Hierarchy Theory

Working in Morse-Kelley set theory:

A hierarchy is defined as a class that is the union of sets uniquely indexed by ordinals, called as stages, such that each stage is the power set of the immediately prior indexed stage or if no immediate prior indexed stage exist then its the union of all prior indexed stages. The class of all ordinal indexing its stages is to be called the index class of the hierarchy, and the well ordering relation on that class after which the stages of the hierarchy are ordered is called the well ordering of the hierarchy.

Let H  be the Hierarchy indexed by all and only accessible ordinals (i.e.; ordinals prior to the first weakly inaccessible uncountable cardinal).

Now the class of all indices of stages of H , denoted by IHis the index class of H .

Axiom of stretchability: for every well ordering R  on IH , there is a Hierarchy HR whose stages are indexed by the class IH  but well ordered after relation R, so here HR  would be the union of all stages HRi such that HRminR=∅,, and HRi+R1=P(HRi) and HRj=⋃({HRi|iRj})when j is an R-limit.

Example is let R be the <∗ that is the well ordering on IH defined as:

α<∗β≡dfα>∅∧(β>α∨β=∅)

Now according to the above principle H<∗∅ must exist! And clearly this would be an inaccessible stage of the hierarchy H<∗.

To extend this method further we'd define the unary predicate "stretchable" on hierarchies, and the binary relation Hierarchical birth between hierarchies as:

Define: A hierarchy is stretchable if and only if for every well ordering on its index class there is a hierarchy with the same index class, that is well ordered after that well ordering.

stretchable(K)≡df∀R(R ≀≀ IK→∃X(X=KR))

Where ‘‘≀≀"‘ stands for "well orders".

Define: For any two Hierarchies K,L

L↫K≡dfIL⊆K

Where ‘‘↫"signify "born from"

In English: a hierarchy is born from another hierarchy if and only if the index class of the former is a subclass of the later.

Axiom Stretching by birth: ∀K,L[L↫K∧stretchable(K)→stretchable(L)]

Axiom of Stretching by accessiblity:

            ∀ordinal D⊆⋃(X)∃K(stretchable(K)∧D=IK)]

Where V is the class of all sets.

Axiom of Progeny : ∀K[stetchable(K)→{X|X↫K}∈V]

In English: The class of all hierarchies born from a common stretchable hierarchy, is a set

Writing the above in LaTeX:

Working in Morse-Kelley set theory:

A hierarchy is defined as a class that is the union of sets uniquely indexed by ordinals, called as stages, such that each stage is the power set of the immediately prior indexed stage or if no immediate prior indexed stage exist then its the union of all prior indexed stages. The class of all ordinal indexing its stages is to be called the index class of the hierarchy, and the well ordering relation on that class after which the stages of the hierarchy are ordered is called the well ordering of the hierarchy. 

Let $\mathcal H$ be the Hierarchy indexed by all and only accessible ordinals (i.e.; ordinals prior to the first weakly inaccessible uncountable cardinal).

Now the class of all indices of stages of $\mathcal H$, denoted by $I^\mathcal H$ is the index class of $\mathcal H$.

**Axiom of stretchability:** for every well ordering $\mathcal R$ on $I^\mathcal H$, there is a Hierarchy $\mathcal H^{\mathcal R}$ whose stages are indexed by the class $I^\mathcal H$ but well ordered after relation $\mathcal R$, so here $\mathcal H^{\mathcal R}$ would be the union of all stages $\mathcal H^{\mathcal R}_i$ such that $\mathcal H ^\mathcal R_{min^R} = \emptyset, $ and $\mathcal H^\mathcal R_{i+^{\mathcal R}1} = \mathcal P (\mathcal H^\mathcal R _i)$ and $\mathcal H^\mathcal R _j = \bigcup (\{\mathcal H^\mathcal R_i| i \mathcal R j\})$ when $j$ is an $\mathcal R$-limit.

Example is let $\mathcal R$ be the $<^*$ that is the well ordering on $I^\mathcal H$ defined as:

$\alpha <^* \beta \equiv_{df} \alpha> \emptyset \land (\beta > \alpha \lor \beta=\emptyset)$

Now according to the above principle $\mathcal H^{<^*}_\emptyset$ must exist! And clearly this would be an inaccessible stage of the hierarchy $\mathcal H^{<^*}$.

To extend this method further we'd define the unary predicate "*stretchable*" on hierarchies, and the binary relation *Hierarchical birth* between hierarchies as:

Define: A hierarchy is *stretchable* if and only if for every well ordering on its index class there is a hierarchy with the same index class, that is well ordered after that well ordering. 

$stretchable(\mathcal K) \equiv_{df} \forall \mathcal R (\mathcal R  \ \wr\wr \  I^\mathcal K \to \exists \mathcal X( \mathcal X= \mathcal K^\mathcal R))$

Where $``\wr\wr"$ stands for "well orders".

Define: For any two Hierarchies $\mathcal K, \mathcal L$

$ \mathcal L  \looparrowleft \mathcal K \equiv_{df} I^{\mathcal L} \subseteq \mathcal K$

Where $``\looparrowleft"$ signify "*born from*" 

In English: a hierarchy is born from another hierarchy if and only if the index class of the former is a subclass of the later.

**Axiom Stretching by birth:** $\forall \mathcal K, \mathcal L [\mathcal L \looparrowleft \mathcal K \land stretchable (\mathcal K) \to stretchable(\mathcal L)]$

**Axiom of Stretching by accessiblity:** $$ \forall X \in V [\forall Y \in X (stretchable(Y)) \to \forall \text{ordinal} \ D \subseteq \bigcup (X) \exists \mathcal K (stretchable (\mathcal K) \land D=I^\mathcal K)  ]$$

Where $V$ is the class of all sets.

**Axiom of Progeny :** $\forall \mathcal K[ stetchable (\mathcal K) \to \{\mathcal X| \mathcal X \looparrowleft \mathcal K\} \in V]$

In English: The class of all hierarchies born from a common stretchable hierarchy, is a set