## Simple Cardinal Characteristics Of The Continuum (1993)

Venue: | 63–90, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat |

Citations: | 13 - 1 self |

### BibTeX

@INPROCEEDINGS{Blass93simplecardinal,

author = {Andreas Blass},

title = {Simple Cardinal Characteristics Of The Continuum},

booktitle = {63–90, Israel Math. Conf. Proc., 6, Bar-Ilan Univ., Ramat},

year = {1993},

pages = {63--90}

}

### OpenURL

### Abstract

. We classify many cardinal characteristics of the continuum according to the complexity, in the sense of descriptive set theory, of their definitions. The simplest characteristics (\Sigma 0 2 and, under suitable restrictions, \Pi 0 2 ) are shown to have pleasant properties, related to Baire category. We construct models of set theory where (unrestricted) \Pi 0 2 -characteristics behave quite chaotically and no new characteristics appear at higher complexity levels. We also discuss some characteristics associated with partition theorems and we present, in an appendix, a simplified proof of Shelah's theorem that the dominating number is less than or equal to the independence number. 1. Introduction Cardinal characteristics of the continuum are cardinal numbers, usually between @ 1 and c = 2 @0 inclusive, that give information about the real line R or the closely related sets P(!) (the power set of the set ! of natural numbers), [!] ! (the set of infinite subsets of !...

### Citations

506 | Set Theory - Jech - 2003 |

198 | Descriptive Set Theory - MOSCHOVAKIS - 1980 |

100 | A combinatorial theorem
- Erdős, Rado
- 1950
(Show Context)
Citation Context ...quals c, aSIMPLE CARDINAL CHARACTERISTICS 21 It is clear that one can similarly associate characteristics with other partition theorems, for example the canonical partition theorem of Erdős and Rado =-=[4]-=- or the finite sum theorem of Hindman [10]. We shall discuss only two more analogs of par and two analogs of hom, associated to very weak partition theorems. The first of these theorems can be viewed ... |

67 |
Finite sums from sequences within cells of a partition of
- Hindman
- 1974
(Show Context)
Citation Context ...S 21 It is clear that one can similarly associate characteristics with other partition theorems, for example the canonical partition theorem of Erdős and Rado [4] or the finite sum theorem of Hindman =-=[10]-=-. We shall discuss only two more analogs of par and two analogs of hom, associated to very weak partition theorems. The first of these theorems can be viewed as the “canonical partition theorem for si... |

44 |
Prikry K., Borel sets and Ramsey’s theorem
- Galvin
- 1973
(Show Context)
Citation Context ...ent of H ′ after n. So Yn does not split H ′ . □ Going beyond Ramsey’s theorem, we can define analogous characteristics associated with the partition theorems of Nash-Williams [19], Galvin and Prikry =-=[6]-=-, and Silver [22]. Little is known about these characteristics, but we list for reference some elementary facts. As we go from weaker to stronger partition theorems, the hom characteristics weakly inc... |

39 |
uncountable cardinals and topology, in: Open Problems of Topology
- Vaughan, Small
- 1990
(Show Context)
Citation Context ...power set of the set ω of natural numbers), [ω] ω (the set of infinite subsets of ω), and ω ω (the set of functions from ω to ω). We give a few examples here (others will be given later) and refer to =-=[3, 23]-=- for more examples and an extensive discussion. The most obvious characteristic is c, the cardinality of R and of the other sets, P(ω), etc., mentioned above. Baire category gives rise to several char... |

33 |
Combinatorial aspects of measure and category
- Bartoszynski
(Show Context)
Citation Context ...ne can do considerably better by using combinatorial characterizations of these cardinals. Specifically, by writing out the characterizations of add(L) and cof(L) implicit in Theorems 0.9 and 0.10 of =-=[1]-=- (and using Lemma 0.5 to eliminate a quantifier), we find that add(L) is a uniform Π0 2-characteristic and cof(L) is a uniform Σ02 - (hence Π01 -)characteristic. (Actually, Theorems 0.9 and 0.10 are c... |

