nLab
sharply smaller cardinal

Sharply smaller cardinals

Sharply smaller cardinals

Definition

Theorem

For regular cardinals λμ\lambda\le\mu, the following are equivalent:

  • Every λ\lambda-accessible category is μ\mu-accessible.
  • For every μ<μ\mu'\lt\mu, the set P λ(μ)P_\lambda(\mu') of subsets of μ\mu' of cardinality <λ\lt\lambda has a cofinal subset? of cardinality <μ\lt\mu.

For a proof, see Theorem 2.11 of Adamek-Rosicky or section 2.3 of Makkai-Pare.

If these equivalent conditions hold, we write λμ\lambda\unlhd \mu. If λμ\lambda \unlhd \mu and λ<μ\lambda\lt\mu, we write λμ\lambda\lhd \mu and say that λ\lambda is sharply smaller than μ\mu.

Examples

  • For any uncountable regular cardinal λ\lambda we have 0λ\aleph_0\lhd \lambda. (In fact 0\aleph_0 is the only infinite regular cardinal with this property; see this question.)

  • For any regular cardinal λ\lambda we have λλ +\lambda\lhd \lambda^+ (its successor cardinal).

  • If λμ\lambda\le\mu then λ(2 μ) +\lambda \lhd (2^\mu)^+. Thus, for any set SS of regular cardinals there is a regular cardinal μ\mu such that λμ\lambda\lhd \mu for all λS\lambda\in S.

  • We write λμ\lambda\ll\mu if for every λ<λ\lambda'\lt\lambda and μ<μ\mu'\lt\mu we have (μ) λ<μ(\mu')^{\lambda'} \lt\mu. (This is Higher Topos Theory, Definition A.2.6.3.) Then if λμ\lambda\ll\mu, then λμ\lambda\lhd \mu.

    The converse claim (λμ\lambda \lhd \mu implies λμ\lambda\ll\mu) is independent of ZFC. On one hand it implies the generalized continuum hypothesis (GCH) for regular cardinals (and in particular the ordinary continuum hypothesis (CH)), since if λ +<2 λ\lambda^+ \lt 2^\lambda then we have λ +λ ++\lambda^+ \lhd \lambda^{++} but not λ +λ ++\lambda^+ \ll \lambda^{++}. Thus it is unprovable in ZFC (if ZFC is consistent), since CH is unprovable. On the other hand, it is implied by the full GCH, as explained by Goldberg, and is thus consistent with ZFC since GCH is.

  • If κ\kappa is an inaccessible cardinal, then every λ<κ\lambda\lt\kappa satisfies λκ\lambda\lhd \kappa.

  • 1 ω+1\aleph_1 \lhd \aleph_{\omega+1} does not hold.

References

  • Michael Makkai, Robert Paré, Accessible categories: The foundations of categorical model theory Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.1989.

Last revised on March 14, 2019 at 16:35:01. See the history of this page for a list of all contributions to it.