User:TheoLaLeo/Sandbox

Bigger Axiom Page
Set axioms, quantifiers restricted to $V$

Axiom:Axiom of Extension/Set Theory

Axiom:Axiom of Pairing/Set Theory

Axiom:Axiom of Unions/Set Theory

Axiom:Axiom of Powers/Set Theory

Axiom:Axiom of Infinity/Set Theory

Class axioms, quantifiers range over classes

Axiom:Axiom of Extension/Class Theory

Axiom:Axiom of Foundation (Classes)

Axiom:Class Comprehension Schema

Axiom:Axiom of Limitation of Size

Smaller Axiom Page
Definition:Von Neumann-Bernays-Gödel Axioms

Axiom:Axiom of Extension/Class Theory

Axiom:Axiom of Specification/Class Theory

Jech
Jech axioms page 70

Axioms A
1. class extensionality

2. all sets are classes

3. all members are sets

4. set pairing

Axioms B
1. class comprehension

Axioms C
1. $\exists$ inductive set

2. sets have unions

3. sets have powersets

4. replacement (with class functions)

Axioms D
1. regularity

Axiom E
1. global choice (with class function)

NBG is A-D and NBGC is A-E

Redundancies
A2, only A3 is needed to characterize distinction

A4, see Axiom of Pairing from Powers and Replacement

C2, see Levy's [|union redundancy result]

Bigger axiom system is Jech minus choice and replacement and plus limitation of size.

Bigger also doesn't have A2 and A3, set/class axioms.

Limitation of size is sort of a replacement, implicitly using the uppercase lowercase convention in saying all all classes $C$ where $|C| \ne |V|$ have a set $x$ where $x=C$.

So Bigger is equivalent to Jech.

Try to find approach with restricted quantifier or predicate?

Results/Other Things To Do
see Levy's [|union redundancy result]

limitation of size implies


 * replacement


 * trivial


 * choice


 * ords are proper, biject with anything proper


 * have to set up ordered classes

Jech NBG implies limitation of size:

Lim:


 * $\alpha$: $V$ bijects then proper


 * $\beta$: proper then $V$ bijects

can weaken each with schroder bernstein for classes:


 * $\alpha$: $V$ injects/surjects to $V$ then proper


 * trivial with replacement


 * $\beta$: proper then $V$ injects/surjects to $V$


 * can inject $V$ to ordinals and ordinals to proper class


 * can well-order arbitrary proper classes

set up ordered class infrastructure