Downward Löwenheim-Skolem Theorems and Choice PrinciplesThere are no comments on this post.
Mar 31 2014, 04:00
I have posted a new note on the Papers page.
It's a short little proof that the classic downward Löwenheim-Skolem theorem is equivalent to \(\DC\), and that for a well-ordered \(\kappa\), the downward Löwenheim-Skolem asserting the existence of models of cardinality \(\leq\kappa\) is in fact equivalent to the conjunction of \(\DC\) and \(\AC_\kappa\).
The proof is quite straightforward and not very long.
Interestingly enough, despite not appearing on the "usual" choice dictionaries, this was known for quite some time. It appeared in a book by George Boolos, and it was independently found by Christian Espindola (not too long ago as well!). You can find his versions on his homepage.
Any comments on the note, or suggestions for improvements and extensions are more than welcome here on this page.
Downward Löwenheim-Skolem Theorems and Choice Principles.