The different types (not sizes!) of infinity

Starting with the natural numbers:

Cardinals: How many are there?

Ordinals: What comes next?

Limits: Where are they going?

Hyperreals: What is bigger than all of them?

So, there are many mathematical posts of this form it would be extremely low-effort for me to write, and I'd be happy to write more such posts, except that I don't know what people are confused about in this way. Anyone have any requests and / or confusions they'd be willing to share?

I think Surreals and Hyperreals are serving a very similar function. The Surreals take a total order as a foundation, and then push that up to infinity, then define algebraic opperations. The Hyperreals start with the large set of indexed lists of real numbers, define algebraic opperations pointwise, and then to make them totally ordered. They both seem like they are about combining ordered set, and infinity, and getting the algebraic stuff for free.

Ordererd set is about $<$ and $>$, which is what defines the Surreals/Hyperreals. Ordinals are not abot that, and are instead about pushing the notion of successor to infinity.

Paul Graham recently wrote a short essay about the value of things that are both general and surprising. I used to think infinities were of different sizes, not types, and I consequently didn't know what the types were. The post cleared up this basic yet deep confusion for me really quickly, and I expect similarly for many others. For these reasons, I've curated this post.

"So, does 1+ω make sense (as something different from ω)? It does, for the ordinals and hyperreals only. "

I am confused because for ordinals, 1+ω = ω. Did you mean ω+1?

I've been keeping 330 browser tabs open with the intention of getting back to each and every one of them some day. And finally one of those tabs comes in handy! This just proves that I should never close anything.

This is a video explaining the distinctions between cardinals and ordinals. This post may be useful in letting people know that there are different types of infinities, but it does nothing for actually explaining them. There are probably other good resources available online for those who want to know, but this is the only one I've ever seen. (Wikipedia is hopeless here.)

Hmm, your post was at the top two days ago, and now it's at the top of today's posts. Does editing adjust the post date forward? This seems like a bug that many people will want to exploit :-)

More generally, don't assume that the same word means the same thing in different fields. Although, in this case, I'd say it's the mathematician's fault for being so incredibly obtuse.

But what about your friend and Cantor's argument? Was she confusing the "size" infinities with the "limit" infinities?

To add to the confusion, there are normally only two symbols for infinity to go around - ω and ∞

This post is the first I recall seeing ω used to refer to a cardinal. (The smallest infinite cardinal is usually called א‎₀.)

And I think hyperreals are probably a much less common topic than the other three, so at least we have different notation for the top three of these four notions (ω for the ordinal, א‎₀ for the cardinal, ∞ for the limit).

I'm not confident in this, but also: It looks to me like ω can be considered the same number in the ordinals, surreals, and hyperreals (since the surreals contain both of the other two). So I guess that's probably why we use the same notation, and why it's not really a serious ambiguity.

So, does 1+ω make sense? It does, for the ordinals and hyperreals only.

It make sense for cardinals (the size of "a set of some infinite cardinality" unioned with "a set of cardinality 1" is "a set with the same infinite cardinality as the first set") and in real analysis (if lim f(x) = infinity, then lim f(x)+1 = infinity) too.

What about -1+ω? That only makes sense for the hyperreals.

And for cardinals (the size of the set difference between "a set of some infinite cardinality" and "a subset of one element" is the same infinite cardinality) and in real analysis (if lim f(x) = infinity, then lim -1+f(x) = infinity) too.

Many of these notions of infinity simply add resolution to other notions of infinity. In the finites, what counts as a distinct number is generally clear (except for the infintesimals), but for the infinites, sometimes it is convenient to merge what could be considered distinct numbers togethers. For example, since there is a bijection from X to X+1, and hence from X to X+n, for any finite n, we are tempted to treat all these entities as the same as we really can't distinguish them very well. Unfortunately, some people conflate an inability to distinguish elements as the seperate elements not having their own existence, but it is well known, for example, that there are numbers that exist, but which can't be explicitly described and no-one says that these elements are all the "one" element.