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

• Pi3.1415
In summary, the conversation is about recommendations for books on Axioms, specifically the completeness of ZF(C) and its relationship to Gödel's theorem. The participants also discuss different definitions of completeness in Axiom systems and the book "Introduction to set theory" by Hrbacek and Jech is suggested as a good resource. Another book, "Set theory" by Jech, is also mentioned as a more advanced option. The conversation ends with a recommendation for lecture notes by Van den Dries on consistency and provability.
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.

Pi3.1415 said:
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?

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.

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

Pi3.1415 said:
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...

Pi3.1415 said:
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".

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

Thank you micromass :)

## 1. What is the significance of Axioms in mathematics?

Axioms are fundamental statements or principles that are accepted as true without proof. They serve as the basic building blocks of a mathematical system and provide a starting point for logical reasoning and theorems.

## 2. What is ZF(C) and why is it important?

ZF(C) refers to the Zermelo-Fraenkel set theory with the Axiom of Choice. It is one of the most widely accepted axiomatic systems in mathematics and provides a foundation for modern set theory. It is important because it allows for the construction of more complex mathematical structures and the development of rigorous proofs.

## 3. What does it mean for a system to be complete?

A system is complete if it is able to prove or disprove every statement within that system. In the case of ZF(C), it is complete in the sense that it can prove or disprove any statement about sets and their properties using the specified axioms.

## 4. How does ZF(C) compare to other axiomatic systems?

ZF(C) is one of the most commonly used axiomatic systems, but there are other systems that have been developed as well. For example, NBG (von Neumann-Bernays-Gödel) set theory is a conservative extension of ZF(C) that allows for the existence of some large sets that are not permitted in ZF(C).

## 5. Can ZF(C) be proven to be consistent?

The consistency of ZF(C) cannot be proven within ZF(C) itself, as this would lead to a contradiction known as Russell's paradox. However, assuming ZF(C) is consistent, it can be shown that ZF(C) cannot prove its own consistency using Gödel's incompleteness theorems.

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