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Consider the language of set theory together with the constant smybols $\langle \Omega_\alpha|\alpha\in Ord\rangle$. Let's add to $ZFC$ the axiom that for every $\alpha$,$n$ there is some $j: V_{\Omeg …

There is some problems with definitions here. The definition you use of $\alpha+1$-Mahlo cardinals is a cardinal $\kappa$ such that $\kappa$ is the $\kappa$th $\alpha$-Mahlo cardinal. This is not the …

In the paper Vickers, J.; Welch, P. D., On elementary embeddings from an inner model to the universe, J. Symb. Log. 66, No. 3, 1090-1116 (2001). ZBL1025.03049. it is stated to that if $Ord$ is R …

There is actually a type of cardinal that satisfies this, called the "stationarily superhuge" cardinals. A cardinal $\kappa$ is called stationarily superhuge if the $\{\lambda|\kappa\text{ is huge wit …

Theorem: If $\kappa$ is the critical point of $j\colon V_\lambda\prec V_\lambda$, then $\kappa$ is $\lambda$-weakly extendible. Furthermore, if $\kappa$ is the critical point of $j\colon V_\lambda\pre …

There is a sense in which rank-into-rank axioms could be considered generilzations of $\omega$. Woodin discussed this in his paper Suitable Extender Models $I$, given the following conjecture, which I …

What is the consistency strength of a cardinal $\kappa$, such that there is some $j: V\prec M$ such that $M^{\lt j^\omega(\kappa)}\subseteq M$; in other words, for every cardinal $\lambda\lt\delta$, $ …

There are three interesting types of upwards reflection. First off, reflecting upwards to an unbounded class of cardinals, or reflecting oneself upwards, or both. The other instance is when you can fi …

I would like to mention, on the second question, that superhuge cardinals are in fact $\Sigma_5$-reflecting, that is to say inaccessible and $V_\kappa\prec_{\Sigma_5} V$. The reason is first that ever …

I would like to add that, for $n>1$, the $\Pi_n-$Bernays cardinals are precisely the $\Pi_n^1-$indescribable cardinals. We can do this really simply by giving a $\Pi_2$ definition of the Axiom of limi …

These large cardinals actually already have a name. They are called $n$-huge* cardinals. Theorem: If $\kappa$ is $n$-huge*, then $\kappa$ is $n$-huge and a limit of $n$-huge cardinals. Proof. Let $j: …

I added some information elsewhere, but it was to long to put in one post: http://metaordinals.azurewebsites.net/?p=261 Part 1: Your ordering is correct, but the size of the gaps are much more complex …

Inspired by this question: What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"? Definitions: $\delta$ is Berkeley iff for every $\alpha\lt\delta$ and transit …

First off, note that being weakly $\Pi^1_0$-indescribable is actually the same as being weakly $\Sigma_1^1$-indescribable. Let $\phi(x_0...x_n,S)$ be some formula, say $\exists X(\psi(X,x_0...x_n,S))$ …

Work in ZF. Let $\lambda$ be strongly inaccessible. Let $\kappa$ be such that for every $\alpha\lt\lambda$, there is some $j: V_\lambda\prec V_\lambda$, with critical point $\kappa$, such that $j(\kap …