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Trees

  • Thomas J. Jech (a1)

Extract

This is an attempt to give a survey of recent results concerning trees. The article is an extended version of our talk in Oberwolfach (Schwarzwald) last May; the forests surrounding the Forschungsinstitut turned out to be a good inspiration.

A tree is a partially ordered set T = (T, ≤) such that for every x ∈ T, the set = {yT: y < x} is well-ordered. The order type of is called the order of x, o(x), and the length of T is sup {o(x) + 1: xT}; an α-tree (where α is an ordinal) is a tree of length α. The αth level of T is the set Uα of all elements of T whose order is α. Tα is the union of all Uβ, β < α; its length is α. A tree (T2, ≤2) is called an extension of (T1, ≤1) if ≤1 = ≤2 ∩ (T1 × T1); T2 is on end-extension of (T1 if T1 = T2α for some α. A maximal linearly ordered subset of a tree T is called a branch of T; an α-branch is a branch of length α.

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[1]Bukovsky, L., Consistency theorems connected with some combinatorial problems, Commentationes Mathematicae Universitatis Carolinae, vol. 7 (1966), pp. 495499.
[2]Cohen, P. J., Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York, 1966.
[3]Hrbáček, K. M., A note on generalized Suslir's Problem, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 307309.
[4]Jech, T. J., Non-provability of Suslir's hypothesis, Commentationes Mathematicae Universitatis Carolinae, vol. 8 (1967), pp. 291305.
[5]Jensen, R. B., Notices of the American Mathematical Society, vol. 15 (1968), 935.
[6]Keisler, H. J. and Tarski, A., From accessible to inaccessible cardinals, Fundamenta Mathematicae, vol. 53 (1963–64), pp. 225308.
[7]Kurepa, D., L'hypothèse de ramification, Comptes Rendus Hebdomadaires des Séances de l' Academie des Sciences. Séries A et B, vol. 202 (1936), pp. 185187.
[8]Marczewski, E. (Szpilrajn, E.), Séparabilité et multiplication cartésienne des espaces topologiques, Fundamenta Mathematicae, vol. 34 (1947), pp. 127143.
[9]Miller, E. W., A note on Suslin's Problem, American Journal of Mathematics, vol. 65 (1943), pp. 673678.
[10]Příkrý, K. L., Ph.D. Thesis, University of California, Berkeley, 1968.
[11]Rowbottom, F., Ph.D. Thesis, University of Wisconsin, Madison, 1965 (notes for 1967).
[12]Scott, D., Lectures on Boolean-valued models for set theory, Proceedings of the Summer Institute on Set Theory, U.C.L.A., 1967 (to appear).
[13]Sierpiński, W., Sur un problème de la théorie générale des ensembles équivalent au problème de Souslin, Fundamenta Mathematicae, vol. 35 (1948), pp. 165174.
[14]Silver, J., Ph.D. Thesis, University of California, Berkeley, 1966.
[15]Silver, J., The independence of Kurepa's conjecture and two-cardinal conjecture in model theory, Proceedings of the Summer Institute on Set Theory, U.C.L.A., 1967 (to appear).
[16]Solovay, R. M. and Tennenbaum, S., Iterated Cohen extensions and Suslir's Problem (to appear).
[17]Specker, E., Sur un problème de Sikorski, Colloquium Mathematicum, vol. 2 (1951), pp. 912.
[18]Stewart, D. H., M.Sc. Thesis, Bristol, 1966.
[19]Souslin, M., Problème 3, Fundamenta Mathematicae, vol. 1 (1920), p. 223.
[20]Tennenbaum, S., Suslin's Problem, Proceedings of the National Academy of Sciences of the United States of America, vol. 59 (1968), pp. 6063.
[21]Vopénka, P. and Hájek, P., Sets, semisets, models, North-Holland (to appear).
[22]Kurepa, D., On A-trees, Publications de l'Institut Mathematique, vol. 8 (22), (1968), pp. 153161.
[23]Gaifman, H. and Specker, E. P., Isomorphism types of trees, Proceedings of the American Mathematical Society, vol. 15 (1964), pp. 17.
[24]Solovay, R., forthcoming paper on Kurepa trees.
[25]Jensen, R., SH is compatible with CH, (mimeographed).
[26]Jensen, R., SH = weak compactness in L, (mimeographed).

Trees

  • Thomas J. Jech (a1)

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