Simplified morass

In mathematics, a (κ,n)-morass is a specific structure M of "height" κ and "gap" n for any uncountable regular cardinal κ and natural number n ≥ 1.

The original definition and applications of gap-1 and higher gap (ordinary) morasses, invented by Ronald Jensen, are complicated ones, see eg.[1]

Velleman [2] defined much simpler structures for n = 1 and showed that the existence of gap-1 morasses is equivalent to the existence of gap-1 simplified morasses.

Roughly speaking: a (κ,1)-simplified morass M = < φ→, F⇒ > contains a sequence φ→ = < φβ : β ≤ κ > of ordinals such that φβ < κ for β < κ and φκ = κ+, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of monotone mappings from φα to φβ for α<β ≤  κ with specific (easy but important) conditions.

Velleman's clear definition can be found in,[3] where he also constructed (ω0,1) simplified morasses in ZFC. In [4] he gave similar simple definitions for gap-2 simplified morasses, and in [5] he constructed (ω0,2) simplified morasses in ZFC.

Higher gap simplified morasses for any n ≥ 1 were defined by Morgan [6] and Szalkai,.[7][8]

Roughly speaking: a (κ,n + 1)-simplified morass (of Szalkai) M = < M→, F⇒ > contains a sequence M→ = < Mβ : β ≤ κ > of (< κ,n)-simplified morass-like structures for β < κ , Mκ is a (κ+,n) -simplified morass, and a double sequence F⇒ = < Fα,β : α < β ≤ κ > where Fα,β are collections of mappings from Mα to Mβ for α < β ≤ κ with specific conditions.

Quagmires are similar, morass-like structures in set theory.[9]

References

  1. ↑ K. Devlin. Constructibility. Springer, Berlin, 1984.
  2. ↑ D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
  3. ↑ D. Velleman. Simplified Morasses, Journal of Symbolic Logic 49, No. 1 (1984), pp 257–271.
  4. ↑ D. Velleman. Simplified Gap-2 Morasses, Annals of Pure and Applied Logic 34, (1987), pp 171–208.
  5. ↑ D. Velleman. Gap-2 Morasses of Height ω0, Journal of Symbolic Logic 52, (1987), pp 928–938.
  6. ↑ Ch. Morgan. The Equivalence of Morasses and Simplified Morasses in the Finite Gap Case, PhD.Thesis, Merton College, UK, 1989.
  7. ↑ I. Szalkai. Higher Gap Simplified Morasses and Combinatorial Applications, PhD-Thesis (in Hungarian), ELTE, Budapest, 1991. English abstract: http://math.uni-pannon.hu/~szalkai/Szalkai-1991d-MorassAbst-.pdf
  8. ↑ I. Szalkai. An Inductive Definition of Higher Gap Simplified Morasses, Publicationes Mathematicae Debrecen 58 (2001), pp 605–634. http://math.uni-pannon.hu/~szalkai/Szalkai-2001a-IndMorass.pdf
  9. ↑ Kanamori, Akihiro (1983). "Morasses in combinatorial set theory". In Mathias, A.R.D. Surveys in set theory. London Mathematical Society Lecture Note Series 87. Cambridge: Cambridge University Press. pp. 167–196. ISBN 0-521-27733-7. Zbl 0525.03036.