2015年2月14日 · (The parallel postulate). A theorem is a logical consequence of the axioms. In Geometry, the "propositions" are all theorems: they are derived using the axioms and the valid rules. A "Corollary" is a theorem that is usually considered an "easy consequence" of another theorem. What is or is not a corollary is entirely subjective.

2022年3月28日 · I’m reading Modern Geometry by Dolciani from 1965. There is no mention of the segment addition postulate, but they do name a definition with the usual given statement of the segment addition postulate seen in newer geometry textbooks. My question is, is the statement for segment addition a definition or postulate?

$\begingroup$ I would suggest finding one book that fills out Hilbert's approach and sticking with it. The three books I have are George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, then Robin Hartshorne, Geometry: Euclid and Beyond, then Marvin Jay Greenberg, Euclidean and Non-Euclidean Geometries, 4th edition.

2016年1月13日 · My question is: does the absence of the (strong) parallel postulate imply a geometry with curvature and does the presence of the parallel postulate imply the absence of curvature? I know that this question may sound really vague. It is motivated by playing with Euclids axioms and I don't have a precise mathematical definition of curvature in mind.

Definition: Theorem, Lemma, Proposition, Corollary, Postulate, Statement, Fact, Observation, Expression, Fact, Property, Conjecture and Principle. Most of the time a mathematical statement is classified with one the words listed above. However, I can't seem to find definitions of them all online, so I will request your aid in describe/define them.

2018年9月15日 · (Now, one could augment Postulate 2 to state that there's one and only one parallel through any given point, noting that if the point is on the given line, then the parallel is the line. But postulates like to be as simple as possible. Separating the notions into Postulate 2 and Fact 3 is cleaner.) $\endgroup$ –

Good question. To begin with, the first part of the question is false. The "fact" that a full rotation is 360 degrees, is in fact a definition of the angular measure of a full rotation, and thus is not equivalent to the parallel postulate. You can think of it as defining what a degree is, namely, one part in 360 of a full rotation.

It seems that axioms and postulates refer to the same thing. From Dictionary.com, definition 6 of postulate: Mathematics, Logic. a proposition that requires no proof, being self-evident, or that is for a specific purpose assumed true, and that is used in the proof of other propositions; axiom.

2012年12月14日 · Postulate and axiom are used interchangeably in modern mathematics and they mean a statement that is assumed to be true within a specified domain. Every deductive mathematical system (such as Euclidean Geometry) normally will have statements that are self-evident (or assumed to be true) and don’t need proofs.

2018年11月15日 · But there is no ruler to be defined on lines in this geometry. In other words, the ruler postulate is (among other things) a completeness axiom: it tells you that every line has all the points that a line in the Euclidean plane would have.