Cofinality spectrum theorems in model theory, set theory, and general topology. (English) Zbl 1477.03125
Summary: We connect and solve two long-standing open problems in quite different areas: the model-theoretic question of whether \(SOP_2\) is maximal in Keisler’s order, and the question from general topology/set theory of whether \( \mathfrak{p} = \mathfrak{t}\), the oldest problem on cardinal invariants of the continuum. We do so by showing these problems can be translated into instances of a more fundamental problem which we state and solve completely, using model-theoretic methods.
MSC:
| 03C20 | Ultraproducts and related constructions |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 03E17 | Cardinal characteristics of the continuum |
| 03E05 | Other combinatorial set theory |
References:
| [1] | Ax, James, The elementary theory of finite fields, Ann. of Math. (2), 88, 239\textendash 271 pp. (1968) · Zbl 0195.05701 |
| [2] | Ax, James; Kochen, Simon, Diophantine problems over local fields. I, Amer. J. Math., 87, 605\textendash 630 pp. (1965) · Zbl 0136.32805 |
| [3] | Balcar, Bohuslav; Pelant, Jan; Simon, Petr, The space of ultrafilters on \({\bf N}\) covered by nowhere dense sets, Fund. Math., 110, 1, 11\textendash 24 pp. (1980) · Zbl 0568.54004 |
| [4] | Bell, Murray G., On the combinatorial principle \(P({\mathfrak{c}})\), Fund. Math., 114, 2, 149\textendash 157 pp. (1981) · Zbl 0581.03038 |
| [5] | Blass, Andreas, Combinatorial cardinal characteristics of the continuum. Handbook of set theory. Vols. 1, 2, 3, 395\textendash 489 pp. (2010), Springer, Dordrecht · Zbl 1198.03058 · doi:10.1007/978-1-4020-5764-9\_7 |
| [6] | Cantor, G., Uber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal f\`“ur die reine und angewandte Mathematik (Crelle”s journal), 77, 2, 258\textendash 262 pp. (1874) |
| [7] | Dow, Alan, Good and OK ultrafilters, Trans. Amer. Math. Soc., 290, 1, 145\textendash 160 pp. (1985) · Zbl 0532.54021 · doi:10.2307/1999788 |
| [8] | van Douwen, Eric K., The integers and topology. Handbook of set-theoretic topology, 111\textendash 167 pp. (1984), North-Holland, Amsterdam · Zbl 0561.54004 |
| [9] | D{\v{z}}amonja, Mirna; Shelah, Saharon, On \(\vartriangleleft^*\)-maximality, Ann. Pure Appl. Logic, 125, 1-3, 119\textendash 158 pp. (2004) · Zbl 1040.03029 · doi:10.1016/j.apal.2003.11.001 |
| [10] | Frayne, T. E.; Scott, D. S.; Tarski, A., Reduced products, Not. Amer. Math.Soc., 5, 673\textendash 674 pp. (1958) |
| [11] | Fremlin, D. H., Consequences of Martin’s axiom, Cambridge Tracts in Mathematics 84, xii+325 pp. (1984), Cambridge University Press, Cambridge · Zbl 1156.03050 · doi:10.1017/CBO9780511896972 |
| [12] | Gaifman, Haim; Dimitracopoulos, Constantine, Fragments of Peano’s arithmetic and the MRDP theorem. Logic and algorithmic, Zurich, 1980, Monograph. Enseign. Math. 30, 187\textendash 206 pp. (1982), Univ. Gen\`eve, Geneva · Zbl 0498.03045 |
| [13] | Hausdorff, F., Summen von \(\aleph_1\) Mengen, Fund. Math., 26, 241\textendash 255 pp. (1936) · JFM 62.0228.03 |
