# Cohenは連続体濃度が到達不能基数だと考えていた

Cantor は $\aleph_1$，Gödel と初期の Woodin は $\aleph_2$，近年の Woodin は $\aleph_1$ と考えたらしい．要は，$\aleph_1$ と $\aleph_2$ の間で揺れ動いているわけだけど，Cohen はどうやらどちらでもなく弱到達不能基数と考えていたという話があった．日本語ではまったく見当たらなかったが，その立場を打ち出したのは 1966 年の Set theory and the continuum hypothesis の出版時点のようだった：

A point of view which the author feels may eventually come to be accepted is that CH is obviously false. The main reason one accepts the Axiom of Infinity is probably that we feel it is absurd to think that the process of adding one set at a time can exhaust the entire universe. Similarly with the higher axioms of infinity. Now $\aleph _ 1$ is the set of countable ordinals and this is merely a special and the simplest way of generating a higher cardinal. The set $C$ is, in contrast, generated by a totally new and more powerful principle, namely the Power Set Axiom. It is unreasonable to expect that any description of a larger cardinal which attempts to build up that cardinal from ideas deriving from the Replacement Axiom can ever reach $C$. Thus $C$ is greater than $\aleph _ n$, $\aleph _ \omega$, $\aleph _ \alpha$ where $\alpha = \aleph _ \omega$ etc. This point of view regards $C$ as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction. Perhaps later generations will see the problem more clearly and express themselves more eloquently. (p. 151)

この立場は 1971 年の “Comments on the foundations of set theory”（Axiomatic set theory 所収）でも同じく打ち出されている：

From the Realist point of view one can speculate as to the fate of CH. It would seem that only axioms such as the Axiom of Constructibility which limit the nature of sets can possibly decide it. On the other hand, there seems little hope that such an axiom will come to be accepted as intuitively obvious. A more likely development is that its negation will be accepted as an axiom. The justification might be that the continuum, which is given to us by the power set construction, is not accessible by any process which attempts to build up cardinals from below by means of a construction based on the Replacement Axiom. Thus $C$ would be considered greater than $\aleph_1$, $\aleph_n$, $\aleph_{\omega}$, etc. Of course, this is sheer speculation. Some attention has already been given to the technical consequences of various axioms which relate to CH. Although such work may be of great esthetic value, it seems highly unlikely that it can lead to basic philosophical clarification. (p. 12)

$\aleph_1$ と $\aleph_2$ のどっちかならまだわかるけど，到達不能基数はちょっと信じがたい．ところで，『キューネン集合論』の270ページに「連続体の濃度は共終数が可算でさえなければどんな値になることも集合論と矛盾しない」と書いてあるが，これは少しミスリーディングであることを（某氏のご教示により）思い知らされた．やっぱり何事もちゃんと勉強を続けないといけないなあ．