User blog:Cheetahrock63/Axiom System of Conglomerates (ACG)

ACG (Axiom system of Conglomerates) is a theory that formalizes objects called conglomerates. Conglomerates are objects that are able to contain any class, including proper classes like ${\rm {Ord}}$. Conglomerates were considered in order to deal with "collections of classes" (which can come up in stuff like category theory) in the first place.

The rationale there was probably like "hey uh what should we call thingies that have proper classes like ${\rm {Ord}}$ inside them? they're not sets or classes since thems proper classes are too dang fat to fit inside those! i know! we'll call them conglomerates because why not! and we'll call thingies too fat to fit inside those proper secondclasses because i've exhausted all of my creative energy when calling the last objects "conglomerates" and am starting to get lazy!" but possibly with bigger and more verbose terms, actual grammar, and maybe undertones of an uppity and stuck-up "i'm smarter than you" attitude.

A common way to treat conglomerates and secondclasses is to consider a theory such as MK+(there exists a Grothendieck universe) and then rename elements of the universe "sets", rename subsets of the universe "classes", rename sets within the theory "conglomerates", and rename classes within the theory "secondclasses". This is a totally valid way to treat them, since MK+(there exists a Grothendieck universe) behaves exactly like a theory such as ACG. However, treating them this way can make conglomerates feel a little bit less special and can raise questions about there really being any fundamental difference between sets or conglomerates. One possible reconciliation is to think of MK+(there exists a Grothendieck universe) as a model of a conglomerate theory.

ACG is a stronger theory than MK and ZFC+(there exists a strongly inaccessible cardinal) and was originally defined in two parts. The first part is literally just NBG—which deals with sets and classes. The second consists of five groups of axioms that talk about conglomerates and secondclasses.

• $x$, $y$ (with or without subscripts) denote sets
• $X$, $Y$ (with or without subscripts) denote classes
• $\mathcal {X}$ (with or without subscripts) denotes a conglomerate
• $\mathcal Y$ (with or without subscripts) denotes a secondclass
• ${\mathfrak {M}}({\mathcal {Y}})$ means that $\mathcal Y$ is a set
• ${\mathfrak {Cla}}({\mathcal {Y}})$ means that $\mathcal Y$ is a class
• ${\mathfrak {Cog}}({\mathcal {Y}})$ means that $\mathcal Y$ is a conglomerate
• ${\mathfrak {Sec}}({\mathcal {Y}})$ means that $\mathcal Y$ is a secondclass

Axioms

Group A

1. All conglomerates are secondclasses: $\forall {\mathcal {X}}{\mathfrak {Sec}}({\mathcal {X}})$
2. Any secondclass contained by another secondclass is a conglomerate: ${\mathcal {Y}}_{1}\in {\mathcal {Y}}_{2}\implies {\mathfrak {Cog}}({\mathcal {Y}}_{1})$
3. Axiom of Extensionality for conglomerates: $\forall {\mathcal {X}}({\mathcal {X}}\in {\mathcal {Y}}_{1}\iff {\mathcal {X}}\in {\mathcal {Y}}_{2})\implies {\mathcal {Y}}_{1}={\mathcal {Y}}_{2}$
4. Axiom of conglomerates' closure under pairing: $\forall {\mathcal {X}}\forall {\mathcal {X}}_{1}\forall {\mathcal {X}}_{2}\forall {\mathcal {X}}_{3}({\mathcal {X}}\in {\mathcal {X}}_{3}\iff {\mathcal {X}}={\mathcal {X}}_{1}\lor {\mathcal {X}}={\mathcal {X}}_{2})$

Group B

2. Axiom of Intersection:
3. Axiom of the Complement:
4. Axiom of the Domain:
5. Axiom of the Direct Product:
6. Existence Axioms of Secondclasses:

Group C

1. Axiom of existence of powerconglomerate:
2. Axiom of existence of union:
3. Axiom of replacement:
4. All classes are conglomerates:
5. Every conglomerate that has only sets as elements is a class:

Additional Axioms often assumed for convenience

• Generalized continuum hypothesis: $\aleph _{\alpha +1}=\beth _{\alpha }$
• Existence of a weakly compact cardinal:

Definitions

A proper secondclass is a secondclass such that it is not a conglomerate.

