What’s an Axiom?

Based on any sort of mathematical knowledge I have, I imagine that an axiom is simply something to believe in relation to mathematics. When something cannot be proved, but seems to always be true, it is assumed to be true. This is what I have believed to be an axiom. Anything I did know essentially came from studying in a course at Grand Valley, and axioms were heavily used in proofs.

Axioms are actually used universally in logic. They stem from our basic sense of reasoning, and are used in the beginning of logic thoughts or arguments. An axiom is specifically a premise that is accepted as a truth since it is very evident. Axioms came from the Greeks, in part to their development of proofs and logic. Euclid was essential in the development of axioms, and deduced many ideas that developed into axioms that believed to always be true without proof. In Euclidian geometry, there are five basic axioms:

- Given any two points, they can be joined by exactly one line.
- Given any finite, non-zero length line segment, it can be extended infinitely into

exactly one line. - Given any line segment, there is exactly one circle with one endpoint of the segment as the center, and with the other endpoint on the circle.
- All right angles are equivalent modulo translation, rotation, and mirroring.
- Given a line l and a point p which is not on l, there is exactly one line that passes through

p but never intersects l.

All other ideas were meant to proven.

Most axioms were used the different kinds of sciences when they were developed, but mathematics has altered that in the last 200 years. Now there are 9 axioms by way the Zermelo-Fraenkel set theory. The axioms are used to develop proofs in modern mathematics. They have been part of other sciences such as physics, and the axiom remains as the expanding set of believed truths that continue to develop the sciences.

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ReplyDeleteCoherent - seems like you're just giving an opinion. What work could you do on this that you could communicate?

Complete - need to show a bit more of a time commitment. Plus it's true that Euclid had 5 postulates, but he also had 5 common notions.

Consolidated - last paragraph feels like a part of argument rather than a summary. Were you trying to illustrate a point or something else?

Correct - so do we need axioms? In modern view are they supposed to be true? What does non-Euclidean geometry mean for these ideas, or the different number of ZF axioms?

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