Kojman menachem begin biography
Set Theory
Roy Shalev (BIU)
-
A dichotomy for transitive lists, part 1
Place: Seminar roomWe present a dichotomy statement concerning a subclass of transitive lists which is a consequence of Martin's Axiom and in fact follows both from K'_2 and from the Martin Axiom for Y-cc posets.
This dichotomy implies that the bounding number is bigger than w1, that every w1-Aronszajn tree is special, moreover, that there are no w1-Souslin lower semi-lattices.
Next, we prove that an higher analog for w2 of the dichotomy is consistent with CH, assuming the existence of a weakly compact cardinal.
This result extends the Laver-Shelah theorem on the consistency of CH with the w2-Souslin Hypothesis, as here one gets a CH model with no w2-Souslin lower semi-lattices.
Based on a joint work with Stevo Todorčević and Boriša Kuzeljević.
Inbar Oren (HUJI)
-
Indecomposable ultrafilters and collapsing successors of singular cardinals
Place: Seminar roomIn a famous paper by Ben David and Magidor, they show the existence of an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1}, in an intermediate model of a forcing that collapses \aleph_{\omega+1}. This is an indication that an (\aleph_0,\aleph_\omega)-indecomposable ultrafilter on \aleph_{\omega+1} might be enough to collapse \aleph_{\omega+1} by a forcing extension.
I will show that if \lambda is a singular cardinal, strong limit and there is an (\aleph_0,\lambda)-indecomposable ultrafilter on \lambda^+, then there is a \lambda^{++} c.c. forcing notion that collapses \lambda^+ and preserves \lambda.
As a corollary we will get that there can't be a (\aleph_0,\lambda)-indecomposable ultrafilter on \lambda^+ if cf(\lambda)>\omega_1, in partcular there is no (\aleph_0,\aleph_{\omega_2})-indecomposable ultrafilter on \aleph_{\omega_2+1}.
Assaf Rinot
-
How to contsruct a Countryman line
Place: Seminar roomWe revisit Shelah's 1976 construction of a
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(Perlman Prize)
.
.
.
.
.
.
(Tsafriri Lecture)
.
.
(Erdos lecture)
.
.
.
Most interesting mathematics mistake?
C.N. Little listing the Perko pair as different knots in 1885 (10161 and 10162). The mistake was found almost a century later, in 1974, by Ken Perko, a NY lawyer (!)
For almost a century, when everyone thought they were different knots, people tried their best to find knot invariants to distinguish them, but of course they failed. But the effort was a major motivation to research covering linkage etc., and was surely tremendously fruitful for knot theory.
Update (2013):
This morning I received a letter from Ken Perko himself, revealing the true history of the Perko pair, which is so much more interesting! Perko writes:
The duplicate knot in tables compiled by Tait-Little [3], Conway [1], and Rolfsen-Bailey-Roth [4], is not just a bookkeeping error. It is a counterexample to an 1899 "Theorem" of C.N. Little (Yale PhD, 1885), accepted as true by P.G. Tait [3], and incorporated by Dehn and Heegaard in their important survey article on "Analysis situs" in the German Encyclopedia of Mathematics [2].
Little's `Theorem' was that any two reduced diagrams of the same knot possess the same writhe (number of overcrossings minus number of undercrossings). The Perko pair have different writhes, and so Little's "Theorem", if true, would prove them to be distinct!
Perko continues:
Yet still, after 40 years, learned scholars do not speak of Little's false theorem, describing instead its decapitated remnants as a Tait Conjecture- and indeed, one subsequently proved correct by Kauffman, Murasugi, and Thistlethwaite.
I had no idea! Perko concludes (boldface is my own):
I think they are missing a valuable point. History instructs by reminding the reader not merely of past triumphs, but of terrible mistakes as well.
And the final nail in the coffin is that the image above isn't of the Perko pair!!! It's the `Weisstein pair' $10_{161}$ and mirror $10_{163}$, described by Perko as "those magenta
General Works on the History of Logic in Western Thought
Angelelli, Ignacio, and Cerezo, María, eds. 1996. Studies on the History of Logic. Proceedings of the III. Symposium on the History of Logic. Berlin: Walter de Gruyter.
Contents: Preface V; List of Contributors XI; Mario Mignucci: Aristotle's theory of predication 1; Robin Smith: Aristotle's regress argument 21, Hermann Weidemann: Alexander of Aphrodisias, Cicero and Aristotle's definition of possibility 33; Donald Felipe: Fonseca on topics 43; Alan Perreiah: Modes of scepticism in medieval philosophy 65; Mikko Yrjönsuuri: Obligations as thoughts experiments 79; Angel d'Ors: Utrum propositio de futuro sit determinate vera vel falsa (Antonio Andrés and John Duns Scotus) 97; Earline Jennifer Ashworth: Domingo de Soto (1494-1560) on analogy and equivocation 117; Allan Bäck: The Triplex Status Naturae and its justification 133; William E. McMahon: The semantics of Ramon Llull 155; Paloma Pérez-Ilzarbe: The doctrine of descent in Jerónimo Pardo: meaning, inference, truth 173; Jeffrey Coombs: What's the matter with matter: Materia propositionum in the post-medieval period 187; Rafael Jiménez Cataño: Copulatio in Peter of capua (12th century) and the nature of the proposition 197; Lynn Cates: Wyclif on sensus compositus et divisus 209; Mauricio Beuchot: Some examples of logic in New Spain (Sixteenth-Eighteenth century) 215; Adrian Dufour: necessity and the Galilean revolution 229; Guy Debrock: Peirce's concept of truth within the context of his conception of logic 241; Pierre Thibaud: Peirce's concept of proposition 257; Jaime Nubiola: Scholarship on the relations between Ludwig Wittgenstein and Charles S. Peirce 281; José Miguel Gambra: Arithmetical abstraction in Aristotle and Frege 295; Herbert Hochberg: The role of subsistent propositions and logical forms in Russell's 1913 Philosophical logic and in the Russell-Wittgenstein dispute 317; Alfonso García Suárez: Are the objects of the Tractatus