My guess would be to collapse some sort of large cardinal to omega1 and make that the first step of an iterative forcing of counterexample to each sequence. Picking support, making sure you stop stuff that crops up along the way, etc. This might be massive overkill though.
1. Re: your sketch of Con(CH + ~Diamond). Wow, yeah that sounds like something that might work. I thought you could get it without LCAs, but maybe not (I don't think I have any evidence for that assertion).
2. Doing Con Diamond via forcing. I should probably learnt the L proof at some point, but I wanted to get an example of a countable approximation forcing on there.
3. New syllabus isn't quite finished: the topology and forcing sections are finished but the "further set theory" section needs more work. There's some stuff that we've definately cut and some stuff we're definately not cutting and then a large amount in between that Leo said he'd be happy for me to have or not have, but I should check with Steel about them. The big issue at the moment is whether I put extenders on or not. I think I should, but it might involve cutting quite a bit of other stuff.
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My guess would be to collapse some sort of large cardinal to omega1 and make that the first step of an iterative forcing of counterexample to each sequence. Picking support, making sure you stop stuff that crops up along the way, etc. This might be massive overkill though.
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Also would definitely be interested in seeing your new syllabus. If you don't intend to soon post it somewhere, any chance you could email it to me?
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2. Doing Con Diamond via forcing. I should probably learnt the L proof at some point, but I wanted to get an example of a countable approximation forcing on there.
3. New syllabus isn't quite finished: the topology and forcing sections are finished but the "further set theory" section needs more work. There's some stuff that we've definately cut and some stuff we're definately not cutting and then a large amount in between that Leo said he'd be happy for me to have or not have, but I should check with Steel about them. The big issue at the moment is whether I put extenders on or not. I think I should, but it might involve cutting quite a bit of other stuff.
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