A well-known theorem about infinite trees that if a tree only grows a finite amount at each node, it must have an infinite branch. This is known as König’s lemma. Obviously, an uncountable tree that only grows a countable amount at each node must have an uncountable branch, right? Amazingly, the answer is no. Counterexamples, now known as Aronszajn trees, were constructed by Nachman Aronszajn. Keith Devlin has more.
Yeah, Aronszajn trees really ought not to exist, but you can get used to them. The really weird ones are Suslin (or Souslin) trees. Not only is each level countable, but so is each anti-chain. I know that it’s consistent that Suslin trees don’t exist - but I think it’s also consistent that they do (at least, assuming the consistency of some relatively small large cardinal).
Is a “relatively small large cardinal” more like a Jumbo Shrimp or a Dwarf Mammoth?
Kenny, that’ll be today’s post.
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Dwarf Mammoth.