A proper conglomerate is a conglomerate such that it is not a class.

A proper class is a class such that it is not a set.

The powerconglomerate of a conglomerate: ${\mathcal {P}}({\mathcal {X}}):=\{{\mathcal {X}}_{1}|{\mathcal {X}}_{1}\in {\mathcal {X}}\}$

The powerconglomerate of a class is known as the class’ powerclass

The powerconglomerate of a set is known as the set’s powerset.

A subsecondclass of a conglomerate is called a subconglomerate.

A subsecondclass of a class is called a subclass.

A subsecondclass of a set is called a subset.

Two secondclasses are considered disjoint if they don't share the same elements.

Ordered pair of conglomerates: $({\mathcal {X}}_{1},{\mathcal {X}}_{2}):=\{\{{\mathcal {X}}_{1}\},\{{\mathcal {X}}_{1},{\mathcal {X}}_{2}\}\}$

The Cartesian product of two conglomerates ${\mathcal {X}}_{1}\times {\mathcal {X}}_{2}:=\{({\mathcal {X}}_{3},{\mathcal {X}}_{4})|{\mathcal {X}}_{3}\in {\mathcal {X}}_{1}\land {\mathcal {X}}_{4}\in {\mathcal {X}}_{2}\}$.

The union of two conglomerates: ${\mathcal {X}}_{1}\cup {\mathcal {X}}_{2}:=\{{\mathcal {X}}|{\mathcal {X}}\in {\mathcal {X}}_{1}\lor {\mathcal {X}}\in {\mathcal {X}}_{2}\}$.

The intersection of two conglomerates: ${\mathcal {X}}_{1}\cap {\mathcal {X}}_{2}:=\{{\mathcal {X}}|{\mathcal {X}}\in {\mathcal {X}}_{1}\land {\mathcal {X}}\in {\mathcal {X}}_{2}\}$

The difference of two conglomerates: ${\mathcal {X}}_{1}\backslash {\mathcal {X}}_{2}:=\{{\mathcal {X}}|{\mathcal {X}}\in {\mathcal {X}}_{1}\land {\mathcal {X}}\notin {\mathcal {X}}_{2}\}$

Ordinals of type two

An ordinal of type two is a conglomerate $\mathcal {X}$ that satisfies

• $\forall {\mathcal {X}}_{1}\in {\mathcal {X}}\forall {\mathcal {X}}_{2}\in {\mathcal {X}}({\mathcal {X}}_{1}\in {\mathcal {X}}_{2}\lor {\mathcal {X}}_{1}={\mathcal {X}}_{2})\in {\mathcal {X}}_{1}$
• $\forall {\mathcal {X}}_{1}\in {\mathcal {X}}\forall {\mathcal {X}}_{2}\in {\mathcal {X}}_{1}({\mathcal {X}}_{2}\in {\mathcal {X}})$

The secondclass of all ordinals of type two is denoted as ${\rm {{Ord}_{2}}}$.

The smallest ordinal of type two that is not a set is the class of all ordinals ${\rm {Ord}}$. Ordinals of type two that are sets are called ordinals.

The smallest ordinal (of type two) is the $0$, $0:=\emptyset$.

The successor of an ordinal of type two $\alpha$, $\alpha+1$, is $\alpha\cup\{\alpha\}$.

$\alpha <\beta \iff \alpha \in \beta$

$\alpha \leq \beta \iff \alpha \subseteq \beta$

An ordinal of type two $\lambda$ that there does not exist an ordinal $\alpha$ such that $\alpha +1=\lambda$ is called a limit ordinal of type two. A limit ordinal of type two that is an ordinal is called a limit ordinal.

