Universal graphs and functions on ω1
Introduction
It is well known that countably saturated models are universal for models of cardinality . However, in the case of graphs, the existence of a saturated model is equivalent to . These two observations immediately raise the question of whether it is possible to have a universal graph of cardinality in the absence of the Continuum Hypothesis. This provided the motivation for the articles [1], [2] and [3] which solved not only this question, but also others about the universality of different structures.
While the results to be presented here have their roots in this work, they are also motivated by considerations that are not model theoretic. A function is said to be (Sierpiński) universal if for any there is such that for all x and y in X. The function e will be called an embedding of G into U. An early reference to this notion can be found in Problem 132 of the Scottish Book [4] in which Sierpiński asked if there is a Borel function which is universal in this sense, when X is the real line. He had already shown in [5] that there is a Borel universal function assuming the Continuum Hypothesis. This notion of universal function is also studied in Rado [6]. More recently, this notion and various generalizations of it were studied in [7], in which a restricted form of the following definition appears as Definition 7.5. Definition 1.1 A function is weakly universal if for every there exist one-to-one functions and such that for all α and β in κ. The pair will be called a weak embedding.
The following is Theorem 5.9 of [7] showing that there is no difference between asking about the existence of universal graphs — in other words, symmetric, irreflexive functions from to 2 — and non-symmetric functions from to 2. Theorem 1.2 For any infinite cardinal κ the following are equivalent: For each there is a universal function from to n. For some with there is a universal function from to n. There is a symmetric, irreflexive function from to 2 universal for all symmetric, irreflexive functions from to 2 There is a universal graph on κ.
Given Theorem 1.2 it is reasonable to focus attention only on symmetric functions from to ω and this will be done from now on. This justifies the following:
Convention: Any function with domain will be assumed to be symmetric. In other words, should be read as .
Moreover, given that there are only two possible values for k when , the validity of Remark 7.7 of [7] should now be clear and, of course, assuming the Continuum Hypothesis there is a Sierpiński universal function from to , and hence there is also a weakly universal function from to 2. Problem 7.8 of [7] asks if the existence of one type of universal function implies the existence of the others in general. To provide a negative answer to the question, it is therefore necessary to consider models where the Continuum Hypothesis fails. This provides an other path to the questions considered in [1], [2] and [3].
When and it was shown in [1] and [2] that it is consistent with the failure of the Continuum Hypothesis that there is a universal function which satisfies all three universality properties. The methods used in the generalization of this result by Mekler to other theories in [3] provide examples of Sierpinski universal functions for λ equal to 2, ω or . In the models of [1] and [2] the cardinal invariants and have the following values respectively: and . One might, therefore ask, whether these values are needed for the existence of universal functions from with the failure of the Continuum Hypothesis. This question becomes even more interesting in light of the positive results to be presented in Lemma 6.1 and Lemma 7.8 which rely on small values of and respectively. It will be shown in Corollary 3.10 that it is consistent with the existence of universal functions from that and in Corollary 4.19 that it is consistent with the existence universal functions from that .
However, the main goal of this paper will be to answer Problem 5.10 of [7]. This is done in Corollary 6.2 which establishes that there is a Sierpiński universal function from to 2 but no such function from to ω. It is worth recalling that Theorem 5.9 of [7] asserts that if then there is a Sierpiński universal function from to 2 if and only if there is a Sierpiński universal function from to n. A solution to Problem 7.8 of [7] will also be presented under the assumption that is small. This and related questions are discussed in §7. That section also contains some arguments extending the methods of [1] to some weak versions of embedding. Finally, if one is only interested in obtaining a model of set theory in which the Continuum Hypothesis fails, yet there is a universal graph of cardinality , then §2 presents an easier argument than the original of [1]. On the other hand, §9 presents a reformulation and simplification of arguments from [1].
The graphs that will be shown to be universal in the arguments to follow will all come from some initial model of set theory in which the Continuum Hypothesis holds. For §2, §3 and §4 the following will be relevant. Definition 1.3 Given any function and define by and then define . A function such that is everywhere non-meagre for each will be called category saturated.
