9

By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For example: whenever $\kappa$ is infinite,

  1. $2^\kappa$ is regular
  2. $2^\kappa$ is a fixed-point of $\aleph$
  3. $2^\kappa$ is weakly inaccessible
  4. $2^\kappa$ is weakly hyper-inaccessible
  5. $2^\kappa$ is weakly Mahlo

etc. (I do not know if these are all consistent).

Let us also include more ambiguous cases in our definition of "large powerset axioms," like:

  1. the map $\kappa \mapsto 2^\kappa$ is injective
  2. for every infinite set $X$, the powerset of $X$ has a subchain of cardinality $2^{|X|}$

Anyway, looking at the literature, there doesn't seem to be a lot of interest in these kinds of axioms, as compared to the amount of attention given to large cardinal axioms.

Question. Is there any particular reason for this lack of interest? For example, are there conservativity results showing that such axioms are "very weak" which might explain the lack of interest?

Joel David Hamkins
  • 224,022
goblin GONE
  • 3,693
  • What is a subchain of a powerset? – Monroe Eskew Apr 28 '14 at 21:01
  • @MonroeEskew, if $X$ is a set, then by a subchain of the powerset of $X$ I just mean a subset $C$ of $\mathcal{P}(X)$ that is totally ordered by $\subseteq$ restricted to $C$. – goblin GONE Apr 28 '14 at 21:03
  • 3
    Is $\kappa\mapsto 2^\kappa$ being injective really a "large powerset axiom" ? This is a more GCH-like property and therefore seems to be in some sense "orthogonal" to largeness questions. – Johannes Hahn Apr 28 '14 at 21:57
  • 1
    @JohannesHahn, I fully agree that it is a dubious axiom to list as "large powerset." However I included it for the following reason; firstly, assuming this axiom, we see that $\kappa < \nu$ implies $2^\kappa < 2^\nu$. Secondly, assuming also that for all infinite $\kappa$, we have that $2^\kappa$ is weakly inaccessible; then if $\kappa < \nu$ are infinite cardinals, we may deduce that $2^\kappa < 2^\nu$, hence since $2^\nu$ is weakly inaccessible, there is a huge wilderness of cardinals between $2^\kappa$ and $2^\nu$. – goblin GONE Apr 28 '14 at 22:51
  • @user18921 Well sort of... If one assumes GCH (which implies injectivity of the power function) than the "wilderness" between $2^\kappa$ and $2^\nu$ is exactly as wild as the cardinals between $\kappa$ and $\nu$, so there is nothing new in that sense. – Johannes Hahn Apr 28 '14 at 23:24
  • @JohannesHahn, oh I fully agree. In the presence of GCH, injectivity of the aforementioned function should not be considered a large powerset sentence. However, in the presence of sufficiently strong large powerset axioms, the injectivity sentence certainly acts like a large powerset axiom. By the way, the same can be said of regularity; GCH proves that every $2^\kappa$ is regular, so this is not really a large powerset sentence. However in the presence of the axioms to the effect that "$2^\kappa$ is a limit cardinal," suddenly it is a large powerset sentence. – goblin GONE Apr 29 '14 at 01:07
  • 1
    Just a minor historical comment: the hypothesis that, for every infinite $\kappa$, $2^\kappa$ is the least weakly inaccessible $ > \kappa$ seems due to Kurepa, Sull'ipotesi del continuo (Italian) Univ. e Politec. Torino. Rend. Sem. Mat. 18 1958/1959 11-20 (downloadable here) So there has always been interest in hypotheses under which the powerset function is "large". – Paolo Lipparini May 05 '14 at 16:45

3 Answers3

8

I suppose there is some interest in propositions implying large powerset. For example, a classical question at the infancy of set theory and real analysis was whether there is a probability measure on the unit interval that extends Lebesgue measure, in which every set is measurable. This implies that the continuum is weakly Mahlo, Rowbottom, etc. It's also equiconsistent with a two-valued measurable cardinal.

The stronger interest in large cardinals is surely due to the empirical fact that they form a coherent system that measures the relative consistency of everything. It's not clear that large powerset type propositions have a similar property. Some of the things you mention have no strength, like the continuum function is injective. It follows from GCH and it's negation is easily forced.

Related fact: If $2^\omega$ is real-valued measurable, the continuum function is not injective. $2^\kappa = 2^\omega$ for all $\kappa < 2^\omega$. This is due to Prikry, and a proof can be found in chapter 22 of Jech.

Monroe Eskew
  • 18,133
7

Contrary to your remarks, I believe that there is interest in these kind of principles. Let me mention a few places where they come up.

One example arises in connection with $\text{MP}_{\text{ccc}}$, the maximality principle for c.c.c. forcing, which implies that the continuum $2^\omega$ is enormous. The maximality principle says that any statement $\varphi$ that is forceable (in this case by c.c.c. forcing) such that it remains true in all forcing extensions (again c.c.c.) is already true in $V$. Under this axiom, since we can add more Cohen reals, and further c.c.c. forcing will not collapse the continuum, it follows under $\text{MP}_{\text{ccc}}$ that $2^\omega$ is larger than any cardinal that we can describe in a c.c.c.-absolute manner (for otherwise we could push it larger than that cardinal and violate maximality). You can read more at:

Another example arises in connection with $\text{RA}(\text{ccc})$, the resurrection axiom for c.c.c. forcing. For example, the following theorem is proved in J.D. Hamkins, T.A. Johnstone, Resurrection axioms and uplifting cardinals, Archive for Math Logic, 52/53, 2014.

Theorem. The resurrection axiom RA(ccc) implies that the continuum $\frak{c}$ is a weakly inaccessible cardinal, even weakly hyper-inaccessible, a limit of such cardinals and so on. In particular, RA(ccc) implies that CH fails spectacularly.

Joel David Hamkins
  • 224,022
7

A number of such "large or complex power set" axioms were considered in the following paper of Gaisi Takeuti:

Hypotheses on power set. In: Axiomatic Set Theory (Proc. Symp. Pure Math, XIII) Part I, pp.439-446 (MR 0300901)

Andreas Blass
  • 72,290