There will be a Very Informal Gathering of
Logicians (VIG) at UCLA, from Friday,
February 1, to Sunday, February 3, 2019.
The 20th in a series of biennial logic meetings at UCLA, this event will
celebrate the 50th anniversary of the 196768 Logic Year at UCLA and the
many influences it had in Mathematical Logic.
The meeting is supported by NSF grant DMS1901676.
Tentative schedule of talks:
Friday, February 1
2:00  2:10 
Opening Remarks 
2:10  3:00 
Menachem Magidor,
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3:30  4:20 
Grigor Sargsyan,
Towards superstrong cardinals
Assume AD\(_{\mathbb R}\) and suppose there is no iteration strategy for a model with a superstrong cardinal. We will introduce the following two conjectures,
 The Direct Limit Independence and
 Bounding.
We will then show how 1 and 2 above imply Generation, which is the statement that every set of reals is Wadge reducible to an iteration strategy of some finestructural, perhaps hybrid, countable model.
We will then show how to prove both conjectures assuming there is no iterable countable structure with a Woodin cardinal that is a limit of Woodin cardinals.
As a consequence, we obtain that Generation holds below a Woodin cardinal that is a limit of Woodin cardinals. We will also explain a method of attacking both 1 and 2.
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4:30  5:20 
Robin TuckerDrob,
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5:30  7:00 
Reception 
7:00  7:50 
Dana Scott,
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Saturday, February 2
8:45  9:30 
Breakfast 
9:30  10:20 
Chris Laskowski,
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10:30  11:20 
Julia Knight,
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11:30  12:20 
Brandon Seward,
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2:00  2:50 
The 2019 Hjorth Lecture
Justin Moore,
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3:30  4:20 
Anush Tserunyan,
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4:30  5:20 
Hugh Woodin,
Two applications of finestructure to the theory of AD\(^+\)
The final synthesis of AD\(^+\) and fine structure will yield many theorems. For example, it will yield the
theorem that assuming AD\(^+\) and \(V = L({\mathcal P}({\mathbb R}))\), GCH holds in HOD.
AD\(^+\) is a technical refinement of the axiom AD and the context is that \(V = L({\mathcal P}({\mathbb R}))\), together with DC for reals and ZF. It is known that AD\(^+\) is not a strengthening of AD in terms of consistency. For example, AD\(^+\) must hold if \(V = L({\mathbb R})\) and AD holds. Further, assuming ZF + DC, and that all real games are determined, then AD\(^+\) must hold in \(L({\mathcal P}({\mathbb R}))\). AD\(^+\) is simply a structural refinement of AD and it is conjectured that AD implies AD\(^+\).
We give two recent examples where one can prove the predicted theorems now, without appealing to that final synthesis. The first example is that assuming AD\(^+\), then there can be at most one Woodin cardinal in HOD below \(\Theta_0\) where \(\Theta_0\) is the supremum of the lengths of the prewellorderings of the reals which are ordinal definable.
To set the context, it is a theorem of AD that \(\Theta_0\) is a Woodin cardinal in HOD. It is also theorem of AD\(^+\) that \(\Theta_0\) is the least Woodin cardinal in HOD\(_x\) for a Turing cone of \(x\). This strongly motivates the conjecture that assuming AD\(^+\), \(\Theta_0\) is the least Woodin cardinal in HOD, and this is actually one of the predicted theorems. While we cannot prove this predicted theorem, we can prove the indicated close approximation. The main application is that if \(V = \)Ultimate \(L\) then the \(\Omega\) Conjecture must hold.
We note that there are many examples from the theory of AD illustrating that the "lightface" version of a "boldface" theorem can be significantly more difficult. To illustrate, assuming AD, CH must hold in HOD\(_x\) for a Turing cone of \(x\). In fact, assuming AD\(^+\) and \(V = L({\mathcal P}({\mathbb R}))\), CH must hold in HOD but that is more subtle since without assuming \(V = L({\mathcal P}({\mathbb R}))\), CH can fail in HOD.
The second example solves a question about just \(L({\mathbb R})\) (where the synthesis is known) and which involves just countable unions of projective sets. But this solution uses methods completely outside the usual methods of descriptive set theory and also gives a general AD\(^+\) theorem.
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5:30  6:20 
Tony Martin,
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7:00  10:00 
Party 
Sunday, February 3
Supported by NSF grant DMS1901676.
