n things a set theorist should have done at least once

So there's a meme going around in the CS blogging community: "Things that a ___ researcher should have done at least once". It started at The Geomblog by Suresh Venkatasubramanian, was picked up for complexity theory by Lance Fortnow at Computational Complexity and for game theory & economics by Noam Nisam at Turing's Invisible Hand.

Now, I'm not really qualified to count up to n, but I thought I'd give it a try anyway.

1. Prove something to be independent of ZFC. (Bonus points if it is independent of CH as well.)
2. Find a "natural" statement and prove it's equivalent to CH.
3. Find a new forcing axiom (that is actually useful).
4. Find a new type of ultrafilter.
5. Find a new large cardinal.
6. Use a non-standard model of PA.
7. Use PCF theory.
8. Find a new cardinal characteristic of the continuum.
9. Collapse some cardinal characteristics of the continuum.
10. Separate some cardinal characteristics of the continuum. (Bonus points if the balance is kept.)
11. Improve a consistency result to a ZFC result.
12. Have an application outside of set theory.
13. Write a paper with Saharon Shelah.

What else?

Update Oct 1, 2012: there's now also a graph theory one by Derrick Stolee at Computational Combinatorics.

Comments

• Thony C, 2012/09/30
  Define an undefinable set.
  • Peter, 2012/09/30
    Nice!
• saf, 2012/09/30
  a variation of 11: prove a ZFC result via forcing
  • Peter, 2012/09/30
    Nice. I thought about adding that but then I decided that 3 should cover that. but you’re probably right – it deserves independence (pardon the pun).
• Asaf, 2012/10/01
  Find a new choice principle.
  • Asaf, 2012/10/01
    Bonus points: find its location within the common principles’ hierarchy. E.g., it implies countable choice but not dependent choice.
    • Peter, 2012/10/01
      Very nice — I didn’t think about choice principles (comes from working with ultrafilters 😉 )
• saf, 2012/10/01
  Have the following statements included somewhere in your proof:
  (a) “take a countable elementary submodel of \langle \mathcal H(\lambda)\,\in\,\lt _ \lambda\,\ldots for a large enough regular cardinal \lambda, containing everything relevant to this proof"
  (b) “note that by absoluteness, it suffices to prove that "1+1=2" holds in L"
  • Peter, 2012/10/01
    Thanks! Good stuff :)
• David Roberts, 2012/12/09
  What about proving something independent of ZF? (over ZF!)
  • Peter, 2012/12/09
    Not my cup of tea, but nice!