Toronto Set Theory

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What is the theory ZFC-Powerset?

by Joel Hamkins | The CUNY Graduate Center and College of Staten Island
Time: 13:30 — 15:00  (Friday, Apr. 22, 2011)
Location: FI210, Fields Institute, 222 College St.
Abstract:
The theory ZFC-, consisting of the usual axioms of ZFC but with the Powerset axiom removed, when axiomatized by Extensionality, Foundation, Pairing, Union, Infinity, Separation, Replacement and the Axiom of Choice, is weaker than commonly supposed, and suffices to prove neither that a countable union of countable sets is countable, nor that $\omega_1$ is regular, nor that the Los theorem holds for ultrapowers, even for well-founded ultrapowers on a measurable cardinal, nor that the Gaifman theorem holds, that is, that every $\Sigma_1$-elementary cofinal embedding $j:M\to N$ between models of the theory is fully elementary, nor that $\Sigma_n$ sets are closed under bounded quantification. Nevertheless, these deficits of ZFC- are completely repaired by strengthening it to the theory obtained by using Collection rather than Replacement in the axiomatization above. These results extend prior work of Zarach. This is joint work with Victoria Gitman and Thomas Johnstone.

Dates in this series

· Friday, Feb. 12, 2010: On the order theory of local bases (David Milovich)
· Friday, Feb. 19, 2010: On maximal resolvability of monotonically normal spaces (Menacham Magidor)
· Friday, Mar. 05, 2010: Productively Lindelof Spaces (Frank Tall)
· Friday, Mar. 26, 2010: Gap Structure of Coherent Aronszajn Trees (continued) (Carlos Azarel)
· Friday, Apr. 09, 2010: Bounded forcing axioms and Pi-2 statements (Ralf Schindler )
· Friday, Apr. 16, 2010: Strongly D-spaces (Leandro Aurichi)
· Friday, Apr. 23, 2010: Reflection principle implies the singular cardinal hypothesis, a simplified version of a proof by Shelah (Brice Minaud)
· Friday, Apr. 30, 2010: Aronszajn trees and weak fragments of Martin's Axiom (Teruyuki Yorioka)
· Friday, May. 07, 2010: Stationary Set Reflection implies the Singular Cardinal Hypothesis (Brice Minaud)
· Friday, May. 07, 2010: Productively Lindelof spaces may all be D (Franklin D Tall)
· Friday, Jul. 23, 2010: A Hausdorff real without dominating reals (Dilip Raghavan)
· Friday, Jul. 30, 2010: PFA(S)[S] and applications (Franklin D Tall)
· Friday, Aug. 06, 2010: PFA(S)[S] and applications II (Franklin Tall)
· Friday, Aug. 13, 2010: Fragments of Martin's Maximum in the Pmax extension (Paul Larson)
· Friday, Aug. 20, 2010: A very short introduction to Woodin's Pmax forcing (Ilijas Farah)
· Friday, Sep. 10, 2010: More applications of PFA(S)[S] (Franklin D Tall)
· Friday, Oct. 29, 2010: Partition properties of the dense local order (joint with Claude Laflamme and Norbert Sauer, University of Calgary) (Lionel Nguyen Van Thé)
· Friday, Nov. 19, 2010: Some set-theoretic motives in statistical learning theory (Vladimir Pestov)
· Friday, Jan. 14, 2011: Boron Tree Structures (Jakub Jasinski)
· Friday, Jan. 21, 2011: A higher-dimensional theory of gaps in P(N)/Fin (Stevo Todorčević)
· Friday, Jan. 28, 2011: A higher-dimensional theory of gaps in P(N)/Fin, part II (Stevo Todorčević)
· Friday, Feb. 04, 2011: A higher-dimensional theory of gaps in P(N)/Fin, part III (Stevo Todorčević)
· Friday, Feb. 11, 2011: A Lindelof $T_1$ space that is not a D-space (Paul Szeptycki)
· Friday, Feb. 18, 2011: The Uniform Box Product Problem (Jocelyn Bell)
· Friday, Feb. 25, 2011: Order property of II_1 factors and its applications,  (Ilijas Farah)
· Friday, Mar. 04, 2011: The Order Property of II_1 factors and its applications, II (Ilijas Farah)
· Friday, Mar. 18, 2011: Unconditional sequences in Banach spaces of high density (Jordi Lopez-Abad)
· Friday, Mar. 25, 2011: Unconditional sequences in Banach spaces of high density (Jordi Lopez-Abad,)
· Friday, Apr. 08, 2011: Research glimpses 2 (Natasha May, Saeed Ghaseemi, Amit Gupta, et al)
· Friday, Apr. 15, 2011: Concrete mathematical incompleteness (Harvey Friedman)
· Friday, Apr. 22, 2011: What is the theory ZFC-Powerset? (Joel Hamkins)
