As an answer to the question (1), one can produce a series of natural orderings on large cardinal axioms using the operator $Con(ZFC+\dots)$.

$\sigma \leq_0 \theta \Longleftrightarrow ZFC\vdash \theta \longrightarrow \sigma$

$\sigma \leq_1 \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta) \longrightarrow Con(ZFC+\sigma)$

$\sigma \leq_2 \theta \Longleftrightarrow ZFC\vdash Con(ZFC+Con(ZFC+\theta)) \longrightarrow Con(ZFC+Con(ZFC+\sigma))$

$\cdots$

We have $\leq_0 \subseteq \leq_1 \subseteq \leq_2\subseteq \cdots$ and one can think about an **animated diagram** which shows the change of positions of each large cardinal axiom in the tree when its ordering varies over $\langle \leq_i:i\in \omega\rangle$.

An important property of these orderings on large cardinal axioms is that by the definitions we have $\forall \sigma,\theta \in \text{Large Cardinal Axioms}~~\forall i,j\in \omega~~((i\leq j~~\wedge~~\sigma=_i \theta )\Longrightarrow \sigma =_j \theta)$ and so maybe there is a (possibly ordinal valued) step $k$ such that **all main large cardinal axioms are equivalent in the sense of $\leq_k$ ordering**. Note that the same phenomenon happens between $WI$ (existence of a weakly inaccessible cardinal) and $SI$ (existence of a strongly inaccessible cardinal) because we have $WI<_0 SI$ but $\forall i\geq 1~~WI=_i SI$.

Also one can ask some natural questions about the interactions of these orderings with each other as follows.

**Definition.** Let $\sigma,\theta$ be two large cardinal axioms. Define the **convergence number of $\sigma, \theta$** ($\alpha_{\sigma, \theta}$) to be $min\{\gamma\in Ord~|~\sigma =_\gamma \theta\}$ if $\{\gamma\in Ord~|~\sigma =_\gamma \theta\}\neq \emptyset$ and $\infty$ if$\{\gamma\in Ord~|~\sigma =_\gamma \theta\}=\emptyset$.

**Example. $\alpha_{WI,SI}=1$**

**Question 1.** Is $\alpha_{\sigma,\theta}<\infty$ for each two large cardinal axioms $\sigma, \theta$?

**Question 2.** Is there an ordinal $\alpha_0$ such that for each two large cardinal axiom $\sigma,\theta$ we have $\alpha_{\sigma,\theta}\leq \alpha_0$?

**Question 3.** What is the convergence number of "existence of a strongly compact cardinal" and "existence of a superstrong cardinal"?

**Question 4:** Consider $\mathbb{L}$ be the set of large cardinal axioms. Is there any ordinal $\alpha$ such that the $\langle \mathbb{L}, \leq_{\alpha}\rangle$ be a *linear order*?

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