33 |
Douwen, The integers and topology, Handbook of set-theoretic topology
- van
- 1984
(Show Context)
Citation Context ...pothesis of the lemma is satisfied because I is independent. So we obtain Bx such that ⋂ ∗ (1) Bx ⊆ k<n Dx(k) k for all n, and (2) Bx ∩ W(X, Y) is infinite for every W(X, Y) ∈ A. Notice that, by (1), =-=(3)-=- Bx ∩ By is finite when x ̸= y. Fix two disjoint, countable, dense (in the usual product topology) subsets Q and Q ′ of ω2. We can remove finitely many elements from Bx, for x ∈ Q ∪ Q ′, so that (3*) ... |

32 |
Some properties of measure and category
- Miller
- 1981
(Show Context)
Citation Context ...need can be proved exactly the same way. The same remark applies to other results cited below.) For the Baire category characteristics, we have that add(B) is either cov(B) or b, whichever is smaller =-=[15]-=-, and cof(B) is either unif(B) or d, whichever is larger [16]. So both add(B) -characteristics. We remark that Bartoszyński’s elegant and cof(B) are uniform Π0 2 characterization [1, Thm. 1.7] of cov(... |

26 |
Consequences of martin’s axiom
- Fremlin
- 1984
(Show Context)
Citation Context ...lary 7. Martin’s axiom implies that every Π0 1-characteristic is equal to c. Proof. Martin’s axiom, even when weakened to apply only to countable (rather than ccc) partial orders, implies cov(B) = c. =-=[5,14]-=-. □ 4. Consistency Results for Π 0 2 and Higher Characteristics All the familiar cardinal characteristics of the continuum — those defined in Section 1 above as well as numerous others [3,23] — are eq... |

24 |
On better-quasi-ordering transfinite sequences
- Nash-Williams
- 1968
(Show Context)
Citation Context ...eater than the next element of H ′ after n. So Yn does not split H ′ . □ Going beyond Ramsey’s theorem, we can define analogous characteristics associated with the partition theorems of Nash-Williams =-=[19]-=-, Galvin and Prikry [6], and Silver [22]. Little is known about these characteristics, but we list for reference some elementary facts. As we go from weaker to stronger partition theorems, the hom cha... |

22 |
Vive la différence I: Nonisomorphism of ultrapowers of countable models
- Shelah
- 1992
(Show Context)
Citation Context ...characteristics and that one might be was that there might be very few uniform Π0 1 able to classify them all. Shelah promptly informed me that, by a countable-support product of forcing notions from =-=[21]-=-, he can produce models with infinitely many uniform Π0 1-characteristics, all of the form “the smallest number of g-branching subtrees of <ωω needed to cover all the paths through an f branching subt... |

22 |
Every analytic set is
- Silver
(Show Context)
Citation Context ... n. So Yn does not split H ′ . □ Going beyond Ramsey’s theorem, we can define analogous characteristics associated with the partition theorems of Nash-Williams [19], Galvin and Prikry [6], and Silver =-=[22]-=-. Little is known about these characteristics, but we list for reference some elementary facts. As we go from weaker to stronger partition theorems, the hom characteristics weakly increase and in part... |

17 |
Infinite combinatorics and definability
- Miller
- 1989
(Show Context)
Citation Context ...istics in the forcing extension. The first part of the proof uses Hechler’s technique [9] for forcing maximal almost disjoint families of different sizes. The second is related to a theorem of Miller =-=[17]-=- that, when many independent Cohen reals are added to a model of GCH, no cardinal strictly between ℵ1 and c is a Borel-characteristic; we strengthen the conclusion from “Borel” to “ODR”, and we work w... |

11 | Initial segments of models of Peano’s axioms. In: Set theory and hierarchy theory - Kirby, Paris - 1977 |

8 |
On covering of real line by null sets
- Bartoszyński
- 1988
(Show Context)
Citation Context ...ation. So the claim is proved. Notice that p ′′ agrees with p ′ on the common part of their domains, (F ′ ∩J)×n ′ , because q extends them both. Extending p ′′ if necessary, we assume n ′′ > k; then, =-=(2)-=- (∀η ∈ X) p ′′ (κ, η, k) = 0 because p ′′ forces (1). Define a function s : F ′ × n ′′ → 2 by making s agree with p ′ on F ′ × n ′ , making s agree with p ′′ on (F ′ ∩ J) × n ′′ , setting s(κ, ξ, k) =... |

