Abstract
Let S be an infinite set. A measure on S (a nontrivial Σ-additive measure, to be more precise) is a real-valued function μ on P(S) such that:
It follows from (ii) that μ(X), the measure of X, is nonnegative for every X ⊆ S; in a special case of (iv) we get μ(X ∪ Y) = μ(X) + μ(Y) whenever X ∩ Y = Ø (finite additivity).
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Historical Notes and Guide to the Bibliography
The study of measurable cardinals originated around 1930 with the work of Banach, Kuratowski, Tarski, and Ulam. Ulam showed in [1930] that measurable cardinals are large, that the least measurable cardinal is at least as large as the least inaccessible cardinal. Around 1960 Keisler and Tarski introduced the method of ultraproducts in the study of measurable cardinals, and it was established that the least measurable cardinal is greater than the least inaccessible cardinal. Scott [1961] used the method of ultrapowers to prove that existence of measurable cardinals contradicts the axiom of constructibility. In the 1960s two methods have been successfully applied to measurable cardinals: Rowbottom and Silver initiated applications of infinitary combinatorics (developed by Erdös and his collaborators), and Gaifman and Kunen introduced the method of iterated ultrapowers.
The main result on measurable and real-valued measurable cardinals (Theorem 66) is due to Ulam [1930]. The fact that a measurable cardinal is inaccessible (Lemma 27.2) was discovered by Ulam and Tarski (cf. Ulam [1930]). Prior to Ulam, Banach and Kuratowski proved in [1929] that if 2𝔑o = 𝔑1, then there is no measure on the continuum; their proof is as in Lemma 27.9. Real-valued measurable cardinals were introduced by Banach in [1930].
The notion of a κ-saturated ideal was introduced and studied by Tarski in [1939–1945, II].
The example of a Lebesgue nonmeasurable set in Exercise 27.1 was given by Vitali in [1905].
For further information on measure theory, we refer the reader to Halmos’ book [1950].
The fundamental theorem on ultraproducts (Lemma 28.1) is due to Łoś [1955]. Keisler and Tarski [1964] introduced ultrapowers in the study of large cardinals. Theorem 67 is due to Scott [1961], and Theorem 68 is due to Kunen [1971a].
In [1964a] Hanf studied compactness of infinitary languages (see Sections 32 and 33); his work led to the systematic study of Keisler and Tarski. Hanf proved that the least inaccessible cardinal is not measurable (in fact not weakly compact); Erdös and Hajnal then pointed out (cf. [1962]) that the same result can be proved using infinitary combinatorics. Keisler and Tarski introduced the Mahlo operation (28.23) and showed that the least measurable cardinal is much greater than, e.g., the least Mahlo cardinal, etc. (Lemma 28.12).
In [1962] Vopénka gave another proof of Scott’s theorem: Observing that 𝔐 = HOD𝔑, it follows that the cardinals in the model Ult are not absolute and hence V ≠ L.
Lemma 28.8: Erdös and Hajnal [1966].
Exercise 28.12: For a related result see Ketonen [1973].
Exercise 28.14: Solovay.
Lemma 28.13 follows from Lemma 28.15, which appears in the abstract [1961] of Hanf and Scott. The result in Exercise 28.18 was announced by Vopěnka in [1970]; more applications of a measurable cardinal to the singular cardinal problem (including Exercise 28.20) are in Jech [1973b].
Lemma 29.1 (Ramsey’s theorem) is due to Ramsey [1930]. The theory of partition relations has been developed by Erdös, who has written a number of papers on the subject, some coauthored by Rado, Hajnal, and others. The arrow notation is introduced in Erdös and Rado [1956]. Other major comprehensive articles on partition relations are Erdös, Hajnal, and Rado [1965] and Erdös and Hajnal [1971].
Theorem 69 appears in Erdös and Rado [1956]. The present proof incorporates ideas from a model-theoretical proof of the Erdös-Rado theorem due to Simpson [1970], as published in Chang and Keisler’s book [1973].
