Axiom . Statement that was considered in the past, “evident” and accepted without requiring prior proof. In a deductive system it is any proposition not deduced (from others), but rather constitutes a general rule of logical thought by dialectical contradiction to the theorems. [1] . The second step in forming an axiom system consists of making a list of all propositions for which no proof is given. These propositions are the postulates or axioms . [2]
Summary
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- 1 Etymology
- 2 Definition
- 3 Concepts
- 4 In Logic
- 5 Logical Criterion
- 6 Characteristics of Axioms
- 7 Axiomatic systems
- 8 Axiom of syllogism
- 9 Examples in mathematics
- 10 References
- 11 Sources
Etymology
The word axiom comes from the Greek αξιωμα, which means “what seems fair” or what is considered evident and without the need for proof. The word comes from the Greek αξιοειν (axioein) which means “to value”, which in turn comes from αξιος (axios) which means “valuable” or “worthy”. Among the ancient Greek philosophers, an axiom was that which appeared to be true without any need for proof.
Definition
In mathematics , an axiom is a proposition, as part of a basic theoretical corpus, for the convenience of initial structuring, it is accepted without proof, as a starting point to prove other results generically known as theorems. Traditionally, the axioms were chosen from those considered “evident affirmations”, as stated in Euclid’s “Elements”.
In mathematics two types of propositions are distinguished: axioms and theorems, these in turn are considered lemmas, short theorem that gives introduction; and a corollary that is an immediate consequence. Currently in disuse, scolium, a small theorem as a result linked to another of more importance
In Mathematical Logic , a postulate is a proposition, not necessarily self-evident: a well-formed Formula of a formal Language used in a Deduction to arrive at a Conclusion.
Concepts
The axiom is one of the fundamental concepts of the way of knowing that we call Scientific way of knowing or acquisition of scientific knowledge. Epistemology deals with the analysis of this way of knowing .
The concept of axiom has not remained unchanged throughout history, but has been modified as a consequence of our greater understanding of the possibilities of knowing and the scope of scientific knowledge itself.
The axiom is a Primitive proposition of a Scientific System, that is, a proposition that is admitted without proof. All the other propositions of the scientific system are rigorously deduced from the Set of axioms . In addition to the axioms themselves, the axiom system consists of primitive terms and rules. The primitive terms lack definition; from them all other terms are defined.
The Rules are of two types, those of formation and those of transformation; these are sometimes also called Inference. The training rules are like the grammar of the scientific system in question: they tell us what a meaningful proposition is within the system. The rules of transformation or inference tell us how to obtain or deduce new propositions from already possessed propositions.
In Logic
The Logic of the axiom is to start from a Premise qualified by itself (the axiom) and to infer on it, other Propositions by means of the Deductive Method, obtaining conclusions consistent with the axiom. Axioms have to fulfill only one requirement: from them, and from Rules of inference, all other propositions of a given theory must be deduced.
Logic Criterion
For a hundred years, logicians have doggedly pursued the axiomatization of their science, a task that in the Middle Ages had been approached rather partially. The Apotheosi of axiom systems is Mathematical Principle of Whitehead and Russell. There from a few axioms one tries to deduce all logic and mathematics, because following the German logician Gottlob Frege, Whitehead and Russell believed that mathematics was simply a part of logic.
There were several subsequent modifications to Principia Matemática; for example, David Hilbert showed how it is possible to do without one of the five axioms of the Propositional Logic of Mathematical Principle, which turned out not to be independent of the others.
Lukasiewicz, as a result of a meditation on Aristotle’s Peri Hermeneias , described a system in which Propositions, in addition to being true or false, could be indeterminate; in technical terms he built a trivalent Logic instead of Bivalent. Authors like Post wrote about logics of many truth values, that is, Polyvalent. Other logicians experimented with systems that did not include the concept of negation.
With all this the same concept of axiom is modified. In the case of Mathematical Principia, the axioms are not what is better known than the conclusions. Mathematical relationships, for example, 3 + 5 = 8, are much more obvious than the definition of number or the propositions about sets.
Likewise, the principle of non-contradiction is more evident than the so-called Summation Principle, although formally the principle of non-contradiction is not axiomatic, but derived from Principia Matemática. The axiom has become a means of intellectual economy. A kind of logical game is made in which the aim is to obtain as many conclusions as possible from the fewest possible principles.
Characteristics of Axioms
Axioms are certain formulas in a Language that are universally valid, that is, formulas that are satisfied by any structure and by any variable Function . In colloquial terms, they are statements that are true in any possible world, under any possible interpretation and with any assignment of values. A minimum Set of Tautologies is usually taken as axioms that are sufficient to prove a theory.
Axiomatic systems
Kurt Göde demonstrated in the mid- 20th century that axiomatic systems of a certain complexity, however defined and consistent, have serious limitations. In any system of a certain complexity, there will always be a proposition P that is true, but not provable. In fact, Gödel proves that, in any formal System that includes Arithmetic , a Proposition P can be formed that affirms that this statement is not demonstrable.
Alogy of syllogism
Basic principle of syllogism; Aristotle formulated it as follows:
“” When something is attributed to a thing as a subject, everything that is said about the predicate will also be said about the subject. ”
Instead of the words “attributed”, Aristotle often used the term “inherent” and considered the expressions “A attributed to B” and “B contained in A” to be equivalent. In this way, the axiom of the syllogism is susceptible of interpretation both for its content (intensive) and for its extension (extensive). In traditional formal logic, the meaning of the syllogism axiom is revealed by reducing all syllogisms to those of the first figure ( Syllogistics ). In modern formal logic, the problem concerning the axiom of syllogism is solved in the context of a larger problem, that of the axiomatization of syllogistics.
Examples in mathematics
- For the definition of the system of natural numbers, the Italian mathematician Giusseppe Peano proposed two systems, somewhat parallel. In one case it included 0, in the other it included 1. [3] [4]
- For the definition of real numbers, Richard Dedekind proposed an axiomatic system, focused on three aspects: the algebraic properties on the two operations of addition and multiplication. The order properties based on the relation <= and its links with addition and multiplication; and the topological properties on continuity, specifically, in the axiom of the supreme. [5]
- An axiomatic presentation on the group structure is that of AG Kurosch through the associativity of the internal operation in G, the existence of solutions for ax = b, as well as ya = b. [6]
