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Why are they called "axioms"? Shouldn't they be called "definitions"?
Axioms are the traditional name for the elements of a generating set of a mathematical theory.Why are they called "axioms"? Shouldn't they be called "definitions"?
Why are they called "axioms"? Shouldn't they be called "definitions"?
Why are they called "axioms"? Shouldn't they be called "definitions"?
I mean that every vector space exists as pure sets under ZFC already. You just need to define them, not postulate their existence.
You titled this "Axioms of Vector Spaces". The axioms for vector spaces have nothing to do with postulating the existence of vector spaces!
ZFC just gives you a set of points without any additional structure, e.g., the ability to "add" two points together, or to "multiply" a point by a scalar from some field. The additional structure comes from the vector space axioms.
I too have never understood why the defining properties of a vector space are sometimes called the "axioms" of a vector space. That use of the word "axiom" does not seem to agree with the definition of the dictionary. http://dictionary.reference.com/browse/axiom
Logic, Mathematics. a proposition that is assumed without proof for the sake of studying the consequences that follow from it.
From that dictionary:
How does that not agree?
Axioms are propositions that are taken as true without proof. It makes no sense to think of the "axiom of additive closure" as an axiom, as a set with addition may or may not satisfy it. You are DEFINING what a vector space is!
You define "add" and "multiply" in ZFC as functions, or ordered tuples. This "additional structure" already exists in ZFC. No need for further axioms. Just definitions.
A vector space is defined to be something satisfying the axioms of a vector space. Amongst other things one of the axoims is that x+y=y+x for all x,y. The 'proposition' would be - if x,y are in V then x+y=y+x.
You're confusing a model with the axioms.
I'm curious to know whether Dragonfall would consider the "axioms" of geometry, like the parallel postulate and so forth, as legitimate axioms, or merely the definition of a particular geometry (Euclidean, Poincare, etc).