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

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    Axioms Books
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Discussion Overview

The discussion revolves around the topic of axioms, specifically focusing on Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC) and its completeness. Participants explore the origins of ZFC and the implications of Gödel's incompleteness theorem, as well as alternative definitions of completeness in axiomatic systems.

Discussion Character

  • Debate/contested
  • Conceptual clarification
  • Technical explanation

Main Points Raised

  • One participant inquires about good books on axioms, particularly regarding the development and completeness of ZFC.
  • Another participant asserts that ZFC is known to be incomplete due to Gödel's theorem, questioning the notion of completeness mentioned by the original poster.
  • A different viewpoint suggests an alternative definition of completeness in axiomatic systems, indicating that no new axioms can be added that would lead to inconsistency.
  • Several participants recommend specific books on set theory, including "Introduction to Set Theory" by Hrbacek and Jech, and "Set Theory" by Jech.
  • One participant attempts to clarify the definition of completeness, suggesting that it should mean no new axioms can be added that will not make the new system inconsistent.
  • Another participant expresses a need for books to better understand the topic.
  • A resource is shared, pointing to lecture notes by Van den Dries as a valuable reference concerning consistency and provability.

Areas of Agreement / Disagreement

Participants express differing views on the completeness of ZFC, with some asserting it is incomplete while others propose alternative definitions of completeness. The discussion remains unresolved regarding the implications of these definitions and the consensus on ZFC's completeness.

Contextual Notes

There are limitations in the discussion regarding the definitions of completeness and the implications of Gödel's theorem, which are not fully explored or agreed upon by participants.

Pi3.1415
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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.
 
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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 :)
 

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