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The Hierarchy of Ramsey-Like Cardinals

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The Hierarchy of Ramsey-Like Cardinals The hierarchy of Ramsey-like cardinals Victoria Gitman [email protected] http://victoriagitman.github.io Algebra and Logic Seminar University of Denver January 14, 2019 Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 1 / 30 Large cardinals Models of set theory A universe V of set theory satisfies the Zermelo-Fraenkel axioms ZFC. It is the union of the von Neamann hierarchy of the Vα's. V V0 = ; . Vα+1 = (Vα). Vλ P . S . Vλ = Vα for a limit ordinal λ. Vα+1 α<λ Vα S . V = Vα. α2Ord V !+1 V! Definition: A class M is an inner model of V if M is: M V transitive: if a 2 M and b 2 a, then b 2 M, Ord ⊆ M, M j= ZFC. G¨odel'sconstructible universe L is the smallest inner model. L L0 = ; V Lα+1 = \All definable subsets of Lα" S Lλ = α<λ Lα for a limit λ. S L = α2Ord Lα. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 2 / 30 Large cardinals A hierarchy of set-theoretic assertions There are many set theoretic assertions independent of ZFC. Some of these assertions are stronger than others. A theory T is stronger than a theory S if from a model of T we can construct a model of S, but not the other way around. Examples: ZFC + CH and ZFC + :CH are equally strong. Given a model of ZFC + CH, we can use forcing to build a model of ZFC + :CH and visa-versa. ZFC and ZFC + V = L are equally strong. The axiom V = L: V is equal to its constructible universe. The constructible universe L j= V = L. ZFC + Con(ZFC) is stronger than ZFC. The axiom Con(ZFC): G¨odel'sarithmetized assertion that ZFC is consistent. The strength is a consequence of G¨odel'sSecond Incompleteness Theorem. ZFC + Con(ZFC + Con(ZFC)) is stronger than ZFC + Con(ZFC). Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 3 / 30 Large cardinals Large cardinal axioms A large cardinal axiom LC asserts the existence of some very large infinite object. ZFC + LC is stronger than ZFC. Large cardinal axioms form a hierarchy of increasingly strong set theoretic assertions. The strength of every known set theoretic assertion can be measured against the large cardinal hierarchy. Motivating themes: ! is not unique: generalize combinatorial relationships between finite numbers and ! to uncountable cardinals. Reflection: properties of large Vα should reflect down. Elementary embeddings: the universe should elementarily embed into large inner models. Smaller large cardinals are defined via combinatorial properties. Larger large cardinals are defined via existence of elementary embeddings. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 4 / 30 Small large cardinals Inaccessible cardinals Definition: (Sierpi´nski,Tarski) An uncountable cardinal κ is inaccessible if: κ is regular: there is no cofinal function f : α ! κ for any α < κ, κ is a strong limit: jP(α)j < κ for every α < κ. Observations: ! is inaccessible to the finite numbers. If κ is inaccessible, then Vκ j= ZFC. ZFC + Inaccessible is stronger than ZFC. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 5 / 30 Small large cardinals Weakly compact cardinals Definition: (Erd¨os,Tarski) An uncountable cardinal κ is weakly compact if every coloring f :[κ]2 ! 2 has a homogeneous set of size κ. [κ]2 = fhα; βi j α < β < κg. Equivalently we can use α-many colors for any α < κ. Definition: Suppose κ is a cardinal. A κ-tree is a tree of height κ with levels of size less than κ. The infinitary logic Lκ,κ allows conjunctions, (disjunctions) and quantifier blocks of size less than κ. Theorem: (Erd¨os,Tarski) An inaccessible cardinal κ is weakly compact iff Konig's Lemma holds for κ-trees (every κ-tree has a cofinal branch). Theorem: (Kiesler, Tarski) An inaccessible cardinal κ is weakly compact iff every <κ-satisfiable theory of size κ of Lκ,κ is satisfiable. weakly compact Theorem:A weakly compact cardinal is a limit of inaccessible cardinals. inaccessible Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 6 / 30 Small large cardinals Ramsey cardinals Definition: (Erd¨os,Hajnal) A cardinal κ is Ramsey if every coloring f :[κ]<! ! 2 has a homogeneous set of size κ. <! S n [κ] = n<![κ] . A set H ⊆ κ is homogeneous for a coloring f :[κ]<! ! 