Cardinal invariants and the generalized Baire spaces Diana Carolina Montoya Kurt Gödel Research Center Universität Wien 11th Young set theory workshop Bernoulli Center. Laussane, June 25, 2018
Contents Classic cardinal invariants and their generalizations A word on Cichón’s diagram The generalized ultrafjlter number The independence number 2
Section 1 Classic cardinal invariants and their generalizations 3
Motivation “Cardinal invariants are simply the smallest cardinals ≤ 𝔡 for which various results, true for ℵ 0 , become false...” Andreas Blass, Combinatorial cardinal characteristics of the continuum , 2010 and understood. In fact, it is possible to directly abstract several defjnitions from 𝜕 to an arbitrary uncountable cardinal 𝜆 . 4 Classical cardinal invariants of the Baire space 𝜕 𝜕 have been extensively studied
there exists 𝑢 ⊇ 𝑡 such that [𝑢] ∩ 𝐵 = ∅ . The generalized Baire spaces 5 Let 𝜆 be an uncountable regular cardinal satisfying 𝜆 <𝜆 = 𝜆 . The generalized Baire space is just the set of functions 𝜆 𝜆 endowed with the topology generated by the sets of the form: [𝑡] = {𝑔 ∈ 𝜆 𝜆 ∶ 𝑔 ⊇ 𝑡} for 𝑡 ∈ 𝜆 <𝜆 . Denote NWD 𝜆 to be the collection of nowhere dense subsets of 𝜆 <𝜆 with respect to this topology, recall that a set 𝐵 ⊆ 𝜆 𝜆 is nowhere dense if for every 𝑡 ∈ 𝜆 <𝜆
𝜆 -ideal means an ideal that in addition is closed under unions of size ≤ 𝜆 ). It is well known that the Baire category theorem can be lifted to this context, i.e. it holds that the intersection of 𝜆 -many open dense sets is open (Friedman, Hyttinen, Kulikov, 2014). 6 Then it we defjne the generalized 𝜆 -meager sets in 𝜆 𝜆 to be 𝜆 -unions of elements in NWD 𝜆 and denote ℳ 𝜆 to be the 𝜆 -ideal that 𝜆 -meager sets determine (here
“The beginning” Since 1995, with the paper “Cardinal invariants above the continuum ” from Cum- mings and Shelah, the study of the invariants associated to these spaces and their interactions has been developing. It is also important to mention that the study of these spaces has been also ap- proached from the point of view of Descriptive Set Theory (Calderoni, Friedman, Hyttinen, Kulikov, Moreno, Motto Ros) and Topology (Korch). 7
Some cardinal invariants Defjnition for all 𝛾 > 𝛽 , 𝑔(𝛾) < (𝛾) . In this case, we say that eventually dominates 𝑔 . Defjnition Let 𝔊 be a family of functions from 𝜆 to 𝜆 . 8 If 𝑔, are functions in 𝜆 𝜆 , we say that 𝑔 < ∗ , if there exists an 𝛽 < 𝜆 such that ▶ 𝔊 is dominating , if for all ∈ 𝜆 𝜆 , there exists an 𝑔 ∈ 𝔊 such that < ∗ 𝑔 . ▶ 𝔊 is unbounded , if for all ∈ 𝜆 𝜆 , there exists an 𝑔 ∈ 𝔊 such that 𝑔 ≮ ∗ .
The unbounding and dominating numbers Defjnition 𝔠(𝜆) = min {|𝔊|∶ 𝔊 is an unbounded family of functions in 𝜆 𝜆 } 𝔢(𝜆) = min {|𝔊|∶ 𝔊 is a dominating family of functions in 𝜆 𝜆 } 9 ▶ The unbounding number: ▶ The dominating number:
Cardinal invariants associated to an ideal Let ℐ be a 𝜆 -ideal (closed under 𝜆 -sized unions) on 𝜆 𝜆 : Defjnition 10 ▶ The additivity number: add (ℐ) = min {|𝒦|∶ 𝒦 ⊆ ℐ and ⋃ 𝒦 ∉ ℐ}. ▶ The covering number: cov (ℐ) = min {|𝒦|∶ 𝒦 ⊆ ℐ and ⋃ 𝒦 = 𝜆 𝜆 }.