Weakly compact cardinals are investigated in the paper [1961] of Erdös and Tarski. The investigations started in that paper were subsequently completed by others; see the notes for Section 32. For applications of partition relations in model theory, we refer the reader to Chang and Keisler’s book [1973].
Lemma 29.3: Sierpiński [1933].
Exercise 29.2: Kurepa [1962].
Exercises 29.3 and 29.4: Erdös and Rado [1952].
The partition relation 𝔑1 → (α)2 for α ω 1 was proved by Baumgartner and Hajnal [1973]; the consistency ℵ2 (ω1+ω)2 by forcing is due to Hajnal, and in L due to Rebholz [1974]. The partition relation ω→ (ω) ω is consistent with ZF (without AC), by Mathias [1968] and [1977] (using an inaccessible cardinal). For further results on the partition relation ω→ (ω)ω for Borel and analytic partitions, see Galvin and Prikry [1973], Silver [1970b], and Ellentuck [1974].
That every measurable cardinal is a Ramsey cardinal had been proved by Erdös and Hajnal (see their paper [1962]). The stronger version of that result (Theorem 70) is due to Rowbottom [1971], as are Theorem 71 and Lemma 29.10.
No 𝔑n is a Jónsson cardinal: Erdös and Hajnal [1966]. The least Jónsson cardinal is weakly inaccessible or has cofinality ω: Rowbottom. If V = L, then there is no Jónsson cardinal: Keisler and Rowbottom [1965]. A model where every Jónsson cardinal is a Ramsey cardinal: Kunen [1970]. A model in which there is a Rowbottom cardinal of cofinality ω: Prikry [1970a]. Consistency of Chang’s conjecture: Silver.
Exercise 29.14: Silver; cf. Benda and Ketonen [1974].
The generalization of Aronszajn’s construction for successors of regular cardinals (assuming the GCH) is due to Specker [1951]. A model in which there exists no special Aronszajn tree on ω 2 was constructed by Mitchell [1972], who also showed that Mahlo cardinals are necessary for the construction. An analogous result for Aronszajn trees and weakly compact cardinals is due to Silver, see Mitchell [1972].
The generalization of forcing construction producing Suslin trees of larger heights is due to Hrbácčk [1967]. Jensen’s construction of a Suslin tree in L generates to larger cardinals, using ◊ and ǹ; see Jensen [1972]. The construction of a Suslin co2-tree using ǹ and the GCH is due to Gregory [1976]. The model without Suslin ω 2-trees is due to Laver.
Theorem 72 was discovered by Gaifman. Gaifman’s results were announced in [1964b] and the proof was published in [1974], Gaifman’s proof used iterated ultrapowers (see also Gaifman [1967]). Silver in his thesis (1966, published in [1971a]) developed the present method of proof, using infinitary combinatorics, and proved the theorem under the weaker assumption of existence of κ with the property k →(𝔑1)<ω. Gaifman proved that if there is a measurable cardinal, then there exists A ⊆ ω such that the conclusion of Theorem 72 holds in L[A]. Solovay formulated 0≠ and proved that it is a Δ3 1 set of integers; Silver deduced the existence of 0# under weaker assumptions.
Construction of models with indiscernibles was introduced by Ehrenfeucht and Mostowski in [1956]; for further details and applications in model theory; see Chang and Keisler [1973].
The equivalence of existence of 0≠ with existence of a nontrivial elementary embedding of L (Theorem 73) is due to Kunen; the present proof is due to Silver.
Theorem 74 (and its corollaries) is due to Jensen. A proof of the theorem appeared in Devlin and Jensen [1975].
Exercise 30.5: Solovay.
Exercise 30.7: Magidor.
For further work on Silver indiscernibles, see Paris [1974].
Most of the results in this section are due to Kunen, who in [1970] developed the method of iterated ultraproducts invented by Gaifman (cf. [1964b] and [1974]). Kunen found the representation of iterated ultraproducts (Lemma 31.11) and generalized the construction for M-ultrafilters. Kunen applied the method to obtain the main results on the model L[D] (Theorem 76).
Theorem 75 (the proof of the GCH in L[D]) is due to Silver [197 lc] ; using a similar technique, Silver proved in [1971d] that in L[D] there is a Δ1/3 well-ordering of the reals.
