# Article

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Keywords:
axiom of choice; compact; consistent; prime ideal; system of finite character; subbase
Summary:
\font\jeden=rsfs7 \font\dva=rsfs10 We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if $\text{\jeden S}$ is a system of finite character then so is the system of all collections of finite subsets of $\bigcup \text{\jeden S}$ meeting a common member of $\text{\jeden S}$), the Finite Cutset Lemma (a finitary version of the Teichm"uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.
References:
[1] Artin E., Schreier O.: Algebraische Konstruktion reeller Körper. Abh. Math. Sem. Hamb. Univ. 5 (1926), 85-99.
[2] Aubert K.E.: Theory of $x$-ideals. Acta Math. 107 (1962), 1-52. MR 0148773 | Zbl 0108.26002
[3] Banaschewski B.: The power of the ultrafilter theorem. J. London Math. Soc. (2) 27 (1983), 193-202. MR 0692524 | Zbl 0523.03037
[4] Banaschewski B.: Prime elements from prime ideals. Order 2 (1985), 211-213. MR 0815866 | Zbl 0576.06010
[5] Banaschewski B.: A new proof that Krull implies Zorn''. Mathematical Logic Quarterly 40 (1994), 478-480. MR 1301940 | Zbl 0813.03032
[6] Banaschewski B., Erné M.: On Krull's separation lemma. Order 10 (1993), 253-260. MR 1267191 | Zbl 0795.06005
[7] Crawley P., Dilworth R.P.: Algebraic Theory of Lattices. Prentice-Hall, N.J., 1973. Zbl 0494.06001
[8] Davey B.A., Priestley H.A.: Introduction to Lattices and Order. Cambridge University Press, 1990. MR 1058437 | Zbl 1002.06001
[9] de Bruijn N.G., Erdös P.: A colour problem for infinite graphs and a problem in the theory of relations. Indag. Math. 13 (1951), 371-373. MR 0046630
[10] Ebbinghaus H.-D., Flum J., Thomas W.: Mathematical Logic. Springer-Verlag, New York, 1991. MR 1278260 | Zbl 1139.03001
[11] Engeler E.: Eine Konstruktion von Modellerweiterungen. Z. Math. Logik Grundlagen Math. 5 (1959), 126-131. MR 0109124 | Zbl 0087.00904
[12] Erné M.: Semidistributivity, prime ideals and the subbase lemma. Rend. Circ. Math. Palermo II -XLI (1992), 241-250. MR 1196618
[13] Erné M.: A primrose path from Krull to Zorn. Comment. Math. Univ. Carolinae 36 (1995), 123-126. MR 1334420
[14] Erné M.: Prime ideal theorems for universal algebras. Preprint Univ. Hannover, 1995.
[15] Erné M., Gatzke H.: Convergence and continuity in partially ordered sets and semilattices. in: R.-E. Hoffmann and K.H. Hofmann (eds.), Continuous lattices and their applications, Lecture Notes in Pure and Appl. Math. 101, Marcel Dekker Inc., New York-Basel, 1985, pp.9-40. MR 0825993
[16] Frink O.: Topology in lattices. Trans. Amer. Math. Soc. 51 (1942), 569-583. MR 0006496 | Zbl 0061.39305
[17] Fuchs L.: Partially Ordered Algebraic Systems. Pergamon Press, Oxford, 1963. MR 0171864 | Zbl 0137.02001
[18] Gähler W.: Grundstrukturen der Analysis I. Akademie-Verlag and Birkhäuser Verlag, Berlin-Basel, 1977. MR 0519344
[19] Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S.: A Compendium of Continuous Lattices. Springer-Verlag, Berlin-Heidelberg-New York, 1980. MR 0614752 | Zbl 0452.06001
[20] Grätzer G.: General Lattice Theory. Birkhäuser, Basel, 1978. MR 0504338
[21] Halpern J.: The independence of the axiom of choice from the Boolean prime ideal theorem. Fund. Math. 55 (1964), 57-66. MR 0164891 | Zbl 0151.01002
[22] Halpern J., Lévy A.: The Boolean prime ideal theorem does not imply the axiom of choice. in: D. Scott (ed.), Axiomatic set theory, Proc. Symp. Pure Math., Univ. of California, Los Angeles 13 (1), (1967), 83-124. MR 0284328