Definition 1.4 Let ν be an atomic probability measure on ω and let be the Fubini product of this measure on for any . A function such that has outer measure 1 for each will be called ν-saturated. The notion defined here will not be needed in this full generality, but is does find an application in [8]. For the purposes of this article a function will be called measure saturated if it is ν-saturated where ν is the measure on 2 giving each point equal measure.
Lemma 1.5 Assuming the Continuum Hypothesis, there is a symmetric function from to ω that is category saturated and ν-saturated for every atomic probability measure ν.
Indeed, using the Continuum Hypothesis it is easy to construct such that for each . Note, however, that adding a real will destroy this stronger property; nevertheless, in certain generic extensions the weaker properties of being category saturated or ν-saturated may persist.
Since trees will play a central role in the following discussion, it may be worthwhile reviewing some notation and terminology, even though most of this is standard and almost all of the notation used will follow that of Sections 1.1.D and 7.3.D of [9]. By a tree T will be meant a subset that is closed under initial segments — in other words, if and then . If T is a tree and then will denote the tree defined by and will denote the set .
A tree T will be called infinite splitting if for each . Define and define If T is infinite splitting then let be the unique bijection from to preserving the lexicographic ordering. For let — the reader is warned that this notation differs from [7]. To emphasize that only elements of a tree are being considered, the notation will be used to indicate that for some .
Let enumerate in such a way that if then there is such that . Then for infinite splitting trees T and S the ordering is defined by if and for all . Define and let denote .
Finally, recall that Miller forcing, or rational perfect set forcing, is denoted by PT in [9] and consists of all infinite splitting trees ordered by inclusion. Laver forcing, on the other hand, is denoted by LT in [9] and consists of all infinite splitting trees T such that , also ordered by inclusion. In the case of Laver forcing the notion of a front is useful: If then is a front if it consists of incomparable elements of T and every maximal branch of T contains an element of F.
For a tree let denote the maximal elements of W with respect to inclusion, the natural ordering on the tree. As usual, and in contradiction to common sense, a tree with no infinite branches will be called well founded. If F is a front, then there is a well founded tree W such that .
Definition 1.6 If define to be with inclusion as a tree ordering; the special case when f has constant value ω will be denoted . If then a function satisfying that unless will be said to have disjoint range. In later sections additional requirements will be imposed on functions with disjoint range. If G is a filter of subtrees of and ψ has disjoint range define noting that when G is sufficiently generic then ⋂G is a branch through . If G is a generic filter of trees over a model V define
It will be shown that in various generic extensions is a P-ideal and this will be used in conjunction with the P-ideal dichotomy which is the following the statement from [10]. Definition 1.7 The statement says that for every P-ideal on one of the following two alternatives holds: there is an uncountable such that can be decomposed into countably many sets orthogonal to .
Theorem 1.8 Abraham & Todorcevic
Let be a P-ideal on that is generated by a family of countable sets — in particular, this will hold if and . Then there is a proper partial order , that adds no reals, even when iterated with countable support, such that there is a -name for a subset of such that for any which is not the union of countably many sets orthogonal to
Proof
The proof is the same as that in [10]. To begin, given a non-principal P-ideal on , fix such that
- •
- •
- •
if then
- •
every member of is almost included in some .
- •
(The reader should not be confused by the claim in [10] that can be any countable subset of .)
- •
- •
.
- •
- •
- •
for every
To see this suppose that there is no member of extending p. Then for each there is some such that is countable. For let and note that the hypothesis on Y implies that . It therefore suffices to show that each is orthogonal to . To see that this is so, suppose that b is an infinite subset of and that . Then is clearly a co-countable subset of X because of the cofinality of . Moreover and hence there is some such that is uncountable. Then if this contradicts that and because . □
Section snippets
Category saturated graphs are universal after adding Miller reals
Lemma 2.1 If then there is and such that Proof Given find and ψ with disjoint range such that . Now construct and such that: the domain of is if then .
Then if it is clear that if then ψ and witness that satisfies the lemma.
