philosophies -- epistemic justification -- epistemology -- ethical re ...
See also:List of basic philosophical topics, List of basic philosophical topics - General philosophical topics, List of basic philosophical topics - Branches of philosophy, List of basic philosophical topics - Subdisciplines of philosophy, List of basic philosophical topics - Philosophical movements, List of basic philosophical topics - Philosophical movements of the ancient world, List of basic philosophical topics - Philosophical movements of the modern world, List of basic philosophical topics - Influential philosophers, List of basic philosophical topics - Basic philosophical concepts, List of basic philosophical topics - The Isms doctrines schools and principles of philosophy, List of basic philosophical topics - Philosophical topics by region, List of basic philosophical topics - Potential emergent philosophies, List of basic philosophical topics - General philosophy lists, List of basic philosophical topics - General philosophy topics, List of basic philosophical topics - General online philosophy resources Read more here: » List of basic philosophical topics: Encyclopedia II - List of basic philosophical topics - Basic philosophical concepts |
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| | | | | List of axioms: Encyclopedia II - Logical argument - The mathematical paradigm In mathematics, an argument can be formalized using symbolic logic. In that case, an argument is seen as an ordered list of statements, each one of which is either one of the premises or derivable from the combination of some subset of the preceding statements and one or more axioms using rules of inference. The last statement in the list is the conclusion. Most arguments used in mathematical proof are rigorous, but not formal. In fact, strictly formal proofs of all but the most trivial assertions are extremely hard to c ...
See also:Logical argument, Logical argument - Overview, Logical argument - Argument validity, Logical argument - The mathematical paradigm, Logical argument - Theories of arguments, Logical argument - Argumentative dialogue, Logical argument - Other theories Read more here: » Logical argument: Encyclopedia II - Logical argument - The mathematical paradigm |
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| | | | | List of axioms: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory The important idea of Cantor's, which got set theory going as a new field of study, was to define two sets A and B to have the same number of members (the same cardinality) when there is a way of pairing off members of A exhaustively with members of B. Then the set N of natural numbers has the same cardinality as the set Q of rational numbers (they are both said to be countably infinite), even though N is a proper subset of Q. On the other hand, the set R of real numbers d ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - The origins of rigorous set theory |
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| | | | | List of axioms: Encyclopedia II - Axiomatic set theory - Axioms for set theory The axioms for set theory now most often studied and used, although put in their final form by Skolem, are called the Zermelo-Fraenkel set theory (ZF). Actually, this term usually excludes the axiom of choice, which was once more controversial than it is today. When this axiom is included, the resulting system is called ZFC.
An important feature of ZFC is that every object that it deals with is a set. In particular, every element of a set is itself a set. Other familiar mathematical objects, s ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Axioms for set theory |
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| | | | | List of axioms: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets In 1917, Dmitry Mirimanov (also spelled Mirimanoff) introduced the concept of well-foundedness:
a set, x0, is well founded iff it has no infinite descending membership sequence:
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In ZFC, there is no infinite descending ∈-sequence by the axiom of regularity (for a proof see Axiom of regularity). In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC- (that is, ZFC wit ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Well-foundedness and hypersets |
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| | | | | List of axioms: Encyclopedia II - Axiomatic set theory - Objections to set theory Since its inception, there have been some mathematicians who have objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set th ...
See also:Axiomatic set theory, Axiomatic set theory - The origins of rigorous set theory, Axiomatic set theory - Axioms for set theory, Axiomatic set theory - Independence in ZFC, Axiomatic set theory - Set theory ZFC foundations for mathematics, Axiomatic set theory - Well-foundedness and hypersets, Axiomatic set theory - Objections to set theory Read more here: » Axiomatic set theory: Encyclopedia II - Axiomatic set theory - Objections to set theory |
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| | | | | List of axioms: Encyclopedia II - Theory - Mathematics In mathematics, the word theory is used informally to refer to certain distinct bodies of knowledge about mathematics. This knowledge consists of axioms, definitions, theorems and computational techniques, all related in some way by tradition or practice. Examples include group theory, set theory, Lebesgue integration theory and field theory.
The term "theory" also has a formal usage in mathematics, particularly in mathematical logic and model theory. A theory in this sense is a set of statements closed under certain rul ...
See also:Theory, Theory - Etymology, Theory - Science, Theory - Models, Theory - Types of theories, Theory - Further explanation of a scientific theory, Theory - Characteristics, Theory - Mathematics, Theory - Other fields, Theory - List of famous theories, Theory - Reference Read more here: » Theory: Encyclopedia II - Theory - Mathematics |
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| | | | | | List of axioms: Encyclopedia II - Naive set theory - Specifying sets The simplest way to describe a set is to list its elements between curly braces. Thus {1,2} denotes the set whose only elements are 1 and 2. (See axiom of pairing.) Note the following points:
Order of elements is immaterial; for example, {1,2} = {2,1}.
Repetition (multiplicity) of elements is irrelevant; for example, {1,2,2} = {1,1,1,2} = {1,2}.
(These are consequences of the definition of equality in the previous section.)
This notation can be informally abused by saying something like {dogs} to indicate the set of all dogs, but this example would usually be read by mathematicians as ...
See also:Naive set theory, Naive set theory - Introduction, Naive set theory - Sets membership and equality, Naive set theory - Specifying sets, Naive set theory - Subsets, Naive set theory - Universal sets and absolute complements, Naive set theory - Unions intersections and relative complements, Naive set theory - Ordered pairs and Cartesian products, Naive set theory - Some important sets, Naive set theory - Paradoxes, Naive set theory - Footnote Read more here: » Naive set theory: Encyclopedia II - Naive set theory - Specifying sets |
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| | | | | List of axioms: Encyclopedia II - Unifying theories in mathematics - Mathematical theories The term theory is used informally within mathematics to mean a self-consistent body of definitions, axioms, theorems, examples, and so on. (Examples include group theory, Galois theory, control theory, and K-theory.) In particular there is no connotation of hypothetical. Thus the term unifying theory is more like sociological term used to study the actions of mathematicians. It may assume nothing conjectural, that would be analogous to an undiscovered scientific link. There is really no cognate within mathematics to ...
See also:Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory Read more here: » Unifying theories in mathematics: Encyclopedia II - Unifying theories in mathematics - Mathematical theories |
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| | | | | List of axioms: Encyclopedia II - Unifying theories in mathematics - Through-axiomatisation Early in the 20th century, parallel to the development of mathematical logic as a stand-alone branch of mathematics, many parts of mathematics began to treated by delineating useful sets of axioms and then studying their consequences. Thus for example the studies of "hypercomplex numbers", popular at the turn of the century, were put onto an axiomatic footing as branches of ring theory (in this case, sp ...
See also:Unifying theories in mathematics, Unifying theories in mathematics - Mathematical theories, Unifying theories in mathematics - Geometrical theories, Unifying theories in mathematics - Through-axiomatisation, Unifying theories in mathematics - Bourbaki, Unifying theories in mathematics - Category theory as a rival, Unifying theories in mathematics - Uniting theories, Unifying theories in mathematics - Reference list of major unifying concepts, Unifying theories in mathematics - Recent developments in relation with modular theory, Unifying theories in mathematics - Isomorphism conjectures in K-theory Read more here: » Unifying theories in mathematics: Encyclopedia II - Unifying theories in mathematics - Through-axiomatisation |
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