User blog:DontDrinkH20/Some Explanations of Certain Large Cardinals

(I'm sorry I haven't made a post in so long. School + extracurriculars have been a nightmare recently, and that won't change until after the beginning of November.)

Why Do I Need to Know More About Large Cardinals?
I've noticed recently that people on this wiki seem to make some pretty strange (from a set theorist's perspective) claims about large cardinals. For example, there was the whole "stage cardinal" idea, which I found strange and very counterintuitive (which is not the ideal adjective I would want to describe my intuition-based OCF).

Of course, you know why you need to know about large cardinals. They are pretty valuable in OCFs, and in fact reverse mathematics as well as fast-growing hierarchies in general. They are also valuable to set theory that isn't quite relevant to googology, for example ultrapowers, consistency strength, saturated ideals, determinacy and descriptive set theory, infinite graph theory, model theory, partitions, infinite combinatorics, and in a couple cases, category theory (Woodin-For-Supercompactness cardinals have a nice category theoretical definition, and the existence of a proper class of measurables as well).

Most of that stuff is pretty irrelevant to googology, so it makes sense that the public involved in large numbers wouldn't know much about them, just as I have no idea what the laws of derivatives or calculating integrals or cones in category theory are. People don't know things that they don't know, and that's OK. The problem is that some people on this site may tend to (subconsciously) act as though they know things that they don't (which is OK also, because they don't know that they don't know it). This is only a problem in practicality, which I aim to fix with this blog post.

The Problems
The problems are all based on when people have intuition about things even though they don't quite know the true nature of them. I think this breaks down into three problems: Underestimation, Lack of Formalization, and Misunderstanding about consistency strength.

The First Problem: Underestimation
There's a lot of intuition about large cardinals floating around in the OCF-making community. Recently we saw the "stage" cardinals in the UNOCF (and related OCFs). These appeared strictly out of the intuition about large cardinals present in the community. In reality, to a set theorist they are very counterintuitive and seem to be impossible to formalize.

I think stage cardinals are a perfect example of how underestimation made its way into the community. The idea is that "inaccessibles are to Mahlo cardinals as Mahlo cardinals are to weakly compacts" and the question became "how do I generalize that further?"

This assumption was based off of the concept that weakly compact cardinals are the 'smallest kind of cardinal suitable for a Mahlo-diagonalizing OCF.' The problem with this is that it's nowhere close to true, and people often GREATLY underestimate the true size of even \(1\)-Mahlo cardinals.

COMMON MISCONCEPTIONS:
 * 1) "\(1\)-Mahlo cardinals are just Mahlo limits of Mahlo cardinals." In reality, the least \(1\)-Mahlo cardinal \(\kappa\) is a Mahlo limit of Mahlo limits of Mahlo limits of ... (\(\kappa\)-many times the word "limit" is said) Mahlo cardinals. The least \(1\)-Mahlo cardinal \(\kappa\) is also a limit a Mahlo limit of those kinds of cardinals, and so on and so forth. The amount of Mahlo-ness is ridiculous already.
 * 2) "The least weakly compact is weakly inaccessible but not strongly inaccessible." As far as we know, there is no 'version of weakly compact cardinals which doesn't imply strong limit-ness.' Weakly compact cardinals inherently are strong limits by definition, and there's no way to change it.
 * 3) "Weakly compact cardinals are to Mahlo cardinals as Mahlo cardinals are to inaccessibles." If you study a bit of large cardinal theory, you may eventually come to understand that in reality \(1\)-Mahlo cardinals are to Mahlo cardinals as Mahlo cardinals are to inaccessibles, \(2\)-Mahlo cardinals are to \(1\)-Mahlo cardinals as \(1\)-Mahlo cardinals are to Mahlo cardinals, and so on. The least weakly compact cardinal \(\kappa\) is \(\kappa^+\)-Mahlo and a limit of cardinals \(\lambda\) which are \(\lambda^+\)-Mahlo, and still more than that.
 * 4) "Mahlo cardinals appear in the lower forms of indescribability." Indescribability can only give you inaccessibles and weakly compacts, but nothing in between. The "connection between Mahlo cardinals and weakly compact cardinals" doesn't rely on indescribability, and in fact in my opinion doesn't really exist. Weakly compact cardinals are just too massive to even be compared to Mahlo cardinals and inaccessible cardinals.
 * 5) "Indescribable cardinals are useful in OCFs." Indescribability is inherently a property which states roughly 'combinatorial principles which apply at these cardinals apply many times below it.' Any notion of 'being useful' in an OCF is a combinatorial property, and therefore indescribable cardinals are never the smallest kind of cardinal you could use to make the OCFs. Indescribable cardinals are too... indescribable for that, I guess.

'''Take note of that last bullet point. I think the next thing that should be used in OCFs are other combinatorial cardinals such as subtle cardinals, ethereal cardinals, ineffable cardinals, etc. Eventually, it should reach Ramsey, and I believe it should be difficult to get past measurable.''' (This is all based upon my intuition about OCFs, which happens to be about as strong as your intuition about large cardinals.)

The Second Problem: Lack of Formalization
When it comes to large cardinals on this wikia, people tend not to do proper research on what a large cardinal is actually defined as. This is understandable. After all, the definitions of the cardinals given in any website or textbook rely heavily on set theoretical jargon and are very confusing (it would be like reading an entire real analysis book just to get the definition of a perfect set for set theory).

However, without understanding the definition of a large cardinal, you can't understand it's nature either. This is the reason (I believe) why people so badly misunderstand the nature of Mahlo cardinals and weakly compact cardinals. As a result, people don't actually use the definitions of the cardinal nor do they use the nature of the cardinal, but rather they use the intuition they have based solely on OCF theory.

This is a difficult problem to fix. I recommend just learning enough set theory to do the job (like Nish), or at least just asking a set theorist for help in these situations. Emlightened and I know stuff about large cardinals. Feel free to ask us about it if you are having trouble with it or need to formalize something.

The Third Problem: Misunderstanding Consistency Strength
If you look on Cantor's attic without understanding what the 'height' of a cardinal in the diagram truly means, you may just assume it means "size." In reality, "the least cardinal with property A is smaller than the least cardinal with property B" is not what it means by having property A lower than property B on the table.

Large cardinals are 'large' when they have lots of consistency strength. Roughly speaking, this means their existence implies many, many things, going up meaning it implies more and more things (until it implies a contradiction).

A great example of this is Jónsson cardinals. The existence of a Jónsson cardinal implies that "\(x^{\#}\) exists for every real \(x\)" which, for those who don't understand what the sharp of a set is, just implies that there are a lot of models of ZFC which are small compared to \(V\) but very big compared to set-sized models.

Jónsson cardinals are consistency-wise very, VERY strong (equiconsistent with Ramsey cardinals). As a result, they are 'large' in this sense. On the other hand, the least uncountable singular cardinal \(\aleph_\omega\) may be Jónsson (you can't prove that it is, but you probably can't prove that it's not either)

Another example is "huge" cardinals. Due to the name, people often assume that they are in fact, the "biggest" cardinal you can get to. In reality, an extendible cardinal, if it exists, is MUCH MUCH bigger, although consistency-wise weaker. The reason they are called huge is merely because they imply so much more things than other large cardinals do. (Note: Superhuge cardinals are actually incredibly large size-wise, and are about as big as you can get to without going to contradictory notions [although Ultrahuge cardinals are bigger]).

What Now?
I don't know how to end this blog post. I'm just writing bullcrap as if it's a conclusion and there's some big purpose to this post. Have a good day I guess, and thanks for reading this far!