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By "tree," it is meant "single-rooted tree" in this sense. An $\omega$-tree is a tree of height $\omega$ with all of its levels finite. An antichain is a set of mutually incomparable elements of the tree.

This clearly implies the second of G & T's statement above (with "either" removed), and I suspect that it is what was intended by G & T, in the first place. Most notably, according to this site, H & R's text cites that Form 10 is stronger than Form 216, but that it was not known to be strictly stronger. This leads me to suspect (even more strongly) an error on G & T's part. Obviously, if it holds in $\mathsf{ZF}$ that every $\omega$-tree with an infinite antichain must have an infinite chain, then it isn't strictly stronger, but I'm unable to prove this or find any information confirming/denying this. Does anyone know whether this is true, false, or independent of $\mathsf{ZF}$?

By "tree," it is meant "single-rooted tree" in this sense. An $\omega$-tree is a tree of height $\omega$ with all of its levels finite. An antichain is a set of mutually incomparable elements of the tree.

This clearly implies the second of G & T's statement above (with "either" removed), and I suspect that it is what was intended by G & T, in the first place. Most notably, according to this site, H & R's text cites that Form 10 is stronger than Form 216, but that it was not known to be strictly stronger. This leads me to suspect (even more strongly) an error on G & T's part. Obviously, if it holds in $\mathsf{ZF}$ that every $\omega$-tree with an infinite antichain must have an infinite chain, then it isn't strictly stronger, but I'm unable to prove this or find any information confirming/denying this. Does anyone know whether this is true, false, or independent of $\mathsf{ZF}$?

Each of the following statements imply those beneath it.

 

• The countable union of finite sets is countable.

 
• Every $\omega$-tree has either (sic) an infinite chain or an infinite antichain.

 
• Every countable collection of [non-empty] finite sets has a choice function.

Each of the following statements imply those beneath it.

• The countable union of finite sets is countable.

• Every $\omega$-tree has either (sic) an infinite chain or an infinite antichain.

• Every countable collection of [non-empty] finite sets has a choice function.