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It is of special interest within the educational area, especially at the university level, that science and engineering students can assimilate the importance of being able to understand some considerations about the concept of axioms in mathematics, since as many of us know, one of the peculiarities of mathematics is that every process involves a demonstration.

However, in order to demonstrate, mathematics starts from some propositions that do not require demonstration and that are based on the most logical and fundamental principles under which mathematics can have as a starting point and basis for everything that can be demonstrated from these axioms.

Being everything as I have been explaining, then it is convenient to consider the axioms as a fundamental and basic source to build the basis of everything that is mathematics and especially its demonstrations, if we concatenate the demonstrative process we must say that the axioms allow us to perform mathematical demonstrations where the veracity of any information founded on the axioms mentioned is deduced.  

Not only in mathematics we need fundamental principles where we can sow our bases, the important thing is that to consider the axioms as the principle of everything that a priori is demonstrated in mathematics, we must say that the axioms are statements, that is, propositions that do not require demonstration, therefore they are postulates that must be considered as true, under this concept one can fall into an error of wanting to demonstrate an axiom from another axiom, the correct thing is to consider an axiom as something true that does not require demonstration.
   
I make special emphasis on students at university level, since for example when we are in basic education, it is very common that we have heard and seen the so-called Pythagorean theorem, we should also have been told that being a theorem requires demonstration, the demonstration of a theorem is based mainly on the citation of axioms, although it is not the subject of this post, I must say that the immediate consequence of a theorem is called corollary. 

From my experience in the educational area at the university level, I can tell you that my calculus classes were based mainly on demonstrations in which I quoted axioms of incidence that although they are of geometric origin, they are also linked to infinitesimal calculus.  

 Although many branches of mathematics are axiomatized, I can say that in the study of Euclidean geometry is where more application of axioms, theorems and corollaries must be used to demonstrate, for example, the similarity between triangles.

In conclusion, I want to emphasize the importance of the axiomatic study to understand later that they are the fundamental basis for the process of demonstration of many objects of study within classical and modern mathematics. 

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