Books on Axioms: ZF(C) & Why It's Complete

[Pi3.1415]

Does anyone know of any good books on Axioms. Such as how was ZF(C) came up with and why it is that the general consensus is that it is complete.

 

[Hurkyl]

why it is that the general consensus is that it is complete.

It's well known that ZFC is incomplete, by Gödel's theorem. Or do you mean something else by complete?

 

[Pi3.1415]

Oh i see what you mean. I think another definition of completeness in Axiom systems is that no new axioms can be added that will make the new system inconsistent.

 

[micromass]

The book "Introduction to set theory" by Hrbacek and Jech is a good one.
A more advanced book is "Set theory" by Jech.

 

[Hurkyl]

Oh i see what you mean. I think another definition of completeness in Axiom systems is that no new axioms can be added that will make the new system inconsistent.

That's the same meaning of completeness I just referred to...

 

[HallsofIvy]

Oh i see what you mean. I think another definition of completeness in Axiom systems is that no new axioms can be added that will make the new system inconsistent.

Surely you mean "no new axioms can be added that will not make the new system inconsistent".

 

[Pi3.1415]

yep sorry now you see why i need the books...

 

[micromass]

A great resource concerning consistency, provability, etc. is the lecture notes of Van den Dries: http://www.math.uiuc.edu/~vddries/410notes/main.dvi
It's a real gem!

 

[Pi3.1415]

Thank you micromass :)