| [14] | Keisler, H. Jerome, Good ideals in fields of sets, Ann. of Math. (2), 79, 338\textendash 359 pp. (1964) · Zbl 0137.00803 |
| [15] | Keisler, H. Jerome, Ultraproducts and saturated models, Nederl. Akad. Wetensch. Proc. Ser. A 67 = Indag. Math., 26, 178\textendash 186 pp. (1964) · Zbl 0199.01101 |
| [16] | Keisler, H. Jerome, Ultraproducts which are not saturated, J. Symbolic Logic, 32, 23\textendash 46 pp. (1967) · Zbl 0162.01501 |
| [17] | Kochen, Simon, Ultraproducts in the theory of models, Ann. of Math. (2), 74, 221\textendash 261 pp. (1961) · Zbl 0132.24602 |
| [18] | Koppelberg, Sabine, Cardinalities of ultraproducts of finite sets, J. Symbolic Logic, 45, 3, 574\textendash 584 pp. (1980) · Zbl 0497.03016 · doi:10.2307/2273424 |
| [19] | Kramer, Linus; Shelah, Saharon; Tent, Katrin; Thomas, Simon, Asymptotic cones of finitely presented groups, Adv. Math., 193, 1, 142\textendash 173 pp. (2005) · Zbl 1139.22010 · doi:10.1016/j.aim.2004.04.012 |
| [20] | Kunen, Kenneth, Ultrafilters and independent sets, Trans. Amer. Math. Soc., 172, 299\textendash 306 pp. (1972) · Zbl 0263.02033 |
| [21] | \L o{\'s}, Jerzy, Quelques remarques, th\'eor\`“emes et probl\`emes sur les classes d\'”efinissables d’alg\`ebres. Mathematical interpretation of formal systems, 98\textendash 113 pp. (1955), North-Holland Publishing Co., Amsterdam · Zbl 0068.24401 |
| [22] | Malliaris, M., Persistence and regularity in unstable model theory (2009), Ph. D. thesis, University of California, Berkeley |
| [23] | Malliaris, M. E., Realization of \(\phi \)-types and Keisler’s order, Ann. Pure Appl. Logic, 157, 2-3, 220\textendash 224 pp. (2009) · Zbl 1159.03022 · doi:10.1016/j.apal.2008.09.008 |
| [24] | Malliaris, M. E., Hypergraph sequences as a tool for saturation of ultrapowers, J. Symbolic Logic, 77, 1, 195\textendash 223 pp. (2012) · Zbl 1247.03048 · doi:10.2178/jsl/1327068699 |
| [25] | Malliaris, M. E., Independence, order, and the interaction of ultrafilters and theories, Ann. Pure Appl. Logic, 163, 11, 1580\textendash 1595 pp. (2012) · Zbl 1279.03060 · doi:10.1016/j.apal.2011.12.010 |
| [26] | Malliaris, M.; Shelah, S., Constructing regular ultrafilters from a model-theoretic point of view, Trans. Amer. Math. Soc. (2015) · Zbl 1387.03029 |
| [27] | Malliaris, M.; Shelah, S., Model-theoretic properties of ultrafilters built by independent families of functions, J. Symbolic Logic, 79, 1, 103\textendash 134 pp. (2014) · Zbl 1338.03055 · doi:10.1017/jsl.2013.28 |
| [28] | Malliaris, M.; Shelah, S., A dividing line within simple unstable theories, Adv. Math., 249, 250\textendash 288 pp. (2013) · Zbl 1323.03042 · doi:10.1016/j.aim.2013.08.027 |
| [29] | Malliaris, M.; Shelah, S., Saturating the random graph with an independent family of small range. Logic Without Borders, xvi, 422 pp. (2015), DeGruyter |
| [30] | Malliaris, M.; Shelah, S., An axiomatic approach to cofinality spectrum problems · Zbl 1477.03176 |
| [31] | Malliaris, M.; Shelah, S., Existence of optimal ultrafilters and the fundamental complexity of simple theories · Zbl 1431.03048 |
