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

Smullyan
p. vi

"One general remark about style: Our approach is rather leisurely (particularly at the beginning) and free-wheeling, and is definitely semantic rather than syntactic. But we do indicate how it can all be formalized."

p. 13

"Conceptually, Zermelo-Fraenkel set theory is a simple one, but technically it is in many ways quite awkward and inelegant. A far more attractive system was developed by Von Neumann, later revised by Robinson, Bemays, and Godel and is now known as NBG (sometimes VNB). This is the main system that we study in this book. (It is a very unusual and nonstandard approach to NBG) The basic idea here is that certain collections of things are called classes and certain collections are called sets. The term "class" is the more comprehensive one, since every set is also a class, but not every class is a set. Which classes are sets? Rather than attempt an absolute answer to this (which some authors have done with dubious success), we regard it as philosophically more honest to take these notions as only relative to any given model of the axioms of class-set theory. That is, a collection V is called a model of class-set theory if it satisfies the axioms of NBG, which will be given in the next several chapters. The elements of V are called the sets of the model and the subcollections of V are called the classes of the model. When the model V is fixed for the discussion, then the sets of the model are more briefly called "sets" and the classes of the model are simply called "classes." This is the procedure that we will adopt. And now we turn to a more formal development."

P0: $V$
p. 15

"We now consider a class $V$ - our so-called "universal class" which will be fixed for this volume. The elements of $V$ will be called sets - each set being itself a class of elements of $V$ - and $V$ itself will be called the class of all sets. We do not specify what this class is; it can be any class which satisfies axioms A1-A8 (the axioms of NBG) which will be given. These axioms will provide $V$ with a sufficiently rich structure so that virtually all of mathematics can be done within the universe $V$. For the rest of this volume, "class" will mean subclass of $V$. (Any collection of objects which is not a subclass of $V$, though it may exist, will have no relevance to what we will be doing.) We henceforth use capital letters, $A,B,C,D,E$ as standing for classes (sets included) and small letters, $x,y,z,a,b,c...$ as standing for sets (elements of $V$). By a first-order property of sets we mean one defined by a formula in which we quantify only over sets. (Thus we allow $\forall x$, $\exists x$, for $x$ a set variable, but we do not allow $\forall A$, $\exists A$ where A is a class variable.)"

Smullyan says things here that should be added to his formal exposition.

"$V$ itself will be called the class of all sets" can be added as an alternative definition for sets, since the implicit definition in his exposition is essentially "it's a set iff it's in $V$".

"For the rest of this volume, "class" will mean subclass of $V$" should also be formalized as something like:
 * $\exists x : \forall y : y \subseteq x$

with a proof showing its uniqueness (from extension) and a definition of $V$ as this unique element, similar to what's done with the empty set.

Without an axiom like the above, his theory does not syntactically entail this, which he clearly states in natural language and intends to be the case.

I also don't see a way to prove eg:
 * $\neg \exists V' : V \in V'$

in his theory without P0.

Theorems can just always be stated with the caveat that "for all subclasses of $V$" for classes, but the other class content on the site doesn't have that, and probably never will, since most class theories just take the domain to consist only of classes since that's kind of the whole point of the theory.

P1: Extensionality
p. 14


 * $\forall x : \paren { x \in A \iff x \in B } \implies A = B$

This is standard, though this site has $... \iff A = B$ instead of $... \implies A = B$. They are trivially equivalent in a theory with equality so doesn't really matter.

P2: Separation
p. 15

Let $\map \phi {A_1, ..., A_n, x}$ be a formula whose class variables are $A_1, ..., A_n$ for finite $n$, and whose only free set variable is $x$:


 * $\forall A_1 \ldots \forall A_n : \exists B : \forall x : \paren{ x \in B \iff \map \phi {A_1, ..., A_n, x} }$

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?

my ideal NBG style axioms
define $V$ such that all $x$ are $x \subseteq V$


 * (1): $V$ exists
 * (2): extensionality
 * (3): comprehension schema (restricted to $V$)
 * (4): $V$ contains inductive set
 * (5): $V$ closed under powerset
 * (6): regularity
 * (7): limitation of size

long form:


 * $V$ transitive and swelled
 * from (A1)


 * $V$ swelled ie $\forall x: \forall y: \paren {x \subseteq y \land y \in A \implies x \in A}$
 * $y \in A$ implies $y \in V$, thus $\size x \leq \size y < \size V$ so $x \in V$ by (A7)

thus super complete, grothendieck U (sort of by definition), etc


 * Empty Set
 * comprehension gives empty class, $\size V > 0$ because of inductive set, set by (A7)


 * Replacement
 * image is class by comprehension, image $\leq$ domain size $< \size V$ if set


 * Separation
 * from empty set and replacement


 * pairing
 * replacement from some $\size S = 2$ eg $\powerset {\powerset \O}$


 * strong well ordering
 * every class injects to $\On$ by (A7)


 * global choice
 * from well order


 * ordinals
 * intersection of inductive sets, has injection to (A4) is thus set


 * union
 * levy's paper

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

go through hughes and cresswell or something similar


 * Find good visual or other representation of nested theories


 * Find deduction rule version? probably unnecessary w/ it existing for prop log


 * set up possible worlds and accessibility relation semantics


 * hughes and cresswell probably have


 * make single page with reference table for relation properties and axioms


 * pages for alternative interpretations of modal theories, eg epistemic or temporal


 * johnson's probability stuff in modal logic

transcribe johnson proofs

make equivalence pages for set theory axioms with page about different "equivalence classes" of axioms, eg
 * Z (ZFC - replacement regularity and choice)
 * ZF (ZFC - choice)
 * ZFC
 * ZF - R (ZFC - regularity and choice)
 * ZFC - R (ZFC - regularity)
 * NBG - C (NBG without choice)
 * NBG + AC (NBG with regular choice)
 * NBG + GC (NBG with global)
 * NBG + Lim (NBG with limitation of size instead of GC, regularity, and replacement)
 * Morse Kelley (NBG with bound proper classes allowed in predicates)
 * ZFC + Groth (ZFC with grothendieck universes)
 * ZFC + Card (ZFC with various inaccessible cardinal axioms)

higher order logics

develop frege's actual original set theory


 * from skimming it seems like a lot of philosophy interspersed with a bit of math, of historical interest for sure but not sure how much actual content there is

try Begriffsschrift? probably pointless and very painful

see mendelson on alternative set theories


 * type theory


 * quine's NF


 * urelements