It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large cardinals and the Continuum Hypothesis"--look under title on the Web). Yet in both versions of his paper, "What is Cantor's continuum problem" (both in the 1947 edition and the revised 1964 edition), he never (explicitly) mentioned the 1929 paper of Banach and Kuratowski ("Sur une generalisation du probleme de la mesure") found in Fundamenta Mathematicae, 14: 127-131 (a downloadable version of which can be found in the Wikipedia entry, "Measurable cardinal"). Jech, in his text, Set theory (the third millennium edition), has, in Chapter 10 (Lemma 10.16 and Corollary 10.17 respectively), their proof of the following theorem (Corollary 10.17 of Jech):

If there is a measure on $2^{\aleph_0}$, then $2^{\aleph_0} \gt \aleph_1$.

I find this omission very puzzling in light of the following result:

$ZFC$ + "The cardinality of the continuum is a real-valued measurable cardinal" is equiconsistent with $ZFC$ + "There exists a two-valued measurable cardinal"

Though this is 20\20 hindsight. it is interesting to note the Solovay-Jech result (Theorem 22.1 (ii) of The third millenium edition):

If $\kappa$ is a (two-valued) measurable cardinal, then there exists a generic extension in which $\kappa=2^{\aleph_0}$ and $\kappa$ is real-valued measurable.

for it shows that there is a model of $ZFC$ for which

The cardinality of the continuum is a real-valued measurable cardinal.


Furthermore, since the following results hold (results courtesy of Joel David Hamkins from his answer to David Reid's mathoverflow guestion, "a measurable cardinal & a real-valued measurable cardinal in the same model?"):

Theorem. If there are two measurable cardinals, then there is a forcing extension in which the smaller one becomes the continuum and real-valued measurable, and the larger one remains measurable.

Proof. If $\kappa \lt \lambda$ are both measurable cardinals in $V$, then the forcing to add $\kappa$ many random reals will make $\kappa$ into the continuum and still real-valued measurable (by a result of Solovay), and the larger measurable cardinal $\lambda$ remains measurable, by the Levy-Solovay theorem, because the forcing was much smaller than $\lambda$. $QED$

Corollary. The following are equiconsistent:

  1. There are two measurable cardinals.

  2. There are two real-valued measurable cardinals.

  3. There is a non-measurable real-valued measurable cardinal and a
    measurable cardinal.

  4. The continuum is real-valued measurable and there is another measurable cardinal.

Proof. Statement 1 implies 4 in a forcing extension, by the argument above. Statement 4 implies statement 3 directly. Statement 3 implies statement 2 directly. Statement 2 implies statement 1 in an inner model. $QED$

The reason I am interested in this theorem and its corollary is that these give an indication of the relative 'size' of the continuum as a real-valued measurable cardinal and the size of a two-valued measurable cardinal (by Ulam's dichotomy, the cardinality of the continuum cannot be a two-valued measurable cardinal). Indeed, under the forcing which adds $\kappa$ many random reals so that $\kappa$ is the cardinality of the continuum, $\kappa$ can be considered a non (two-valued) measurable real-valued measurable cardinal (or, in another parlance, an accessible weakly inaccessible cardinal because it is not strongly inaccessible). Yet it is known that $ZFC$ + "The cardinality of the continuum is real-valued measurable" implies that $V \neq L$ (since the constructible reals under the existence of a real-valued measurable cardinal form a countable set), so by one definition of 'large' cardinal (a cardinal is 'large' if $ZFC$ + "The cardinal exists" proves $V \neq L$), real-valued measurable cardinals are 'large' (a definition Gödel, I think, would have approved of, since in his paper (both versions) he writes

As for the continuum problem, there is little hope of solving it by means of those axioms of infinity which can be set up on the basis of principles known today (the above-mentioned proof for the undisprovabiity of the continuum hypothesis, e.g., goes through for all of them without any change) [1947, pg.520; 1964, pg.261]).

Another means of showing forth the 'largeness' of real-valued cardinals is given by D. H. Fremlin in the introductory paragraph of Section 4 (titled "The enormity of real-valued-measurable cardinals") from his preprint, "Real-valued measurable cardinals" (look under title on the Web):

Under this title I collect together results of the form 'If $\kappa$ is real-valued-measurable, there are many complex cardinals below it.' Ulam's theorem that a real-valued-measurable cardinal must be weakly inaccessible (1Dc-d) is the first step: if $\kappa$ is real-valued-measurable, there are $\kappa$ cardinals below it. But enormously more can be said. To develop these ideas, we need labels for some of the intermediate stages. First 'weakly Mahlo' and 'greatly Mahlo' cardinals (4A), and then 'weakly $\Pi^{1}_{n}$-indescribable cardinals' (A4C, 4D). Up to the weakly $\Pi^{1}_{1}$ indescribable cardinals , we can use ordinary infinitary combinatorics (4A-4L); but thereafter we shall need the apparatus of (elementary) model theory from [section] A4 and forcing from [section] 2. The culminating result is Theorem 4P: if $\kappa$ is real-valued-measurable, there are many $\Pi^{2}_{0}$-indescribable cardinals below it. The proof of this theorem includes essentially everything required to prove another remarkable fact: if $\kappa$ is atomlessly-measurable, and $I$ any structure of cardinal less than $\kappa$, then the first- and second-order properties of $I$ are unaffected by random real forcing (Corollary 4Oa). The same arguments provide a general method for proving results of the form 'if $\kappa$ is real-valued-measurable, there are many $\alpha \lt \kappa$ such that $\alpha \vDash \varphi$' when we have found a proof that $\kappa\vDash\varphi$ for every real-valued-measurable cardinal $\kappa$ (4Ob).

