“An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments….
“As used in mathematics, the term axiom is used in two related but distinguishable senses: “logical axioms” and “non-logical axioms”.” Logical axioms seems to be about things that are always true, such as tautologies. Non-logical axioms seem to be about things which are true in certain situations, such as a + b = b + a. (“Axiom,” Wikipedia, retrieved 10/17/20, emphasis added)
When referring to non-logical axioms, the words “axiom”, “postulate”, and “assumption” are interchangeable. (Ibid)
“In the modern understanding, a set of axioms is any collection of formally stated assertions from which other formally stated assertions follow — by the application of certain well-defined rules. In this view, logic becomes just another formal system. A set of axioms should be consistent; it should be impossible to derive a contradiction from the axiom. A set of axioms should also be non-redundant; an assertion that can be deduced from other axioms need not be regarded as an axiom.”
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Disclaimer:
I am not a professional in this field, nor do I claim to know all of the jargon that is typically used in this field. I am not summarizing my sources; I simply read from a variety of websites until I feel like I understand enough about a topic to move on to what I actually wanted to learn. If I am inaccurate in what I say or you know a better, simpler way to explain a concept, I would be happy to hear from you :).