7 | S.: Many simple cardinal invariants
- Goldstern, Shelah
- 1993
(Show Context)
Citation Context ...l property, enjoyed by the models in Theorem 9, that cardinals not explicitly made to be characteristics are not even ODR-characteristics. This work of Shelah will appear (with some modifications) in =-=[7]-=-. It is clear from Theorem 9 (or even from Theorem 8) that we cannot expect restrictive results about Π0 2-characteristics in ZFC. But these theorems leave open the possibility of restrictive results ... |

7 |
Eine Äquivalenz zwischen der Kontinuumhypothese und der Existenz der Lusinschen und Sierpińskischen Mengen, Fundamenta Mathematicae 30
- Rothberger
- 1938
(Show Context)
Citation Context ...ame as for ωω. So we must have κ ≥ cov(B). □ Corollary 6. All of d, r, and unif(L) are ≥ cov(B). □ This corollary is, of course, well-known, but the usual proof of unif(L) ≥ cov(B), due to Rothberger =-=[20]-=-, uses more specific information about Lebesgue measure. Our proof, by contrast, uses only that unif(L) is a Σ0 2- (and therefore -)characteristic. Specifically, if we replace the ideal L of measure z... |

7 |
Short complete nested sequences in βN \ N and small maximal almost-disjoint families
- Hechler
- 1972
(Show Context)
Citation Context ...g a maximal almost disjoint family of κ subsets of ω and second showing that no cardinals λ /∈ C are ODR-characteristics in the forcing extension. The first part of the proof uses Hechler’s technique =-=[9]-=- for forcing maximal almost disjoint families of different sizes. The second is related to a theorem of Miller [17] that, when many independent Cohen reals are added to a model of GCH, no cardinal str... |

6 |
Additivity of measure implies dominating reals
- Miller
- 1984
(Show Context)
Citation Context ...ies to other results cited below.) For the Baire category characteristics, we have that add(B) is either cov(B) or b, whichever is smaller [15], and cof(B) is either unif(B) or d, whichever is larger =-=[16]-=-. So both add(B) -characteristics. We remark that Bartoszyński’s elegant and cof(B) are uniform Π0 2 characterization [1, Thm. 1.7] of cov(B) as the smallest cardinality of any X ⊆ ωω such that (∀y ∈ ... |

4 | On the existence of P-points in the Stone-Čech compactification of the integers, Fund - Ketonen - 1976 |

4 | Set-theoretic characteristics of summability of sequences and convergence of series - Vojtáš - 1987 |

3 |
Projective Well orderings
- Harrington
(Show Context)
Citation Context ... Π0 2-characteristics in ZFC. But these theorems leave open the possibility of restrictive results in stronger theories, perhaps ZFC + MA. The following theorem, a corollary of a result of Harrington =-=[8]-=-, shows that even Martin’sSIMPLE CARDINAL CHARACTERISTICS 15 Theorem 13. It is consistent with Martin’s axiom that c be arbitrarily large and that every uncountable cardinal ≤ c be a ∆ 1 1 -character... |

3 | Set-theoretic characteristics of summability of sequences and convergence of series - Vojtáˇs - 1987 |

2 |
Cardinalities of noncentered system of subsets of ω which reflect some qualities of ultrafilters, p-points and rapid filters
- Vojtaˇs
- 1987
(Show Context)
Citation Context ...mpting to dualize the preceding argument, to obtain hom = max{d,r}, we succeed only partially. To state the result that we obtain, we need the following variant of r introduced and studied by Vojtáˇs =-=[24,25]-=-. rσ = smallest cardinality of any X ⊆ [ω] ω such that, for any countably many sets Yn ∈ [ω] ω , n ∈ ω, there is X ∈ X not split by any Yn. It is clear that rσ is a uniform Π 0 3 -characteristic and t... |

1 | complete nested sequences in fiN \Gamma N and small maximal almost-disjoint families - Hechler, Short - 1972 |

1 | Cardinalities of noncentered systems of subsets of ! which reflect some qualities of ultrafilters, p-points and rapid filters, preprint - Vojt'as |