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EDIT: it appears that my original question has some confusion between auto-morphisms and elementary embeddings as it is obvious from the answer below, therefore I'll clarify here what I exactly want. …

If $\psi$ is a predicate that is definable in $FOL(\in,=)$ by a formula from parameters in $V$, then if $\psi$ hold of the class $\small ORD$ of all ordinals in $V$, then the class of all cardinals in …

In posting about a reflection principle coupled with a limitation of size axiom over Ackermann set theory, the answer is that the theory is blown up to a Mahlo cardinal. I'm here just wondering if th …

I want to coin the notion of heaviness of a class as a function from classes to ordinals such that $$heaviness(x)=|TC(x)|$$, i.e. the heaviness of a class is the cardinality of its transitive closure. …

Let $K_2^+(W)$ be the following theory in the language $L(\in,W)$ with the constant symbol $W$. Extensionality: $\forall x (x \in a \leftrightarrow x \in b) \to a=b$ $\mathcal{Define:} \ set(x) \iff …

In a prior posting, I've posed the idea of reducing set theory to an extended kind of second order ordinal arithmetic $\small \sf`` 2 oO A"$. The idea was to have a domain of ordinals and sets of ordi …

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 immediatel …

Axiom scheme of Ordinal Reflection: if $\phi$ is a formula that doesn't use the symbol $V$, whose parameters are among $x_1,..,x_n$; then: $$\forall x_1 \in V,\dotsc,\forall x_n \in V: \phi(On) \to \\ …

The following is a theory that uses a reflection principle similar to Ackermann but on a size notion that is definable in the language of set theory, my question is about its consistency strength real …

Lets denote any set that is "the set of all strictly smaller subsets of it" as size- unreachable. Formally: $$\operatorname {size-unreachable}(X) \iff \\\forall Y (Y \in X \iff Y \subset X \land |Y|<| …

Working in the language of Ackermann set theory: Let $\phi(\vec{P},\vec{x})$ be a formula not using the symbol $V$, where $\vec{P}$ are predicate symbols definable in the language of set theory, and $ …

if $\kappa$ is a cardinal such that $V_\kappa \models \sf ZFC$, then $\kappa$ is called a worldly cardinal and this is not necessarily downward absolute [Hamkins], i.e. there is a model $V$ of $\sf ZF …

A cardinal κ is a Berkeley cardinal, if for any transitive set $M$ with $κ∈M$ and any ordinal $α<κ$ there is an elementary embedding $j : M → M$ with $\alpha<\text{crit}(j)<\kappa$. My question i …

Sometimes when one tries to capture the abstract aspect of some notion that is intuitively considered as being truly of abstract nature, in set theoretic terms, then this can extend the theory in a hu …

[EDIT] This posting had been edited to assert that we are speaking about regular mutual stages. Let $H_{\kappa}$ be the set of all sets that are hereditarily strictly smaller in cardinality than car …