| [32] | Malliaris, M.; Shelah, S., Keisler’s order has infinitely many classes · Zbl 1402.03045 |
| [33] | Malliaris, M.; Shelah, S., Model-theoretic applications of cofinality spectrum problems · Zbl 1423.03177 |
| [34] | Piotrowski, Zbigniew; Szyma{\'n}ski, Andrzej, Some remarks on category in topological spaces, Proc. Amer. Math. Soc., 101, 1, 156\textendash 160 pp. (1987) · Zbl 0634.54002 · doi:10.2307/2046568 |
| [35] | Wilkie, A. J.; Paris, J. B., On the scheme of induction for bounded arithmetic formulas, Ann. Pure Appl. Logic, 35, 3, 261\textendash 302 pp. (1987) · Zbl 0647.03046 · doi:10.1016/0168-0072(87)90066-2 |
| [36] | Rothberger, F.; Paris, J. B., Une remarque concerenant l’hypoth\`ese du continu, Fund. Math., 31, 3, 224\textendash 226 pp. (1939) · JFM 64.0933.02 |
| [37] | Rothberger, Fritz, On some problems of Hausdorff and of Sierpi\'nski, Fund. Math., 35, 29\textendash 46 pp. (1948) · Zbl 0032.33702 |
| [38] | Shelah, Saharon, Every two elementarily equivalent models have isomorphic ultrapowers, Israel J. Math., 10, 224\textendash 233 pp. (1971) · Zbl 0224.02045 |
| [39] | Shelah, Saharon, Simple unstable theories, Ann. Math. Logic, 19, 3, 177\textendash 203 pp. (1980) · Zbl 0489.03008 · doi:10.1016/0003-4843(80)90009-1 |
| [40] | Shelah, Saharon, Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics 92, xvi+544 pp. (1978), North-Holland Publishing Co., Amsterdam\textendash New York · Zbl 0388.03009 |
| [41] | Shelah, S., Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics 92, xxxiv+705 pp. (1990), North-Holland Publishing Co., Amsterdam · Zbl 0713.03013 |
| [42] | Shelah, Saharon, Toward classifying unstable theories, Ann. Pure Appl. Logic, 80, 3, 229\textendash 255 pp. (1996) · Zbl 0874.03043 · doi:10.1016/0168-0072(95)00066-6 |
| [43] | Shelah, Saharon, Two cardinal invariants of the continuum \((\mathfrak{d}<\mathfrak{a})\) and FS linearly ordered iterated forcing, Acta Math., 192, 2, 187\textendash 223 pp. (2004) · Zbl 1106.03044 · doi:10.1007/BF02392740 |
| [44] | Shelah, Saharon, A comment on “\( \mathfrak{p}<\mathfrak{t} \)”, Canad. Math. Bull., 52, 2, 303\textendash 314 pp. (2009) · Zbl 1169.03038 · doi:10.4153/CMB-2009-033-4 |
| [45] | Shelah, Saharon; Usvyatsov, Alexander, More on \({\rm SOP}_1\) and \({\rm SOP}_2\), Ann. Pure Appl. Logic, 155, 1, 16\textendash 31 pp. (2008) · Zbl 1154.03015 · doi:10.1016/j.apal.2008.02.003 |
| [46] | Todorcevic, Stevo, Coherent sequences. Handbook of set theory. Vols. 1, 2, 3, 215\textendash 296 pp. (2010), Springer, Dordrecht · Zbl 1198.03052 · doi:10.1007/978-1-4020-5764-9\_4 |
| [47] | Todor\v{c}evi\'c, S.; Veli\v{c}kovi\'c, B., Martin’s axiom and partitions, Compos. Math., 63, 391\textendash 408 pp. (1987) · Zbl 0643.03033 |
| [48] | Tsaban, B., Selection principles and special sets of reals (2007), Elsevier, New York |
| [49] | Vaughan, Jerry E., Small uncountable cardinals and topology. Open problems in topology, 195\textendash 218 pp. (1990), North-Holland, Amsterdam |
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