11 octobre 2022

Definition of an Axiomatic System

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A common attitude to the axiomatic method is logicism. In their book Principia Mathematica, Alfred North Whitehead and Bertrand Russell tried to show that all mathematical theories can be reduced to a collection of axioms. More generally, the reduction of a set of statements to a specific collection of axioms underlies the mathematician`s research program. This was very important in twentieth-century mathematics, especially in subjects based on homological algebra. A model for an axiomatic system is a well-defined set that assigns meaning to the indefinite terms represented in the system that is correct with the relationships defined in the system. The existence of a concrete model proves the coherence of a system [controversial – discuss]. A model is said to be concrete when the assigned meanings are real-world objects and relationships [clarification required], as opposed to an abstract model based on other axiomatic systems. An axiomatic system is a set of axioms or statements on indefinite terms. You can create proofs and theorems from axioms. Logical arguments are created from axioms. Not all coherent sets of statements can be captured by a writable collection of axioms. In recursion theory, a collection of axioms is called recursive when a computer program can detect whether a given statement in the language is a theorem. Gödel`s first incompleteness theorem then tells us that there are certain coherent sentences without recursive axiomatization.

Typically, the computer can recognize axioms and logical rules to derive theorems, and the computer can recognize whether a proof is valid, but determine whether a proof exists for a statement can only be solved by « waiting » for the proof or rebuttal to be generated. The result is that you will not know which statements are theorems and the axiomatic method collapses. An example of such a theorem is natural number theory, which is only partially axiomatized by Peano`s axioms (see below). Other challenges to the so-called self-test of axioms arose from the fundamentals of real analysis, Georg Cantor`s set theory, and the failure of Frege`s work on fundamentals. Russell was able to derive a paradox – a kind of contradiction – from Frege`s axioms for set theory, thus showing that Frege`s axiomatic system was not coherent, which showed that the alleged self-proof of Frege`s axioms was false. As an example, consider the following axiomatic system, based on first-order logic with additional semantics of countable addition axioms (these can easily be formalized as a scheme of axioms): It is easy to see that the axiomatic method has limitations outside of mathematics and set theory. For example, in political philosophy, axioms that lead to unacceptable conclusions are likely to be categorically rejected; so no one really agrees with version 1 above. In mathematics and set theory, an axiomatic system is a set of specified axioms from which some or all of these axioms can be used in conjunction with derivation rules or procedures to logically derive theorems. A mathematical theory or set theory consists of an axiomatic system and all the theorems derived from it.

An axiomatic system, which is fully described, is a special type of formal system; Typically, however, the effort of full formalization results in diminishing returns on safety and a lack of legibility for humans. Therefore, the discussion of axiomatic systems is usually only semi-formal. A formal theory usually means an axiomatic system that has been formulated, for example, in model theory. A formal proof is a complete representation of an overall mathematical or theoretical proof within a formal system. Mathematical methods evolved to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without applying the axiomatic method. « Axiom System. » dictionary Merriam-Webster.com, Merriam-Webster, www.merriam-webster.com/dictionary/axiom%20system. Retrieved 11 October 2022. This approach, in which the axioms should be obvious and therefore undeniable, was swept away during the nineteenth century. An important episode was the development of non-Euclidean geometry, based on the negation of Euclid`s parallel postulate (or axiom). It has been found that coherent geometries can be constructed by denying this postulate, taking as an axiom that more than one parallel to a given line can be drawn through a point outside that line, or another axiom that no parallel can be established – both resulting in different and coherent geometric systems that may or may not be applicable to an experienced world. In order to formulate definitions and statements in such a way that any new term can be formally eliminated by the previously introduced terms, primitive terms (axioms) are needed to avoid infinite regressions. This way of doing mathematics is called the axiomatic method.

[4] This is the « parallel postulate, » but it is also a new version of the fifth axiom. The reason for the controversy over the fifth axiom is that axiomatic systems usually meet three conditions or have three properties. For an axiomatic system to be valid, from our robot paths to Euclid, the system must have only one property: coherence. Informally, this infinite set of axioms asserts that there are an infinity of different elements. However, the concept of an infinite set cannot be defined in the system – let alone the cardinality of such a set. A model is said to be concrete when the assigned meanings are real-world objects and relationships, as opposed to an abstract model based on other axiomatic systems. The first axiomatic system was Euclidean geometry. Two models are said to be isomorphic when a one-to-one match can be found between their elements in a way that preserves their relationship. [3] An axiomatic system in which each model is isomorphic to another is called categorical (sometimes categorical).

The property of categoricality ensures the completeness of a system, but the reverse is not true: completeness does not guarantee the categoricalness of a system, since two models can differ in properties that cannot be expressed by the semantics of the system. An axiomatic system is said to be coherent when there is no contradiction. That is, it is impossible to derive both a statement and its negation from the axioms of the system. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would prove any claim (explosion principle). An axiomatic system must have coherence (an internal logic that is not inherently contradictory). It is preferable that it also has independence, in which the axioms are independent of each other; You can`t get one axiom from another. All axioms are fundamental truths that do not depend on each other for their existence. They may refer to indefinite terms, but they do not come from each other. Zermelo–Fraenkel set theory, the result of the axiomatic method applied to set theory, allowed the « correct » formulation of the problems of set theory and avoided the paradoxes of naïve set theory.

One of these problems was the continuum hypothesis. Zermelo–Fraenkel set theory, including the historically controversial axiom of selection, is usually abbreviated to ZFC, where « C » stands for « choice. » Many authors use ZF to refer to the axioms of Zermelo–Fraenkel set theory, excluding the selection axiom. [5] Today, ZFC is the standard form of axiomatic set theory and, as such, the most common basis of mathematics. In an axiomatic system, an axiom is said to be independent if it cannot be proven or refuted by other axioms of the system. A system is said to be independent when each of its underlying axioms is independent. Unlike consistency, independence is not a necessary requirement for a functional axiomatic system – although it is generally sought to minimize the number of axioms in the system. In mathematics, axiomatization is the process of taking a set of knowledge and working backwards toward one`s axioms. It is the formulation of a system of statements (i.e. axioms) that connect a number of primitive terms – so that a coherent set of statements can be derived from these statements. After that, the proof of each claim should in principle be traced back to these axioms.

You can create your own artificial axiomatic system, like this one: the most important criterion for evaluating an axiomatic system is the consistency of that particular system. Inconsistency in an axiomatic system is generally considered a fatal flaw for that system. Independence is also a desirable quality, but its deficiency is not a fatal mistake. Lack of independence means that the system has redundancy in its axioms, which means that one or more of its axioms are not needed. This is generally considered a mistake, because reducing the number of axioms of a system to the minimum required to derive all the necessary or desired theorems of that system is considered a virtue, since axioms are not proven and not provable; Having as few as possible means that as few unproven assumptions as possible are made in this system.

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