What are the axioms in ZFC set theory?

Click For Summary

Discussion Overview

The discussion revolves around the axioms of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). Participants express varying opinions on the necessity of listing these axioms for reference in discussions about set theory, as well as the implications of favoring one formalism over another.

Discussion Character

  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • Some participants request a sticky post listing the axioms of ZFC for easier reference during discussions.
  • Others argue that checking external sources like MathWorld or Wikipedia is trivial and chaotic, suggesting a need for a centralized reference.
  • There is a concern that favoring ZFC could lead to misconceptions about its status as a foundational theory.
  • Some participants assert that discussions about set theory typically assume ZFC unless stated otherwise.
  • One participant questions the practicality of proving theorems directly from the axioms of ZFC, suggesting that it is uncommon.
  • Another participant cites Patrick Suppes' work as an example of proving theorems from ZFC axioms, challenging the notion that such proofs are rare.
  • There is a disagreement regarding the appropriateness of certain comments made about mathematicians and the nature of set theory discussions.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the necessity and utility of listing ZFC axioms, as well as differing opinions on the nature of formalism in set theory discussions. The discussion remains unresolved with no consensus on these points.

Contextual Notes

Some participants highlight the potential for confusion regarding the assumptions made about formalism in set theory, and the discussion reflects a range of attitudes towards the axioms and their application in proofs.

poutsos.A
Messages
102
Reaction score
1
since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up.
thanx
 
Physics news on Phys.org
You misunderstood him Dragonfall, I think he meant that someone should post a sticky in this subforum underlying all the axioms of ZFC, or something like this.

Cause checking in mathworld or wiki is really a triviality, nowadays.
 
loop quantum gravity said:
You misunderstood him Dragonfall, I think he meant that someone should post a sticky in this subforum underlying all the axioms of ZFC, or something like this.

Cause checking in mathworld or wiki is really a triviality, nowadays.

THANK YOU that is what i really meant.INDEED checking in mathworld or in wiki
although a triviality it is sometimes simply chaotic
 
I don't think we should favor any formalism over another, lest someone thinks that ZFC is gods-given or something.
 
Dragonfall said:
I don't think we should favor any formalism over another, lest someone thinks that ZFC is gods-given or something.

Well any god I worship sure as hell wouldn't use category theory!
 
since we are interested more in the logical conclusions and not in the rules them selfs,i think that any set of rules concerning set theory would do.
Also if we find out that a certain set of rules does not solve certain problems then we can refer to another set of rules

But yes i agree with Mr poutsosA ,we must a have a set of rules to refer to, everytime we start a discussion in set theory
 
If anyone here discusses set theory without explicitly mentioning the formalism then it's assumed to be ZFC. Now if you have something to say about ZFC, chances are you know the axioms by heart anyway. It's not necessary to have them listed as if they were the ten freaking commandments.
 
Dragonfall said:
If anyone here discusses set theory without explicitly mentioning the formalism then it's assumed to be ZFC. .

What that suppose to mean??
 
  • #10
Dragonfall said:
I don't think we should favor any formalism over another, lest someone thinks that ZFC is gods-given or something.

mention couple of formalisms,if you like,please
 
  • #11
evagelos said:
What that suppose to mean??

It means "If anyone here discusses set theory without explicitly mentioning the formalism then it's assumed to be ZFC."

evagelos said:
mention couple of formalisms,if you like,please

Morse-Kelley, type theory, category theory, von-Neumann-Godel. You can google the rest yourself.
 
  • #12
Dragonfall said:
It means "If anyone here discusses set theory without explicitly mentioning the formalism then it's assumed to be ZFC."



Morse-Kelley, type theory, category theory, von-Neumann-Godel. You can google the rest yourself.

ZFC first order ,second order,a mixed of the two?

Anyway in this forum i have not seen a lot of proofs coming out straight from ZFC axioms
 
  • #13
Because no person in the right mind would prove things straight from the axioms.
 
  • #14
Dragonfall said:
Because no person in the right mind would prove things straight from the axioms.

what are there for, to be admired at ??

how about Patrick Suppes,what is he doing in his book : Axiomatic Set Theory?
 
  • #15
How about you count the number of published papers with "we will prove this from the axioms of ZFC"?
 
  • #16
First you claim and i quote: no person in the right mind would prove things straight from the axioms.

To that claim i produce the book of Patrick Suppes,Axiomatic Set Theory where he proves from the ZFC axioms all the theorems involved

Now you asking me to produce papers where the theorems in ZFC are proved.

is not one example enough for your claim??
 
  • #17
evagelos, are you being deliberately dense?
 
  • #18
morphism said:
evagelos, are you being deliberately dense?
I think he has a fair objection. Dragonfall has insulted several demographics of mathematicians, computer scientists, students (and probably people in other fields too). While one might assume Dragonfall really just meant something to the effect of "leave set theory to the set theorists", I believe it is quite reasonable to call Dragonfall out on his comment.
 
  • #19
Which comment might that be?
 
  • #20
Because no person in the right mind would prove things straight from the axioms.​
 
  • #21
Oh no, I'm standing by that. That was said to me by my set theory course professor who is a set theorist. I've always found that quote funny.
 

Similar threads

Graduate An additional constraint to the ZFC axioms?
  • · Replies 40 ·
2
Replies
40
Views
5K
Graduate On soundness and completeness of ZFC set theory
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Direct limit of multiverse models of ZFC
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Maybe it is not necessary to define set membership?
  • · Replies 13 ·
Replies
13
Views
2K
MHB Axioms of Set Theory .... and the Union of Two Sets ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB ZFC and the Pairing Principle .... Searcoid Theorem 1.1.5 ....
  • · Replies 1 ·
Replies
1
Views
2K
MHB Set Theory and ZFC - The Axiom of Replacement - Searcoid, Pages 6-7
  • · Replies 14 ·
Replies
14
Views
5K
Graduate Looking for book on ZFC axiomatic set theory (I think)
  • · Replies 12 ·
Replies
12
Views
3K
Graduate A method for proving something about all sets in ZFC
  • · Replies 4 ·
Replies
4
Views
1K
High School Is everything in math either an axiom or a theorem?
  • · Replies 73 ·
3
Replies
73
Views
9K