Question

I don't understand the so-called "Axiomatic method." I need a super patient helper to help me out.
Please, explain me.

Answer 1

@jim_thompson5910

Answer 2

What, in particular, do you have trouble understanding?

Answer 3

Yes, the way they interpret the method is nonsense to me. By definition (table 1.2.1), they said "Any axiomatic system contains a set of statements ....(3)....These are the axioms of the system."
My question: if they are axioms, they must be correct, right?

Answer 4

Yeah, but they are only assumed to be true. A good example could be Classical Geometry. Euclid formulated geometry primarily relying on 4 axioms. The rest of his geometrical frame work then relied on a fifth axiom, the so-called parallel line postulate which essentially supposes that given a line and a point not on this line, then there is exactly 1 line that passes through this point, which doesn't meet the given line. However, the parallel line postulate can be modified to yield different geometries like hyperbolic geometry( more than 1 line which passes through the point and is parallel to the original line) and spherical geometry( no line that passes through the point and is parallel to the original line). So, what axioms we assume to be true, can radically alter our deductions and the results that follow.

Answer 5

So, they give us the axioms (kind of given information, right?) and then 1 statement like theorem or corollary, We have to prove or disprove the theorem base on the given axioms, right?

Answer 6

Yes, precisely. The axioms are the rules to the game. What outcomes come out of the game are entirely constrained by the axioms.

Answer 7

in example 1.2.6. I don't understand the way they define {P,Q} <---> z
How?

Answer 8

Seems as though they are just making a map between the {P,Q,R} and {x,y,z}. They made the decision arbitrarily to show that its not an isomorphism.

Answer 9

I must say I didn't really read everything and I'm just going off my gut.

Answer 10

Whatever it may be, it follows from the relations they have established for Fe's and Fo's

Answer 11

Why do they do that? They tried to prove that the example 1.2.5 and 1.2.6 is not an isomorphism, right?
Is it not that it is trivial? In 1.2.5, only 3 relationships are constructed while there are 6 of them in 1.2.6. They are not isomorphism at the first look. right?

Answer 12

That is the reason why I don't understand the way they interpret the method. :)

Answer 13

Thanks a ton for being here to help me out. I feel better now.

Answer 14

No problem, I'm actually studying the same sort of topic (Algebra and Analysis) for this semester. I have to get off now, but perhaps I may get on later and look further into your text.

Answer 15

What kind of analysis? I am taking complex analysis this semester.

Answer 16

What kind of algebra?

Answer 17

Well, its really a "pre-Analysis" course, here its called advanced calculus. The algebra course is entitled abstract algebra, both are standard upper level undergraduate courses for those pursuing mathematics.

Answer 18

hehehe. good luck. I took both them last year.

Answer 19

Thanks, good luck to you as well.