Set Theory

From MGSA

The Logic page contains more set theory questions.

Adam Booth

My committee was John Steel, Hugh Woodin, Leo Harrington (chair) and George Bergman (outside member).

What can $2^{\aleph_0}$ be? Sketch a proof.
Is the product of ccc forcings necessarily ccc? What about the iteration? How are products and iterations related? Can you use this to prove that under MA+-CH the product of ccc forcings is cc?
What's diamond? Is it equivalent if you change stationary to non-empty? How about unbounded? How about if rather than a Diamond-sequence being a sequence of subsets of $ω 1$ , it's a sequence of countable sets of subsets of $ω 1$ and we just require $A \cap \alpha \in \Sigma \alpha$ for stationarily many $α$ . Does Diamond imply CH? Does CH imply Diamond? Does GCH imply Diamond ${}_{\omega_2}$ ?
Show L models GCH. Are there any $α$ less than $ω 1$ at which no new real appears in $L α$ ? How long a gap can we get between ordinals where new reals occur?

What can you say about subsets of $ω 1$ under AD? How complex can such a subset be to code as a set of reals? This segued onto stuff about $a^\sharp$ under AD. Show subsets of $ω 1$ can be found in $L [a]$ for some real $a$ under AD.