Defjnition cof (ℐ) = min {|𝒦|∶ 𝒦 ⊆ ℐ and for all 𝑁 ∈ ℐ there is a 𝐾 ∈ 𝒦 with 𝑁 ⊆ 𝐾}. non (ℐ) = min {|𝑍 |∶ 𝑍 ⊂ 𝑌 and 𝑍 ∉ ℐ}. 11 ▶ The cofjnality number: ▶ The uniformity number:
Cichón’s diagram Cichón’s diagram summarizes the provable ZFC relationships between some car- dinal invariants related to the 𝜏 -ideals of meager and null sets (with respect to the standard product measure) on the classical Baire space. 12
Cichoń’s Diagram on the Baire space 𝜕 𝜕 cof 𝒪 Figure 1: Cichón’s diagram 𝔡 ℵ 1 add 𝒪 cov 𝒪 non ℳ 𝔠 add ℳ cov ℳ 𝔢 cof ℳ non 𝒪 13 ✲ ✲ ✲ ✲ ✻ ✻ ✻ ✻ ✲ ✻ ✻ ✲ ✲ ✲ ✲
Hasse’s diagram Go to independence number 14
15
Why cardinal invariants of these spaces? I There are some remarkable difgerences between the countable and the uncountable cases that make this study interesting and present new challenges for future research. Here some examples: [ℵ 1 , 𝔡] . However, in the uncountable for instance, the generalization of the splitting number 𝔱(𝜆) can be ≤ 𝜆 , and actually large cardinals are necessary and Gitik found the optimal large cardinal assumption to get 𝔱(𝜆) > 𝜆 + , 2014. Specifjcally: 16 ▶ Expected bounds : Typically, classical invariants take values in the interval to have the expected inequality 𝔱(𝜆) ≥ 𝜆 + (Suzuki, 1998). Also, Ben-Neria
Why cardinal invariants of these spaces? II Theorem equiconsistent to the existence of a measurable cardinal 𝜆 with 𝑝(𝜆) = 𝜇 . extensions in which inequalities 𝔱 < 𝔠 and 𝔠 < 𝔱 hold respectively. Soukup showed (among others) that the same conclusion can be obtained under the hypothesis cf (𝔰(𝜆)) ≤ 𝜆 . 17 Let 𝜆 , 𝜇 be regular uncountable cardinals such that 𝜆 + < 𝜇 . 𝔱(𝜆) = 𝜇 is ▶ New ZFC results : Some examples ▶ Raghavan and Shelah showed that, for uncountable 𝜆 , the inequality 𝔱(𝜆) ≤ 𝔠(𝜆) holds whereas in the countable case, there are two difgerent forcing ▶ They also proved that, if 𝜆 > ℶ 𝜕 then 𝔢(𝜆) ≤ 𝔰(𝜆) . Recently, Fischer and
Why cardinal invariants of these spaces? III developed the method of template iteration forcing to give a model in which question that is still open asks if it is possible to fjnd such a model but in addition having 𝔢 = ℵ 1 . in ZFC for uncountable regular 𝜆 Roitman’s problem can be solved on the positive, i.e. if 𝔢(𝜆) = 𝜆 + , then 𝔟(𝜆) = 𝜆 + . 𝔟 𝑓 (𝜆) = 𝔟 (𝜆) = 𝜆 + . 18 ▶ Roitman’s problem : It asks whether from 𝔢 = ℵ 1 it is possible to prove that 𝔟 = ℵ 1 (still open!). ▶ So far, Shelah gave the best approximation to an answer to this problem: he the inequality 𝔢 < 𝔟 is satisfjed, yet in his model the value of 𝔢 is ℵ 2 ; the ▶ In the uncountable in contrast, Blass, Hyttinen and Zhang (2007) proved that, ▶ Additionally, Fischer and Soukup have proved that from the assumption 𝔢(𝜆) = 𝜆 + other relatives from 𝔟(𝜆) can be decided to have value 𝜆 + . Namely,
Why cardinal invariants of these spaces? IV the following: Theorem Assume GCH, if 𝜆 → (𝛾(𝜆), 𝜀(𝜆), 𝜈(𝜆)) is a class function from the class of all regular cardinals to the class of cardinal numbers, with all 𝜆 . Then, there exists a class forcing ℙ , preserving all cardinals and cofjnalities, such that in the generic extension 𝔠(𝜆) = 𝛾(𝜆) , 𝔢(𝜆) = 𝜀(𝜆) and 𝜈(𝜆) = 2 𝜆 . 19 ▶ Global results: Cummings and Shelah used an Easton-like iteration to prove 𝜆 + ≤ 𝛾(𝜆) = cf (𝛾(𝜆)) ≤ cf (𝜀(𝜆)) ≤ 𝜀(𝜆) ≤ 𝜈(𝜆) and cf (𝜈(𝜆)) > 𝜆 for
Why cardinal invariants of these spaces? V able case the following holds: Theorem (Bastoszyński) Let 𝑔 and be two functions in 𝜕 𝜕 . We say that 𝑔 and are eventually difgerent if Then, if we defjne for arbitrary uncountable regular 𝜆 : 20 ▶ More cardinal invariants via combinatorial characterizations: In the count- there is 𝑜 ∈ 𝜕 , such that for all 𝑛 ≥ 𝑜 𝑔(𝑛) ≠ (𝑛) (and write 𝑔 ≠ ∗ ), then: non ℳ = min {|ℱ|∶ (∀ ∈ 𝜕 𝜕 )(∃𝑔 ∈ ℱ)¬(𝑔 ≠ ∗ )} . cov ℳ = min {|ℱ|∶ (∀ ∈ 𝜕 𝜕 )(∃𝑔 ∈ ℱ)(𝑔 ≠ ∗ )} . ▶ nm (𝜆) = min {|ℱ|∶ (∀ ∈ 𝜆 𝜆 )(∃𝑔 ∈ ℱ)¬(𝑔 ≠ ∗ )} . ▶ cv (𝜆) = min {|ℱ|∶ (∀ ∈ 𝜆 𝜆 )(∃𝑔 ∈ ℱ)(𝑔 ≠ ∗ )} .
Why cardinal invariants of these spaces? VI The following holds: Proposition Moreover, if 𝜆 is strongly inaccessible, the corresponding cardinals coincide. given 𝑔, ∈ 𝜆 𝜆 , we say that 𝑔 < ∗ club 𝐷 on 𝜆 so that, for every 𝛽 ∈ 𝐷 , 𝑔(𝛽) < (𝛽) and defjned 𝔠 cl (𝜆) and 𝔢 cl (𝜆) accordingly. They proved: 21 ▶ 𝔠(𝜆) ≤ nm (𝜆) ≤ non (ℳ 𝜆 ) . ▶ cov (ℳ 𝜆 ) ≤ cv (𝜆) ≤ 𝔢(𝜆) . ▶ Club versions: ▶ Cummings and Shelah defjned the ”club” versions of 𝔢(𝜆) and 𝔠(𝜆) , namely cl ( club dominates 𝑔 ), if there exists a
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