A subset X of the real line is a strong measure zero set(英语:strong measure zero set) if to every sequence (εn) of positive reals there exists a sequence of intervals (In) which covers X and such that In has length at most εn. Borel's conjecture, that every strong measure zero set is countable, is independent of ZFC.
A subset X of the real line is -dense if every open interval contains -many elements of X. Whether all -dense sets are order-isomorphic is independent of ZFC.[2]
序理论
蘇斯林問題(Suslin's problem)提出一個指定的特性列表能否刻画一個實數R的有序集合。這是在ZFC中未決的[3]。 一条 Suslin line 是指一个满足该指定的特性列表但不与R序同构的有序集。鑽石原則證明了Suslin line的存在性,而MA + ¬CH 推導出EATS(every Aronszajn tree is special;每一個Aronszajn tree皆為特別)[4], 而推導出(但不等價於)[5]Suslin line的不存在性。Ronald Jensen證明了CH並不推出Suslin line的存在性[6]。
^Baumgartner, J., All -dense sets of reals can be isomorphic, Fund. Math. 79, pp.101 – 106, 1973
^Solovay, R. M.; Tennenbaum, S. Iterated Cohen extensions and Souslin's problem. Annals of Mathematics. Second Series. 1971, 94 (2): 201–245. JSTOR 1970860. doi:10.2307/1970860.
^Baumgartner, J., J. Malitz, and W. Reiehart, Embedding trees in the rationals, Proc. Natl. Acad. Sci. U.S.A., 67, pp. 1746 – 1753, 1970
^Shelah, S., Free limits of forcing and more on Aronszajn trees, Israel Journal of Mathematics, 40, pp. 1 – 32, 1971
^Devlin, K., and H. Johnsbraten, The Souslin Problem, Lecture Notes on Mathematics 405, Springer, 1974
^Silver, J., The independence of Kurepa's conjecture and two-cardinal conjectures in model theory, in Axiomatic Set Theory, Proc. Symp, in Pure Mathematics (13) pp. 383 – 390, 1967
^Shelah, S., Proper and Improper Forcing, Springer 1992
^Schlindwein, Chaz, Shelah's work on non-semiproper iterations I, Archive for Mathematical Logic (47) 2008 pp. 579 – 606
^Schlindwein, Chaz, Shelah's work on non-semiproper iterations II, Journal of Symbolic Logic (66) 2001, pp. 1865 – 1883