A Problem on Ordered Sets
نویسندگان
چکیده
1. Let S be an ordered set, of power 1 SI and order type 8 = 4, We denote by #* the converse of 4, i.e. the order type obtained from + by replacing every order relation x < y by the corresponding relation y < x, and by wrc the least ordinal number of power N,. It is easy to see that, if ISl=tQ, then S contains a subset S’ such that either s’ = w,, or 8’ = w**. For cardinals N, > w’. the corresponding property, with o,, replaced by o,, no longer holds. Thus, the linear continuum C, ordered by magnitude, satisfies ] C I= Wm >, ~~ but contains no subset of any of the types o,, o,*. If, however, we assume the continuum hypothesis PO = No, then 112 = 1, and the following statement is true. Given any ordinal a < ol, there are subsets Cl and C, of C, of order types a and CC* reqmtively. The question arises whether not only C but every ordered set S of cardinal ~~ contains either (i) a subset of type wl, or (ii) a subset of type 0~8, or (iii) two subsets of types cc and cc* respectively, corresponding to every ordinal cc < wl. We shall show, assuming the continuum hypothesisf and making free use of the axiom of choice, that this is, in fact, true. More generally, we shall obtain, as principal result of this note, a simple characterization of those cardinals ~~ which possess the following
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