[23] Halpern D., Howard P.E.: Cardinals $m$ such that $2m=m$. Bull. Amer. Math. Soc. 76 (1970), 487-490. MR 0268034 | Zbl 0223.02055
[24] Henkin L.: The completeness of the first order functional calculus. J. Symbolic Logic 14 (1949), 159-166. MR 0033781 | Zbl 0034.00602
[25] Hodges W.: Krull implies Zorn. J. London Math. Soc. 19 (1979), 285-287. MR 0533327 | Zbl 0394.03045
[26] Jech T.: The Axiom of Choice. North-Holland, Amsterdam-New York, 1973. MR 0396271 | Zbl 0259.02052
[27] Johnstone P.: Almost maximal ideals. Fund. Math. 123 (1984), 197-209. MR 0761975 | Zbl 0552.06004
[28] Kelley J.L.: The Tychonoff product theorem implies the axiom of choice. Fund. Math. 37 (1950), 75-76. MR 0039982 | Zbl 0039.28202
[29] Klimovsky G.: Zorn's theorem and the existence of maximal filters and ideals in distributive lattices. Rev. Un. Mat. Argentina 18 (1958), 160-164. MR 0132707
[30] Läuchli H.: Coloring infinite graphs and the Boolean prime ideal theorem. Israel J. Math. 9 (1971), 420-429. MR 0288051
[31] Lévy A.: Remarks on a paper by J. Mycielski. Acta Math. Acad. Sci. Hungar. 14 (1963), 125-130. MR 0146088
[32] Los J., Ryll-Nardzewski C.: 0n the application of Tychonoff's theorem in mathematical proofs. Fund. Math. 38 (1951), 233-237. MR 0048795
[33] Los J., Ryll-Nardzewski C.: Effectiveness of the representation theory for Boolean algebras. Fund Math. 41 (1954), 49-56. MR 0065527
[34] Moore G.H.: Zermelo's Axiom of Choice - its Origins, Development and Influence. Springer-Verlag, New York-Heidelberg-Berlin, 1982. MR 0679315 | Zbl 0497.01005
[35] Mycielski J.: Some remarks and problems on the colouring of infinite graphs and the theorem of Kuratowski. Acta Math. Acad. Sci. Hung. 12 (1961), 125-129. MR 0130686
[36] Parovičenko I.I.: Topological equivalents of the Tihonov theorem. Dokl. Akad. Nauk SSSR 184 (1969), 38-39 Soviet Math. Dokl. 10 (1969), 33-34. MR 0238266
[37] Rav Y.: Variants of Rado's selection lemma and their applications. Math. Nachr. 79 (1977), 145-165. MR 0476530 | Zbl 0359.02066
[38] Rav Y.: Semiprime ideals in general lattices. J. Pure and Appl. Algebra 56 (1989), 105-118. MR 0979666 | Zbl 0665.06006
[39] Rubin H., Rubin J.E.: Equivalents of the Axiom of Choice, II. North-Holland, Amsterdam-New York-Oxford, 1985. MR 0798475
[40] Rubin H., Scott D.S.: Some topological theorems equivalent to the prime ideal theorem. Bull. Amer. Math. Soc. 60 (1954), 389 (Abstract).
[41] Sageev G.: An independence result concerning the axiom of choice. Ann. Math. Logic 8 (1975), 1-184. MR 0366668 | Zbl 0306.02060
[42] Scott D.S.: Prime ideals for rings, lattices and Boolean algebras. Bull. Amer. Math. Soc. 60 (1954), 390 (Abstract).
[43] Tarski A.: Prime ideal theorems for Boolean algebras and the axiom of choice. Prime ideal theorems for set algebras and ordering principles. Prime ideal theorems for set algebras and the axiom of choice. Bull. Amer. Math. Soc. 60 (1954), 390-391 (Abstracts).
[44] Teichmüller O.: Braucht der Algebraiker das Auswahlaxiom?. Deutsche Math. 4 (1939), 567-577. MR 0000212
[45] Tukey J.W.: Convergence and uniformity in topology. Annals of Math. Studies 2, Princeton, 1940. MR 0002515 | Zbl 0025.09102
[46] van Benthem J.F.A.K.: A set-theoretical equivalent of the prime ideal theorem for Boolean algebras. Fund. Math. 89 (1975), 151-153. MR 0382003 | Zbl 0363.04010

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