But since you had such fun with ordinals here (and here and here), I better add that Ketonen and Solovay later gave a proof based on the ε_{0} stuff and the hierarchy of fast-growing functions. (The variation due to Loebl and Nešetřil is nice and short.) We should talk about this sometime! I wish I understood all the connections better. (Stillwell’s Roads to Infinity offers a nice entry point, though he does like to gloss over details.)

The main surprise is that ordinal addition is not commutative. We’ve seen that , since

Note the missing in the definition of .

]]>Just so other people know: the problem is that with its usual ordering is not well-ordered. Every countable ordinal is isomorphic to a subset of with its induced ordering. So, if you’re only interested in countable ordinals, you can use subsets of But Cantor, of course, was not only interested in countable ordinals.

]]>But why did Cantor need to invent ordinals, when he could just use the elements of to index transfinite processes? is an ordered set.

]]>• Azimuth blog.

It’s a bit out of date, and it’s a huge pile of stuff, but you’ll see there are different series of posts on different topics, and it will help you navigate these posts.

]]>My aha moment of understanding the ordinal numbers is realizing that the function next(x) is total; but not the function prev(x). This is actually due to the well-ordering even though the relationship is a little counter intuitive at first sight. Since every set has a minimum we can first find a minimum min0 for set A and then find the next minimum min1 for A\min0. But prev(ω) is not defined. This is likely equivalent to your observation that addition is not commutative on the ordinals.

To finish the story: in the order topology of ordinal numbers ω1 is a limit point but there is no countable sequence that converges to it. https://en.wikipedia.org/wiki/Order_topology#Topology_and_ordinals

Sorry for the late comment as I only just discovered this treasure trove of yours from the link to your Nobel physics commentary. I have always found that the chronological nature of the blog format a bit of a hindrance to the discussion of mathematical ideas where timeliness is not high on the agenda.

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