L-like combinatorial principles and level by level equivalence

• Autorzy: Apter, A. W.
• Języki publikacji: EN

Abstrakty

EN

We force and construct a model in which GCH and level by level equivalence between strong compactness and supercompactness hold, along with certain additional "L-like" combinatorial principles. In particular, this model satisfies the following properties: (1) ◊ δ holds for every successor and Mahlo cardinal δ. (2) There is a stationary subset S of the least supercompact cardinal κ 0 such that for every δ ∈ S, [...]δ holds and δ carries a gap 1 morass. (3) A weak version of [...]δ holds for every infinite cardinal δ. (4) There is a locally defined well-ordering of the universe W, i.e., for all κ ≥ ℵ 2 a regular cardinal, W↑H(κ+) is definable over the structure (H(κ+), ∈} by a parameter free formula. The model constructed amalgamates and synthesizes results due to the author, the author and Cummings, and Aspero and Sy Friedman. It has no restrictions on the structure of its class of supercompact cardinals and may be considered as part of Friedman's "outer model programme".

Słowa kluczowe

EN

supercompact cardinal; strongly compact cardinal; strong cardinal; level by level equivalence between strong compactness and supercompactness; diamond; square; morass; locally defined well-ordering

• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 3-4
• Strony: 199-207
• Opis fizyczny: Bibliogr. 17 poz.

Twórcy

autor

Apter, A. W.

• Department of Mathematics, Baruch College of CUNY, New York, NY 10010, U.S.A.

Bibliografia

• [1] A. Apter, Diamond, square, and level by level equivalence, Arch. Math. Logic 44 (2005), 387-395.
• [2] A. Apter, Stationary reflection and level by level equivalence, Colloq. Math. 115 (2009), 113-128.
• [3] A. Apter and J. Cummings, An L-like model containing very large cardinals, Arch. Math. Logic 47 (2008), 65-78.
• [4] A. Apter and S. Shelah, On the strong equality between supercompactness and strong compactness, Trans. Amer. Math. Soc. 349 (1997), 103-128.
• [5] D. Aspero and S. Friedman, Large cardinals and locally defined well-orders of the universe, Ann. Pure Appl. Logic 157 (2009), 1-15.
• [6] A. Brooke-Taylor, Large cardinals and definable well-orders on the universe, J. Symbolic Logic 74 (2009), 641-654.
• [7] A. Brooke-Taylor and S. Friedman, Large cardinals and gap-1 morasses, Ann. Pure Appl. Logic 159 (2009), 71-99.
• [8] J. Cummings and E. Schimmerling, Indexed squares, Israel J. Math. 131 (2002), 61-99.
• [9] M. Foreman and M. Magidor, A very weak square principle, 3. Symbolic Logic 62 (1997), 175-196.
• [10] S. Friedman, Forcing condensation, submitted for publication.
• [11] S. Friedman, Large cardinals and L-like universes, in: Set Theory: Recent Trends and Applications, A. Andretta (ed.), Quad. Mat. 17, Seconda Univ. Napoli, 2006, 93-110.
• [12] J. D. Hamkins, Extensions with the approximation and cover properties have no new large cardinals, Fund. Math. 180 (2003), 257-277.
• [13] J. D. Hamkins, Gap forcing, Israel J. Math. 125 (2001), 237-252.
• [14] J. D. Hamkins, Gap forcing: generalizing the Levy-Solovay theorem, Bull. Symbolic Logic 5 (1999), 264-272.
• [15] A. Levy and R. Solovay, Measurable cardinals and the continuum hypothesis, Israel J. Math. 5 (1967), 234-248.
• [16] T. Menas, On strong compactness and supercompactness, Ann. Math. Logic 7 (1974), 327-359.
• [17] S. Shelah, Diamonds, submitted for publication.