The description of κ-complete ultrafilters over k in L[D] (Lemma 31.18) is due to both Kunen [1970] and Paris [1969]. Lemma 31.3 was first proved by Solovay. Lemma 31.5 is due to Gaifman; cf. [1974]. Lemmas 31.17 and 31.20 are results of Kunen [1970].
Kunen generalized the basic results on L[D] to the model L < D α : α < θ constructed from a sequence of measures (with θ the least measurable cardinal in the sequence). Mitchell in [1974] generalized the construction further; one of the results in Mitchell’s paper is that the model from Exercise 31.13 has a measurable cardinal with exactly two normal measures.
Jensen extended some of his results (e.g., ◊) from [1972] from L to L[D]. 0† was formulated by Solovay, in analogy with 0≠. The model L(j) was considered by Jech in [1971d] and also by Hrbáček.
The result that in L[D] the unique normal measure is the only measure that extends the closed unbounded filter (Exercise 31.12) appeared in Jech [1973a].
For further work on iterated ultrapowers of L[D], see Bukovsky [1973] and Dehornoy [1975].
The equivalence of various formulations of weak compactness is a result of several papers. In [1964a] Hanf initiated investigations of compactness of infinary languages. Erdös and Tarski listed in [1961] several properties that were subsequently shown mutually equivalent (for inaccessible cardinals) and proved several implications. These properties included the partition property k →(k)2 2 , the tree property from Section 29, and the properties from Exercises 32.3, 32.4, and 32.5. Hanf and Scott [1961] introduced indescribability and announced Theorem 77. Further contributions were made in the papers Hanf [1964b], Hajnal [1964], Keisler [1962], Monk and Scott [1964], Tarski [1962], and Keisler and Tarski [1964]. A complete list of equivalent formulations with the proofs appeared in Silver [1971a].
Without inaccessibility, some of the formulations of weak compactness are no longer equivalent; this is discussed by Boos in [1976]. Silver showed that every weakly compact cardinal is weakly compact in L, that every indiscernible is weakly compact in L (both in [1971a]), and that if 𝔑2 has the tree property, then 𝔑2 is weakly compact in L (cf. Mitchell [1972]). Mitchell pointed out that a weakly compact cardinal is not necessarily weakly compact in a submodel. Nonexistence of simple complete Boolean algebras of weakly compact size is proved in Jech [1974a].
Lemma 32.2: Hanf and Scott [1961].
Exercise 32.6: Kunen.
Ineffable cardinals were defined by Jensen (using the property from Lemma 32.7); Kunen proved the equivalence in Lemma 32.7. Kunen also showed that ineffable cardinals are indescribable, and Jensen and Kunen proved that every ineffable cardinal is ineffable in L.
The main results on partition cardinals are due to Rowbottom, Reinhardt, and Silver. Rowbot-tom proved that if ηω1 exists, then there are only countably many constructible reals (see [1971]); Silver then used the same assumption to prove Theorem 72 (i.e., that 0≠ exists; see [1971a]). Theorem 78e was proved by Silver in [1970a]; Theorem 78c and Exercise 32.13 were proved by Reinhardt and Silver (see [1965]).
Lemma 32.8 is due to Rowbottom [1971], and Lemma 32.9 is due to Silver. Lemma 32.10 is in effect proved in Jech and Powell [1971].
Kunen showed that both the existence of Jónsson cardinals and Chang’s conjecture imply that 0≠ exists. Kunen also showed that in L[D] every Jónsson cardinal is a Ramsey cardinal; see [1970]. Kleinberg in [1972] and [1973] showed that existence of Jónsson and of Rowbottom cardinals are equiconsistent. Silver used a Ramsey cardinal to construct a model in which Chang’s conjecture holds and showed that if 𝔑ω is a Jónsson cardinal and a strong limit, then there exists a transitive model with a measurable cardinal.
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Jech, T. (1997). Measurable Cardinals. In: Set Theory. Perspectives in Mathematical Logic. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-22400-7_5
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