UNDER CONSTRUCTION
In mathematics, the axiomatic system refers to the statements and rules used to develop and prove theorems. Explore the definition and properties of the axiomatic system, including consistency, independence, and completeness. Understand how an axiom compares to an axiomatic system. [1]
What exactly is an axiomatic system? I know it sounds like a big word for a complicated system, but it’s actually not all that complicated. Defined, an axiomatic system is a set of axioms used to derive theorems. What this means is that for every theorem in math, there exists an axiomatic system that contains all the axioms needed to prove that theorem. An axiom is a statement that is considered true and does not require a proof. It is considered the starting point of reasoning. Axioms are used to prove other statements. They are basic truths. For example, the statement that all right angles are equal to each other is an axiom and does not require a proof. We know that all right angles are equal to each other and we do not argue that point. Instead, we use this information to prove other things. A collection of these basic, true statements forms an axiomatic system. [1]
References
[1] “The Axiomatic System: Definition & Properties”. 2024. study.com. Accessed March 9. https://study.com/academy/lesson/the-axiomatic-system.html.
Additional References
“Axiom.” 2024. Wikipedia. Wikimedia Foundation. February 13. https://en.wikipedia.org/wiki/Axiom.
“Axiomatic System2.” 2014. SlideShare. Slideshare. February 26. https://www.slideshare.net/JenniferBunquin/axiomatic-system2.
Deductive reasoning takes place in the context of an organized logical structure called an axiomatic ( or deductive) system. One of the pitfalls of working with a deductive system is too great a familiarity with the subject matter of the system. We need to be careful with what we are assuming to be true and with saying something is obvious while writing a proof.
“Axioms and Proofs: World of Mathematics.” 2024. Mathigon. Accessed March 9. https://mathigon.org/world/Axioms_and_Proof.
One interesting question is where to start from. How do you prove the first theorem, if you don’t know anything yet? Unfortunately you can’t prove something using nothing. You need at least a few building blocks to start with, and these are called Axioms.
Mathematicians assume that axioms are true without being able to prove them. However this is not as problematic as it may seem, because axioms are either definitions or clearly obvious, and there are only very few axioms. For example, an axiom could be that a + b = b + a for any two numbers a and b.
Axioms are important to get right, because all of mathematics rests on them. If there are too few axioms, you can prove very little and mathematics would not be very interesting. If there are too many axioms, you can prove almost anything, and mathematics would also not be interesting. You also can’t have axioms contradicting each other.
Mathematics is not about choosing the right set of axioms, but about developing a framework from these starting points. If you start with different axioms, you will get a different kind of mathematics, but the logical arguments will be the same. Every area of mathematics has its own set of basic axioms.
Videos
Join us on an enlightening journey as we delve into the intricate world of axiomatic systems, the fundamental framework that underpins logic and mathematics. Discover the key principles and principles that shape this fascinating discipline and unlock the secrets of logic and mathematical reasoning.
The featured image on this page is from the aXiomatic website.
Mathematical Mysteries 