2 if for every n < !, f is constant on [H]n. Equivalently we can use α-many colors for any α < κ. ! is not Ramsey. Theorem: If κ is a Ramsey cardinal, then every model M in a language Ramsey of size less than κ with κ ⊆ M has a set of indiscernibles of size κ. Theorem: L Ramsey cardinals are weakly compact limits of weakly compact cardinals. weakly compact (Rowbottom) Ramsey cardinals cannot exist in L. inaccessible Next up: large large cardinals... Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 7 / 30 Ultrapowers and elementary embeddings Ultrafilters and ultrapowers Suppose κ is a cardinal and U ⊆ P(κ) is an ultrafilter: If A, B 2 U, then A \ B 2 U (closed under intersections). If A 2 U and B ⊇ A, then B 2 U (closed under supersets), Either A 2 U or κ n A 2 U (ultra). Definition: Suppose f : κ ! V and g : κ ! V . f ∼ g whenever fα < κ j f (α) = g(α)g 2 U, f 2∗ g whenever fα < κ j f (α) 2 g(α)g 2 U. ∗ Fact: ∼ is an equivalence relation with equivalence classes[ f ]U which respects 2 . Definition: Let the ultrapower UltV =U be the structure with universe f[f ]U j f : κ ! V g and membership relation 2∗. Lo´s'Theorem: UltV =U j= '([f ]U ) iff fα < κ j '(f (α))g 2 U. i : V ! UltV =U defined by i(a) = [ca]U is an elementary embedding( ca(x) is the constant function with value a). The model UltV =U is usually ill-founded. Unless... Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 8 / 30 Ultrapowers and elementary embeddings Well-founded ultrapowers Definition: An ultrafilter U ⊆ P(κ) is α-complete if it is closed under intersections of size less than α. !1-complete ultrafilters are called countably complete. Theorem: An ultrafilter U is countably complete iff the ultrapower UltV =U is well-founded. (Mostowski Collapse) A class model with a well-founded set-like membership relation is isomorphic to an inner model. If there is a non-principal countably complete ultrafilter, then there is a non-trivial elementary embedding j : V ! M for some inner model M. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 9 / 30 Ultrapowers and elementary embeddings Well-founded ultrapowers (continued) Definition: The critical point crit(j) of an elementary embedding j : V ! M is the least ordinal moved by j. Theorem: Suppose j : V ! M is an elementary embedding with crit(j) = κ. Then: j Vκ = id. Mκ ⊆ M (every sequence of elements of M of length κ is in M). Vκ+1 ⊆ M. U = fA ⊆ κ j κ 2 j(A)g is a non-principal κ-complete ultrafilter. I U is the ultrafilter generated by κ via j. I U is normal. V M j j(κ) κ κ Theorem: If there is a non-principal κ-complete ultrafilter U ⊆ P(κ), then there is an elementary embedding j : V ! M with crit(j) = κ. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 10 / 30 Ultrapowers and elementary embeddings Normal ultrafilters Definition: Suppose hAα j α < κi is a sequence of subsets of κ. The diagonal intersection \ ∆α<κAα = fξ < κ j ξ 2 Aαg: α2ξ An ultrafilter U ⊆ P(κ) is normal if: I κ n α 2 U for all α < κ, I U is closed under diagonal intersections. Facts: A normal ultrafilter U ⊆ P(κ) is κ-complete. The ultrafilter U ⊆ P(κ) generated from an elementary embedding j : V ! M with crit(j) = κ is normal. κ = [id]U (id(x) = x). Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 11 / 30 Ultrapowers and elementary embeddings Iterated ultrapowers Suppose U ⊆ P(κ) is a countably complete ultrafilter. The ultrapower construction can be iterated along the ordinals producing a strictly decreasing chain V = M0 ⊃ M1 ⊃ M2 ⊃ · · · ⊃ Mξ ⊃ · · · of inner models with elementary embeddings jξη : Mξ ! Mη between them. j0;1 = j : V ! M1 is the ultrapower embedding by U. j1;2 : M1 ! M2 is the ultrapower embedding by j0;1(U) 2 M1. jξξ+1 : Mξ ! Mξ+1 is the ultrapower map by Uξ = j0ξ(U). At limit stages λ, take the direct limit Mλ of the directed system of embeddings hjξ,η : Mξ ! Mη j ξ; η < λi. Theorem: (Gaifman) Countable completeness of the ultrafilter U implies that all direct limits are well-founded. Victoria Gitman The hierarchy of Ramsey-like cardinals Algebra and Logic Seminar 12 / 30 Large large cardinals Very large cardinals Definition: (Banach, Kuratowski, Tarski, Ulam) A cardinal κ is measurable if there exists a non-principal κ-complete ultrafilter on κ. Fact: If κ is measurable, then there exists an elementary embedding supercompact κ j : V ! M with crit(j) = κ, Vκ+1 ⊆ M, M ⊆ M. strong Definition: (Mitchell) A cardinal κ is strong if for every λ > κ, there is measurable an elementary embedding j : V ! M with crit(j) = κ and Vλ ⊆ M.
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