· Friday, May. 06, 2011: Some set-theoretic motives in statistical learning theory II (Vladimir Pestov)
· Friday, May. 27, 2011: Homeomorphism groups of homogeneous compacta need not be minimal (Jan van Mill)
· Friday, Jun. 10, 2011: Choosing Ideals (Paul Larson)
· Friday, Jun. 17, 2011: Weak squares (Dilip Raghavan)
· Friday, Jun. 24, 2011: Union ultrafilters are fascinating -- a survey (Peter Krautzberger)
· Friday, Jul. 08, 2011: Density Theorems for Trees (Kostas Tyros)
· Friday, Jul. 15, 2011: Recent progress and problems concerning Lindelöf products and selection principles, AND Some topological games and selection principles (F. Tall AND R. Dias)
· Friday, Aug. 19, 2011: Mysteries of the Generic Multiverse (Gunter Fuchs)
· Friday, Oct. 28, 2011: The unconditional basic sequence problem, revisited (Stevo Todorčević)
· Friday, Nov. 04, 2011: The unconditional basic sequence problem, revisited, continued (Stevo Todorčević)
· Friday, Nov. 11, 2011: A Noetherian base for scattered linear orders (Natasha May)
· Friday, Nov. 18, 2011: Density theorems for strong subtrees (Konstantinos Tyros)
· Friday, Nov. 25, 2011: A quotient-like construction concerning elementary submodels (Peter Burton)
· Friday, Dec. 02, 2011: A quotient-like construction concerning elementary submodels, II (Peter Burton)
· Friday, Dec. 09, 2011: TBA (Hugh Woodin)
· Friday, Dec. 16, 2011: Indestructibility and selection principles (Rodrigo R. Dias)
· Friday, Jan. 13, 2012: Measurable centres in convolution semigroups (Jan Pachl)
· Friday, Jan. 20, 2012: Logic for metric structures and the number of universal sofic groups (Martino Lupini)
· Friday, Feb. 03, 2012: Generalizing Erdős-Rado to singular cardinals (Assaf Rinot)
· Friday, Feb. 10, 2012: An abstract approach to Ramsey theory with applications to finite trees (Slawomir Solecki)
· Friday, Feb. 24, 2012: A Posner-Robinson Theorem from Axiom I_0 (Xianghui Shi)
· Friday, Mar. 02, 2012: Noetherian type and other topological cardinal invariants of an order-theoretic flavour (Santi Spadaro)
· Friday, Mar. 09, 2012: Automorphisms of Calkin Algebras (Paul McKenney)
· Friday, Mar. 16, 2012: A self-dual Ramsey Theorem (Dimitrios Vlitas)
· Friday, Mar. 23, 2012: The Ramsey property for structures with an arbitrary linear ordering (Miodrag Sokic)
· Friday, Mar. 30, 2012: Lindelof spaces with small pseudocharacter, and an analog of Borel's Conjecture for subsets of uncountable products of [0,1] (Franklin Tall)
· Friday, May. 11, 2012: Constructing a Taller Thin-Thick Space (Jim McGarva)
· Friday, Jun. 08, 2012: Topological Problems for Set Theorists, II (Franklin Tall)
· Friday, Jul. 06, 2012: Easton’s Theorem for Woodin cardinals (Brent Cody)
· Friday, Jul. 13, 2012: Diagonal forms for incidence matrices and zero-sum Ramsey theory (Tony Wong)
· Friday, Jul. 20, 2012: Davies trees and stratified inverse limits (David Milovich)
· Friday, Jul. 27, 2012: Forcing consequences of PFA together with the continuum large (Miguel Angel Mota)
· Friday, Aug. 03, 2012: Ramsey classification theorems for a new class of topological Ramsey spaces, and their applications in the Tukey theory of ultrafilters (Natasha Dobrinen)
· Friday, Aug. 10, 2012: A limit stage for proper iterated forcing of length omega (Miyamoto Tadatoshi)
· Friday, Aug. 17, 2012: A bad scale and not SCH at $\aleph_\omega$ (Dima Sinapova)
· Friday, Sep. 07, 2012: Thompson's group is amenable (Justin Moore)
· Friday, Sep. 07, 2012: Extreme amenability of abelian L0 groups (Marcin Sabok)
· Friday, Sep. 21, 2012: The next best thing to a P-point (Andreas Blass)
· Friday, Sep. 28, 2012: Weak axiom of choice : can the dead be resurrected (Saharon Shelah)
· Friday, Oct. 05, 2012: On Radon-Nikodym compact spaces (Piotr Koszmider)
· Friday, Jan. 11, 2013: Chromatic number of graphs — large gaps (Assaf Rinot)
· Friday, Jan. 18, 2013: More topological consequences of PFA(S)[S] (Frank Tall)
· Friday, Jan. 25, 2013: More topological consequences of PFA(S)[S], part 2 (Frank Tall)
· Friday, Feb. 01, 2013: Automorphisms of Borel quotients of FDD-algebras (Saeed Ghasemi)
· Friday, Feb. 08, 2013: Using T-sequences to create a robust family of topological groups (Mike Pawliuk)