To complete the induction it suffices to note that and that
Measure saturated graphs are universal after adding Laver reals
Many of the ideas developed in §2 an be extended to the context of the Laver model. One exception will be Lemma 3.7 that requires the following definition. Definition 3.1 If ψ has disjoint range and is bounded for each then ψ will be said to have bounded disjoint range. The definition of for ψ with bounded disjoint range is the same as in Definition 1.6, however, if G is a generic filter of trees over a model V define
Lemma 3.2 If is a LT
Measure saturated graphs are universal after adding -bounding reals
This section contains the key consistency result needed to establish the main result of this article, namely that the existence of a universal graph on does not entail the existence of a universal function from to ω. The key concepts are already contained in §3, the only difference being that the partial order of Definition 7.3.3 of [9] will be used in the place of LT. As can be seen in the exposition of in [9], there are a great many analogies between this partial order and
The general framework
The results of the preceding sections were originally obtained by a more complicated argument that was eventually replaced by the simpler arguments described in §2, §3 and §4. However, there is one result for which the simplified argument does not seem to be sufficient; this is the question of finding a universal function from to ω rather than a universal function from to 2. This section will describe an argument showing that it is consistent with — indeed, — that
The key combinatorial lemma
This section describes the combinatorial reason why the existence of a universal graph does not imply the existence of a universal function with range ω. Lemma 6.1 If and there is a sequence of pairs of natural numbers such that for each and then there is no universal . Proof Let be a bijection for each . Suppose that is a universal function. If and let
Allowing embeddings to permute the range
There are various ways of generalizing the notions of embeddings discussed in the introduction. The following two are singled out because something can be said about them. Definition 7.1 A function will be called -weakly universal if for every there exists a one-to-one function and functions and such that The triple will be called a -weak embedding
Set valued universality
This section will consider weakening the notion of universality for functions by attaching a set of potential values to each edge of a graph. The following definition provides the main idea. Definition 8.1 Given a family function is -weakly universal if for every there exists a one-to-one function such that for all α and β in . The function h will be called and -embedding of f.
Definition 8.2 Consider the following assertions about the existence of certain types of
Consistency of for arbitrary k
All the results up to this point have only established that can consistently take on the values 0 and ω. The goal of this section is to correct this and prove the statement of the title, namely that can have any intermediate value. This will be Theorem 9.1. The proof of Theorem 9.1 is ultimately based on arguments in [1] and [3]. However, while some familiarity with those articles may be useful, the main argument will be presented in detail here since various modifications will need.
A -weakly universal function without any -universal function
Note that Theorem 9.1 for the case yields a model of set theory in which there is no universal function from to ω yet there is a -weakly universal function. The main theorem of this section provides a companion result by showing that there is a model of set theory in which there is an -weakly universal function but there is no -universal function for any finite a and b. The proof will require some modifications of §9. To begin, Definition 9.5 needs to be augmented.
Definition 10.1 Assume
Open questions
Question 11.1 Is consistent for all k?
Question 11.2 For which a, b, c and d in does the existence of an -universal function imply the existence of an -universal function? Question 11.3 Does the existence of a -weakly universal function imply that there is a universal function? See Corollary 7.7.
References (18)
On universal graphs without instances of CH
Ann. Pure Appl. Log.
(1984)More set-theory for topologists
Topol. Appl.
(1995)Universal graphs without instances of CH: revisited
Isr. J. Math.
(1990)Universal structures in power
J. Symb. Log.
(1990)Sur une fonction universelle de deux variables réelles
Bull. Int. Acad. Polon. Sci. A
(1936)Universal graphs and universal functions
Acta Arith.
(1964)- et al.
Universal functions
Fundam. Math.
(2014) - et al.
Strong colorings over partitions
Cited by (1)
- 1
The first author's research for this paper was partially supported by the United States-Israel Binational Science Foundation (grant No. 1838/19), and by the National Science Foundation (grant No. DMS 1833363). This is article has been assigned number 1088 in the first author's list of publications that can be found at http://shelah.logic.at.
- 2
The second author's research for this paper was partially supported by NSERC of Canada.