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

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- anonymous

What, in particular, do you have trouble understanding?

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- Loser66

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?

- anonymous

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.

- Loser66

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?

- anonymous

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

- Loser66

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

- anonymous

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.

- anonymous

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

- anonymous

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

- Loser66

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?

- Loser66

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

- Loser66

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

- anonymous

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.

- Loser66

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

- Loser66

What kind of algebra?

- anonymous

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.

- Loser66

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

- anonymous

Thanks, good luck to you as well.

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