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An axiom is a proposition that is believed to be true and is used as a starting point for reasoning and proofs. The axioms are generally simple and can be expressed in the form of elementary propositions.
Definition of an axiom
Axioms are used in mathematics, logic, and other areas of rational thought. In mathematics, axioms are propositions that are accepted as true without proof. The axioms of geometry, for example, are propositions that describe the basic properties of a geometric space. In logic, axioms are propositions that are accepted as true without proof. The axioms of logic are usually expressed as logical symbols.
An axiom is not a conjecture !
Axioms in Mathematics
There are a number of different axioms used in mathematics. Some of them well known are:
- The axiom of choice: there exists a function which associates with each non-empty set an element of this set.
- The axiom of parity: every integer is either even or odd.
- The extension axiom or Zermelo-Fraenkel axiom: for any set A, the inclusion relation is an order relation. It basically states that it is enough to check that two sets have the same elements to show that these two sets are equal.
- The axiom of comparison: if A and B are two non-empty finite sets and a ∈ A, b ∈ B, then A < B if and only if | A | < | B |. And so there is an injection from A into B.
If you want to know a lot more, go see Professor Karré's video which details in particular many elements of Zermelo-Fraenkel set theory:
Axioms in Rational Thought
In areas of rational thought, axioms are fundamental principles that cannot be demonstrated. Axioms are generally accepted as true by the majority of people. The axioms of morality are fundamental principles that describe what is right or wrong. As for those of religion, they are fundamental principles that describe what is sacred or profane.
Axioms in Euclidean Geometry
The 5 postulates of Euclid are the axioms of Euclidean geometry:
- There are dots.
- There are rights.
- Parallel lines meet only at the infinity point.
- Right angles have a measure of 90°.
- The measures of the opposite angles of the same triangle are equal.