Set theory and mathematics
Osnova sekce
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The topic of the class will be large cardinals and their role in set theory and mathematics.
Monday Feb 20, 2023
We discussed the general background for the class, both related to set theory and general mathematics.
For a brief summary of basic facts of set theory, see:
LECTURE NOTES: SELECTED TOPICS FROM SET THEORY (2022/23), sections 2.1, 2.2
For more details regarding the very basics of set theory, see:
SET THEORY, section 1-5.
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For the examination: You should know all defintions, and statements of the theorems, and be able to summarize the topics discussed below. Detailed proofs will only be required if we discussed them in class.
February 27, 2023
We discussed the notion of closed unbounded sets and introduced the notion of the closed unbounded filter. For more details, see
LECTURE NOTES: SELECTED TOPICS FROM SET THEORY (2022/23), section 2.3
March 6, 2023
We continued to discuss the notion of club sets, stationary sets, non-stationary sets and the club filter at kappa and its relation to Frechet filter (the filter of co-bounded subsets of kappa).
For more details, see:
LECTURE NOTES: SELECTED TOPICS FROM SET THEORY (2022/23), section 2.3 (we covered the following: Claim 2.3. Lemma 2.4, Cor 2.5, Cor 2.6, Lemma 2.7, Lemma 2.8 (optional), Remark 2.9, Lemma 2.10)
We also discussed that while AC (Axiom of Choice) is enough to extend the Frechet filter on any regular kappa into an omega-complete ultrafilter (closed under finite intersections), it does not suffice to show that it can be extended into a kappa-complete ultrafilter. If we assume that every kappa-complete filter can be extended into a kappa-complete ultrafilter, we obtain the notion of STRONGLY COMPACT cardinal kappa. In some sense a strong generalization of AC.
March 13, 2023
We returned back to our motivation with inaccessible cardinals and models of set theory. See Section 1 and Section 2 in Honzik-large-cardinals-CH.pdf above. We also discussed generalizations of Ramsey theorem.
March 20, 2023
We discussed some counterexamples to partition relations which may hold for an uncountable kappa and also for omega. See Theorem 6.19 (i)-(iii) in lecture notes Selected topics in set theory above.
March 27, 2023
We discussed further partition relations and proved Theorem 6.19 (iv) and Lemma 6.21 in Selected topics in set theory (the proof of 6.19 (iv) has been extended, so please see the new version above).
Next we moved to discussing kappa-complete non-principal utrafilters and showed that every measurable cardinal must be inaccessible (see Theorem 2.12 in Introduction to large cardinals above).
Finally, we discussed possible generalizations of Q (rational numbers) and R (real numbers) to uncountable kappa. We observed that the lexicographical order on 2^omega is "almost" isomorphic to the ordering of the reals. The lexicographical order on on 2^kappa is therefore a reasonable generalization. The well-known fact that there is no strictly increasing or decreasing sequence of real numbers of order type omega_1 is therefore generalized by Theorem 6.19(iv) which we mentioned above. If you are interested, more details about possible generalizations of Q and R are given in Section 3.1. in paper A survey of special Aronszajn trees above.
April 3, 2023
We discussed the Baire category theorem which states if (X,d) is a complete metric space then the intersection of countably many dense open set is dense. Relatively good proof is here: https://en.wikipedia.org/wiki/Baire_category_theorem
We connected this theorem with Rasiowa-Sikorski theorem which states that if (P,<) is a partially ordered set, then for every countable collection of dense open sets {D_n; n < \omega} in P there exists a filter G on P which meets every D_n. (See the paragraph before Definition 8.1 for terminology and Lemma 8.8 for proof, in Selected topics...)
We mentioned Martin's Axiom (MA) which extends this result to omega_1 many dense sets if (P,<) satisfies some extra condition (that of having the countable chain condition). See Section 8.2, in particular Definition 8.7, in Selected topics...)
In our class on Monday 17.4. we will indicate how to extend this more partial orders and how it relates to large cardinals and also to problems in mathematics (Whitehead's problem, Kaplansky's conjecture).
April 17, 2023
We started to disucss the first application, related to the Suslin problem. The application of MA is simple in this case, but some knowledge of infinite trees is required. A brief introduction is in Selected topics... Section 3.2, more details about Suslin trees in Section 4.2, and the application of MA in Theorem 8.11.
April 24,2023
We reviewed Whitehead's and Kaplansky's conjecture, see Selected topics... Sections 8.2.1 and 8.3.1, respectively.