Apologies for the long title!
So I've been reading Rucker's Infinity and the Mind and came across this:
$$ \theta = \aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}} $$
Previously, theta was defined by Rucker as:
$$ \theta = \aleph_{\theta} $$
I can't really find this terminology anywhere else other than Cantor's attic or Wikipedia, both of which use the uppercase theta instead (Θ), saying that it is the "supremum of the pre-well-orderings on the reals".
Are these two thetas the same thing? i.e., does:
$$ \theta = \aleph_{\theta} = \aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}} = \Theta $$
And finally, where does the first beth-fixed point fit into all of this? Cantor's Attic suggests that it is constructed similarly to $ \aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}} $, saying that $ \kappa = \beth_\kappa $ just as $ \kappa = \aleph_\kappa $.
So I guess the short question is, does:
$$ \theta = \aleph_{\theta} = \aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}} = \Theta = \kappa = \aleph_\kappa = \beth_\kappa = \beth_{\beth_{\beth_{\beth_{\ddots}}}} $$
?
You're right to be confused; Rucker is using very ideosyncratic notation here. These are very different objects.
Let's focus on the cardinals $$\kappa=\aleph_{\aleph_{\aleph_{...}}}\mbox{ and }\lambda=\beth_{\beth_{\beth_{...}}}$$ first (the remaining one is much more complicated). First, we have to unravel what these notations mean. They're shorthand: "$\aleph_{\aleph_{\aleph_{...}}}$" means "the supremum of the sequence $\aleph_0$, $\aleph_{\aleph_0}$, $\aleph_{\aleph_{\aleph_0}}$, ..., and similarly "$\beth_{\beth_{\beth_{...}}}$" means "the supremum of the sequence $\beth_0$, $\beth_{\beth_0}$, $\beth_{\beth_{\beth_0}}$, ...
It's easy to check that $\kappa$ is the least cardinal satisfying the equation $\kappa=\aleph_\kappa$, and $\lambda$ is the least cardinal satisfying $\lambda=\beth_\lambda$; indeed, the construction $$Fix(F)=\sup\{\aleph_0,F(\aleph_0), F(F(\aleph_0)), F(F(F(\aleph_0))), ...\}$$ always gives you the first infinite fixed point of $F$, whenever $F$ is a (continuous nondecreasing) operation on functions.
Note, however, that there are multiple such fixed points! E.g., given any cardinal $\mu$, the limit $\kappa'$ of $\mu, \aleph_\mu, \aleph_{\aleph_{\mu}}, ...$ satisfies $\aleph_{\kappa'}=\kappa'$ and $\kappa'\ge\mu$. Now take $\mu>\kappa$ ... So it's very wrong of Rucker to treat "$\theta=\aleph_\theta$" as a definition of anything; it's an equation which many (indeed, most in a precise sense) cardinals satisfy, not a unique descriptor.
Now what about comparing their sizes? It's easy to show via transfinite induction that $\aleph_\alpha\le\beth_\alpha$ for all $\alpha$; thus, $\kappa\le\lambda$. Moreover, if the generalized continuum hypothesis holds, then $\kappa=\lambda$; however, it is consistent that $\kappa<\lambda$.
Alright, so what about "the supremum of the pre-well-orderings on the reals?" This is what is universally referred to in set theory by "$\Theta$," so that's the symbol I'll use here.
$\Theta$ isn't a very interesting object in ZFC: with the axiom of choice, it's just the successor of the cardinality of the continuum (this is a good exercise). For example, under the continuum hypothesis, $\Theta=\aleph_2$. So in ZFC, $\Theta$ is pretty boring.
If choice fails, however, then things get very interesting! If the reals can't be well-ordered - that is, there is no cardinal in the usual sense corresponding to $2^{\aleph_0}$ - then there is no obvious upper bound on $\Theta$. (Beth numbers in general are wild without choice, and even defining them properly isn't trivial.) Indeed, there might be no uncountable well-orderable sets of reals at all without choice! Looking at pre-well-orderings gets around this, and asking how long these pre-well-orderings can be (that is, asking: what is $\Theta$?) is a decent way to ask about the "cardinality" of the reals in a context where that question doesn't quite mean what it should.
A large part of modern set theory revolves around the idea that the value of $\Theta$ in certain "inner models," often as measured by the "Solovay sequence," is closely connected to both large cardinals and descriptive set theory. This is a very advanced topic, and one that I won't try to summarize here; at a very advanced level, this survey article by Sargsyan is an excellent source (starting in section 2).