I'm largely fine with KingPin's suggestions, yeah. Given both DontTalk and I agreed to postpone our above discussion for later, I don't exactly have much to say beyond that.
Do wish I wasn't.
Thank you for helping out.
A bit of an unlucky formulation on my part there. When I said regular cardinal there, I didn't mean regular in a mathematical sense. I meant regular as in ZFC non-large / accessible cardinal. Replace the word regular with "accessible" maybe. Like:
The point of that part was to make clear that even the amount of levels of infinity we are talking about themselves are of a massively uncountable degree. You know, to make the quantitative difference between the levels more graspable for those with some understanding of how ZFC cardinals work.
Oh, okay. I see what you mean. Tell me how this looks, then:
I'm not sure that you can say that they are bigger than any aleph numbers actually. Do you have some proof of that?
Well, technically, inaccessible cardinals can be written as aleph-k for some inaccessible cardinal k, and the same holds for... pretty much any large cardinal property, really. The issue is that large cardinals, by their general definition, are literally "too big" for ZFC (why do you think they're called large cardinals?), and while the consistency strength of large cardinal axioms =/= the size of the large cardinals that observe them (huge cardinals seem to be smaller than supercompact cardinals, for instance, despite the existence of a huge cardinal being a "stronger statement" than the existence of a supercompact cardinal.) Simply put, the large cardinal hierarchy is kind of just nonlinear.
Now, I do wanna ask: from where do you get your claim that inaccessible cardinals can't be proven to be larger than anything in ZFC? An inaccessible cardinal, at least for our purposes, is a cardinal number k with the following properties:
Well, you asked this earlier:
And since the next large cardinal property after the inaccessible cardinal property (which is currently the basis for High 1-A) is the Mahlo cardinal property, I naturally assumed that that's what you wanted. On the other hand, the system currently doesn't seem to accommodate worldly cardinals, which was an issue I used to have back when the tiering revisions were first being carried out, but I don't know if they're like, relevant enough to be included? What do you think?
I mean, for a start I have been unable to find a statement that says that they actually are actually larger. So obviously I have an interest in checking it's actually true.
When I said that what I had in mind was the taking infinite successor cardinals. Part of that suggestion was to use aleph-omega for 1-A so my suggestion for tier 0 was basically just infinite successors after aleph-omega. (Is it aleph-2*omega or aleph-omega^2? I'm not too good with ordinal arithmetic.)
Because it's not really true to begin with, by the way. An "aleph" is at the core just the cardinality of a set that is both infinite and well-ordered (Cardinals are a type of ordinal, after all), and since the Axiom of Choice determines that any set is well-orderable, then there is no infinite cardinal without an aleph number. You can have one if you ditch Choice, but by then you already let go of a thing that keeps most of standard mathematics in place, anyway.
Yeah, we actually figured out the fact that it isn't beyond all alephs already. What we were talking about is if one could fix the passage you quote by instead saying "Even just the amount of infinite cardinals between the first inaccessible cardinal and aleph-2 (Which defines 1-A) is greater than any non-large cardinal, such as aleph-0, aleph-1, aleph-2, etc."
See, I'm warry of putting that description in there largely because even the notion of what a "large cardinal" is is extremely ill-defined, and I really don't want to force casual readers to go by such nebulous terms, especially since the purpose of the Tiering System page is to first give them a primer of it all that is then complemented by other explanation pages.
This is dipping its toes into Set-Theoretic Universalism vs Potentialism (That is, the question of whether the Universe of Sets is a single, completed collection that we "discover" larger parts of over time, or something that is constantly expanding as we posit stronger and stronger axioms), which is a whole philosophical conundrum that I'd rather not bring into the table. But if you pointed a gun to my head and forced me to give an answer, then I'd say we could treat it as the former, yeah.
Well, in that case what if we just put it as "Even just the amount of infinite cardinals between the first inaccessible cardinal and aleph-2 (Which defines 1-A) is greater than cardinals such as aleph-0, aleph-1, aleph-2, aleph-3,... or even many alephs which's index in itself is an infinite cardinal."
No? This is a strictly mathematical question. No subjectivity or philosophy in it. It's the kind of stuff one could (potentially, if it's decidable) write mathematical proof about.
Yeah, looks alright to me. Given we've already explained what aleph fixed-points are in the Tiering System Explanation Page, we can link that section of it too, so people aren't confused by the bolded tidbit (EDIT: Yeah, I misread that part. Although, nevertheless, alluding to fixed points probably doesn't hurt if we want to illustrate the size of those things)
Apologies my misunderstanding, then. Although that doesn't really change my point much, as far as I can see. We can still compare two axiomatic systems: ZFC + "There are no Inaccessible cardinals" and ZFC + "There exists an inaccessible cardinal," and in the context of the latter model, the least inaccessible is larger than any cardinal whose existence is demonstrable in the former, just like ω is a proper class in ZFC-Infinity, and not a provable set. There is still a cut-off occuring between the two universes, which is more or less what the Generic-Multiverse view that Woodin proposes is all about (In this case, if you call the original universe of sets M, then you could refer to the two aforementioned models as N and N*, which in this case would be separate extensions of it realizing two statements that are consistent with its language)
I'd change this to something like:
I also don't think the "uncountably infinite" part of tier 0's description is necessary, since I'm sure that even casual users can piece together that "infinite" includes "uncountably infinite" from context.
I'm not really here to add anything but just wanna get clarification on something since I haven't followed up on this thread in a while what's gonna happen to characters who reached atleast baseline Outerversal by transcending "all extensions of space time" without the existence of infinity layered hierarchies within their own setting are they gonna remain as outerversal or something what tier would be assigned to the totality of space time since currently 1-A in the tiering system only talks ab the cardinalities representing those tiers but doesn't mention anything about transcending concepts (all extensions) of space time
