In order to fully understand the meaning of the term axiom, the first thing to do is discover what its etymological origin is. In this case, we can state that it is a word that derives from the Greek, more specifically from the word “axiom”. This can be translated as “authority”.
It must be stated that this Latin term was formed from the sum of two clearly delimited components:
– “Axios”, which is equivalent to “valued” or “worthy”.
-The suffix “-ma”, which is used to indicate “result of an action”.
An axiom is a proposition that, due to the degree of evidence and certainty it exhibits, is admitted without proof. In the field of mathematics, an axiom is called a fundamental principle that cannot be proven but is used for the development of a theory.
At a general level it can be said that an axiom is an expression that is accepted or approved regardless of the absence of a proof of its postulate. It is a proposition that cannot be deduced from others: it is the first step in the demonstration of other formulas from a deductive process.
It can be said that an axiom is a postulate that, within the framework of a deduction, allows us to reach a conclusion. This is because the axiom qualifies itself as true even without proof, and allows other propositions to be inferred by deduction that are consistent in this framework.
Continuing with this line of thought, it can be said that the propositions of a theory are inferred from the initial axioms. These axioms are considered true in all possible scenarios, regardless of any interpretation or the adoption of any value.
An axiomatic system is called the series of axioms that, through deductions, is used to prove theorems. An example of an axiomatic system is that used by Euclid, who deduced his theorems of geometry from a set of axioms.
No less important is to establish the existence of what has been called the axiom of choice. This term is used in the field of mathematics, more specifically within what is known as Set Theory. What it determines is that in a family of non-empty sets disjoint two by two, the existence of a set that contains an element belonging to each of them occurs.
Many are the scientists and mathematicians who have not hesitated to work on this axiom. This would be the case, for example, of the American mathematician Paul J. Cohen or the illustrious mathematician Kurt Gödel. However, despite all the work carried out in this regard, there is still no agreement on it, that is, it generates much controversy among experts in the aforementioned field.