The postulate that allows a conclusion to be reached from a deduction is called an axiom.
In order to fully understand the meaning of the term axiom , the first thing to do is discover its etymological origin. In this case, we can state that it is a word that derives from 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”.
What is an axiom
An axiom is a proposition that, due to the degree of evidence and certainty it exhibits, is admitted without demonstration . In the field of mathematics, an axiom is a fundamental principle that cannot be demonstrated but is used to develop a theory.
At a general level it can be said that an axiom is an expression that is accepted or approved beyond the absence of a demonstration of its postulate. It is a proposition that is not deduced from others: it is the first step in demonstrating other formulas from a deductive process .
A postulate that leads to a conclusion
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 demonstration, and allows other propositions that are coherent in this framework to be inferred through deduction.
Continuing with this line of thought, it can be stated that the propositions of a theory are inferred from the initial axioms. These axioms are considered true in all possible scenarios, beyond any interpretation or the adoption of any value.
The axiom is considered true despite the absence of proof.
The series of axioms that, through deductions, serves to prove theorems is called an axiomatic system . An example of an axiomatic system is that used by Euclid , who deduced his theorems of geometry from a set of axioms.
Axiom of choice
No less important is establishing 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 two-by-two disjoint non-empty sets, the existence of a set that contains an element belonging to each of them occurs.
There are numerous scientists and mathematicians who have not hesitated to work with respect to that aforementioned 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 a lot of controversy among experts in the aforementioned field.