I gave an invited talk at the P.O.I Workshop in pure and descriptive set theory, Torino, September 26, 2015.
Title: $\aleph_3$-trees.
Abstract: We inspect the constructions of four quite different $\aleph_3$-Souslin trees.
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Keywords
weak diamond Lipschitz reduction Well-behaved magma Commutative cancellative semigroups Hindman's Theorem Aronszajn tree ZFC construction stick Generalized descriptive set theory Hereditarily Lindelöf space Analytic sets Greatly Mahlo Small forcing Strong coloring Nonspecial tree Subtle cardinal Reduced Power Open Access Weakly compact cardinal super-Souslin tree Ulam matrix Singular cofinality incompactness square Respecting tree Cohen real Diamond Uniformly homogeneous Generalized Clubs Partition Relations Antichain Knaster and friends HOD Reflecting stationary set Square-Brackets Partition Relations SNR Whitehead Problem Diamond for trees Knaster positive partition relation club_AD Diamond-sharp Ineffable cardinal free Souslin tree b-scale tensor product graph higher Baire space OCA stationary reflection Filter reflection PFA(S)[S] Forcing approachability ideal Cardinal Invariants Parameterized proxy principle Hedetniemi's conjecture Minimal Walks full tree square principles P-Ideal Dichotomy coloring number O-space weak square Foundations Erdos Cardinal ccc Non-saturation Luzin set nonmeager set Ramsey theory over partitions Subnormal ideal Iterated forcing Sigma-Prikry Kurepa Hypothesis countably metacompact indecomposable ultrafilter very good scale Sakurai's Bell inequality Club Guessing AIM forcing unbounded function Fast club Large Cardinals sap diamond star Erdos-Hajnal graphs Singular Density S-Space Intersection model reflection principles Ostaszewski square Sierpinski's onto mapping principle Ascent Path Strongly compact cardinal Chang's conjecture Almost Souslin PFA transformations Almost-disjoint family Constructible Universe Commutative projection system regressive Souslin tree Universal Sequences Was Ulam right Rock n' Roll Dowker space Uniformization Microscopic Approach Chromatic number Jonsson cardinal Countryman line Mandelbrot set Uniformly coherent middle diamond C-sequence Closed coloring free Boolean algebra Subadditive Successor of Regular Cardinal xbox specializable Souslin tree Successor of Singular Cardinal GMA Selective Ultrafilter Vanishing levels Postprocessing function Slim tree Souslin Tree Forcing Axioms Prikry-type forcing Subtle tree property Coherent tree Poset Rainbow sets Amenable C-sequence Rado's conjecture Distributive tree Dushnik-Miller projective Boolean algebra weak Kurepa tree Martin's Axiom 54G20 Cardinal function Almost countably chromatic stationary hitting Precaliber Fat stationary set Prevalent singular cardinals Strongly Luzin set Absoluteness L-space strongly bounded groups polarized partition relation Local Club Condensation. Fodor-type reflection Shelah's Strong Hypothesis Axiom R Singular cardinals combinatorics
after my talk, David Aspero was referring to page #103 of my slides, asking why do we appeal to fine structure, where we could also use forcing to obtain the relevant combinatorial objects.
It took me a few minutes to find the right answer (the spontaneous one was – “I just like L”). The deeper answer is that we actually appeal to L to help us *formulate* the right principle. After all, there is no obvious way for a principle combining square and diamond-plus to be defined. It is the iterative process of defining/trying-to-prove-in-L/revisiting-the-definition/retrying-to-prove-in-L/etc’ that led to the “right” definition.