What are the axioms in ZFC set theory?

AI Thread Summary
The discussion centers on the need for a clear reference to the axioms of ZFC set theory in forum conversations. Participants express frustration with the chaos of finding these axioms in external sources like MathWorld or Wikipedia, suggesting a sticky post for easy access. There is a debate about the necessity of proving theorems directly from ZFC axioms, with some arguing that it's impractical while others cite works like Patrick Suppes' "Axiomatic Set Theory" as counterexamples. The conversation also touches on the importance of not favoring one formalism over another, emphasizing that discussions often assume ZFC unless stated otherwise. Overall, the participants advocate for a structured approach to referencing ZFC axioms in discussions.
poutsos.A
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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
 
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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.
 

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