The smallest limit ordinal larger than $0$ is called $\omega$. Any ordinal smaller than $\omega$ is called a natural number. An ordinal is finite if and only if it is a natural number. An ordinal is called infinite if it is not a natural number. An ordinal of type two is called unsettling if it is not an ordinal.

Ordinal of type two arithmetic

• $\alpha+0=\alpha$
• $\alpha+(\beta+1)=(\alpha+\beta)+1$
• $\alpha +\lambda$ is the limit of $\alpha+\delta$ for all $\delta <\lambda$
• $\alpha\cdot 0=0$
• $\alpha \cdot (\beta +1)=(\alpha \cdot \beta )+\alpha$
• $\alpha \cdot \lambda$ is the limit of $\alpha\cdot\delta$ for all $\delta <\lambda$
• $\alpha^0=1$
• $\alpha^{\beta+1}=(\alpha^{\beta})\cdot\alpha$
• $\alpha ^{\lambda }$ is the limit of $\alpha^{\delta}$ for all $\delta <\lambda$

Cardinals of type two

The initial ordinal of type two of the cardinality of a conglomerate $\mathcal {X}$ (the cardinality of $\mathcal {X}$ is denoted $|{\mathcal {X}}|$ or ${\rm {card}}({\mathcal {X}})$) is defined to be the smallest ordinal of type two $\alpha$ such that there exists a bijection between $\mathcal {X}$ and $\alpha$. The secondclass of all such ordinals is denoted ת₂ and elements of ת₂ are called cardinals of type two. A different system of arithmetic called cardinal arithmetic is used with cardinals of type two.

Cardinals of type two that denote the cardinality of a set are called cardinals. The secondclass of all cardinals is denoted ת.

$\aleph _{\alpha }:=\omega _{\alpha }$

$\beth_0 = \aleph_0$

$\beth _{\alpha +1}:=2^{\aleph _{\alpha }}$

${\rm {Card}}:={\rm {Ord}}$

A set or cardinal is called finite if there exists a bijection between it and a natural number. A set or cardinal is called infinite if there does not exist a bijection between it and any natural number. A secondclass or cardinal of type two is called unsettling if there does not exist a bijection between it and any set.

Cardinal of type two arithmetic

• $|{\mathcal {X}}_{1}|+|{\mathcal {X}}_{2}|=|{\mathcal {X}}_{1}\cup {\mathcal {X}}_{2}|$ iff $\mathcal{X}_1$ and ${\mathcal {X}}_{2}$ are disjoint conglomerates
• $|{\mathcal {X}}_{1}|\cdot |{\mathcal {X}}_{2}|=|{\mathcal {X}}_{1}\times {\mathcal {X}}_{2}|$
• $|{\mathcal {X}}_{1}|^{|{\mathcal {X}}_{2}|}=|\{f|f:{\mathcal {X}}_{2}\to {\mathcal {X}}_{1}\}|$

Sursurreals

$({\mathbf {No} }_{2},<,b)$, consisting of a proper secondclass ${\mathbf {No} }_{2}$, a binary relation $<$ (“less than”), and a unary function $b$ (“birthday function”), is a sursurreal number system if

• $<$ is a strict total order over $\mathbf {No} _{2}$
• $b$ is a function from $\mathbf {No} _{2}$ to ${\rm {{Ord}_{2}}}$
• Conway’s Simplicity Theorem: Let A and B be subsecondclasses of $\mathbf {No} _{2}$ such that for all elements x in A and all elements y in B, x < y. There exists a unique z in $\mathbf {No} _{2}$ such that b(z) is minimal and for all elements x in A and all elements y in B, x < z < y.
• If an ordinal of type two α is greater than all b(x) and b(y) such that x is in A, a subsecondclass of $\mathbf {No} _{2}$, and y is in B, a subsecondclass of $\mathbf {No} _{2}$ such that for all y in B, y is greater than all elements of A; then the birthday of the unique minimal birthday sursurreal z such that x < z < y is less than or equal to α.

Sursurreals whose birthdays are equal to ${\rm {Ord}}$ are called surreal gaps. Sursurreals whose birthdays are ordinals are called surreal numbers.