Given the above (though 20\20 hindsight), is there a reason Gödel never mentioned real-valued measurable cardinals (or the Banach-Kuratowski theorem) as a type of 'large' cardinal which validated his thesis (as it stands, he could have placed this type of axiom in the "other (hitherto unknown) axioms of set theory", since real-valued measurability is defined in terms of non-trivial probability measures on $\mathscr P(X))$? Also, is there any known correspondence in which Gödel explicitly mentions or discussed either the Banach-Kuratowski theorem or real-valued measurable cardinals?

Thanks for any help given in this matter.

[Addendum: I will quote Fremlin's version of Ulam's Theorem ((b), (c), (d), and (e) of that theorem) to clarify the distinction between two-valued measurable cardinals and what I call 'real-valued measurable cardinals' (i.e. atomlessly-measurable cardinals--hopefully I made such distinctions in my question so as to help readers distinguish between the two):

Ulam's Theorem (Fremlin's 1D, (b), (c), (d), (e)):

(b) A cardinal is real-valued mesurable iff it is either atomlessly measurable or two-valued measurable.

(c) An atomlessly-measurable cardinal is weakly inaccessible and not greater than $\mathfrak c$

(d) A two-valued-measurable cardinal is strongly inaccessible.

(e) There is an extension of Lebesgue measure to a [non-translation invariant--my comment] measure defined on every subset of $\mathbb R$ iff there is a atomlessly-measurable cardinal.

I hope this helps (Thanks, Andres, for reminding us of the correct distinction.]

[Addendum 2: Though Godel dosn't explicitly mention Ulam (1930) or Banach-Kuratowski (1929), he does make an oblique reference to them in his 1964 version. I quote the relevant part of footnote 20 (interestingly enough, my 1964 version, which comes from Godel's collected works, Vol. II, has, in footnote 20, (in italics), "Revised note of September 1966", made presumably by the editors):

In recent years great progress has been made in the area of axioms of infinity. In particular, some propositions have been formulated which, if consistent, are extremely strong axioms of infinity of an entirely new kind (see Keisler and Tarski 1964 and the material cited there).

If one looks at "Keisler and Tarski 1964" (i.e, "From accessible to inaccessible cardinals"), one finds that they in fact cite Ulam (1930) (Citation 51) and Ulam (1930) cites Banach-Kuratowski (1929). Indeed, if one reads through "From accessible to inaccessible cardinals", one finds Corollary 2.44 (pg. 268):

Assume either of the hypotheses of 2.43 [i.e. "Assume that the continuum hypothesis holds, or at least that $\omega^{+}$ $\le$ $\kappa$ $\le$ $2^{\omega}$ contains no regular limit cardinals--my comment importing the two hypotheses of 2.43 into 2.44]. Then there is no measure on the set of real numbers [see Keisler and Tarski's Theorem 2.41 and note that they distinguish 'measure' from 'two-valued measure' so that the contrapositive of 2.44 is essentially the Jech version of Banach-Kuratowski (1929)].

So the question remains, why did Godel never explicitly mention real-valued measurable cardinals (or better, atomlessly-measurable cardinals) in any version of "What is Cantor's continuum problem" and why does he not discuss the question "does there exist a [non-translation-invariant] measure on the set of real numbers" and its relation to the continuum hypothesis (this question from Keisler-Tarski (1964), pg. 267--in their paper, Keisler and Tarski use the term "measure" as opposed to what I put as a comment in square brackets)?]

  • $\begingroup$ To type Gödel (with an attractive umlaut), try typing it somewhere using HTML, then copy and paste it into the editor here. Or, you could copy and paste it from this comment. Trying to hack it in LaTeX produces an unpleasant result. $\endgroup$ – Todd Trimble Sep 25 '17 at 12:05
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    $\begingroup$ To add to @ToddTrimble's comment, both Godel and Gödel should return the same set of results, so from this view point they are equivalent. But using G$\ddot o$del G$\ddot o$del in your question would mean that it is not found when searching for this keyword. $\endgroup$ – Martin Sleziak Sep 25 '17 at 12:07
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    $\begingroup$ Well, a weakly inaccessible cardinal is a limit cardinal, and $\aleph_1$ is a successor cardinal, so... $\endgroup$ – Todd Trimble Sep 25 '17 at 14:27
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    $\begingroup$ @ThomasBenjamin It wouldn't result in a contradiction. Why do you think it would? And where did weakly compact cardinals enter the picture (or did you mean weakly inaccessible)? I think Godel was well aware of these possibilities, but didn't view them as fitting his goal since they're a priori about the continuum already; I think he was hoping for something like "if there exists a real-valued measurable, then CH holds" since the mere existence of a real-valued measurable a priori has nothing to do with the continuum; "the continuum is real-valued measurable" would not have been as natural. $\endgroup$ – Noah Schweber Sep 25 '17 at 15:46
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    $\begingroup$ You mention one definition of "large cardinal" as "ZFC + 'this cardinal exists' proves $V\neq L$". This definition looks strange to me. There are lots of properties that are usually thought of as "large cardinal" properties yet are consistent with $V=L$: inaccessible, Mahlo, weakly compact, other levels of indescribability, subtle, ineffable. $\endgroup$ – Andreas Blass Sep 25 '17 at 16:33

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