Result Of Shelah Research Articles

Souslin trees and successors of singular cardinals

The questions concerning existence of Aronszajn and Souslin trees are of the oldest and most dealt-with in modern set theory. There are many results about existence of h+-Aronszajn trees for regular cardinals A. For these cases the answer is quite complete. (See Jech [6] and Kanamory & Magidor [8] for details.) The situation is quite different when A is a singular cardinal. There are very few results of which the most important (if not the only) are Jensen’s: V = L implies K-Aronszajn, K-Souslin and special K-Aronszajn trees exist iff K is not weaklycompact [7]. On the other hand, if GCH holds and there are no A+-Souslin trees for a singular h, then it follows (combining results of Dodd-Jensen, Mitchel and Shelah) that there is an inner model (of ZFC) with many measurable cardinals. In 1978 Shelah found a crack in this very stubborn problem by showing that if A is a singular cardinal and K is a super-compact one s.t. cof A < K < A, then a weak version of q ,* fai1s.l The relevance of this result to our problem was found by Donder through a remark of Jensen in [7, pp. 2831 stating that if 2’ = A+, then q ,* is equivalent to the existence of a special Aronszajn tree. Shelah, in [lo], had shown that the situation can be collapsed down to Xo+l so we get Cons(ZFC + there is a super-compact cardinal) implies Cons(ZFC + there are no X,+,-special Aronszajn trees). In the X, case (of Silver and Mitchel) the nonexistence consistency result for Aronszajn trees followed the result for special Aronszajn trees and used similar methods in the proof. A natural hope was that the same scheme will work for the Shelah result. Maybe for a singular A above a large enough cardinal, there can be no A+-Aronszajn trees. If this is true, then maybe by collapsing the large cardinal we could get the consistency of : “there are no rC,+,-Aronszajn trees”. In this paper we show that this is not the case. Theorem 1 shows the consistency with the existence of a super-compact

A mathematical proof of S. Shelah’s theorem on the measure problem and related results

Recently, S. Shelah proved that an inaccessible cardinal is necessary to build a model of set theory in which every set of reals is Lebesgue measurable. We give a simpler and metamathematically free proof of Shelah's result. As a corollary, we get an elementary proof of the following result (without choice axiom): assume there exists an uncountable well ordered set of reals, then there exists a non-measurable set of reals. We also get results about Baire property,K σ-regularity and Ramsey property.

On the bounded monadic theory of well-ordered structures

Monadic (second-order) theories of well-orderings were first studied by Büchi [1], [2], [3] using concepts of automata theory. There it was shown that the monadic theory of ω, the monadic theory of any countable ordinal, and the monadic theory of ω1 are decidable. Expansions of the well-ordering (ω, &lt;) by further relations were considered in [4], [5], [8] and [9], for example. Concerning such expansions, Buchi and Landweber [4] asked whether there is a set P ⊂ ω such that the weak monadic theory of (ω, &lt;, P) is decidable and the (strong) monadic theory of (ω, &lt;, P) is undecidable. In this note we give a negative answer by proving the following general theorem: If α is an ordinal and an n-tuple of subsets of α, then the monadic theory of (α, &lt;, ) is decidable provided the monadic theory of (cf (α), &lt;), i.e. of the cofinality of α, and the bounded monadic theory of (or, &lt;, ) are decidable. (In the bounded monadic theory the second-order variables range only over bounded subsets of α.) Also we show that in the bounded and the (strong) monadic theory of a structure (α, &lt;, ) the same classes of subsets of α are definable. For the proofs we use a result of Shelah [7] and a suitable version of a combinatorial argument which was introduced by Büchi [1] and McNaughton [6] into the study of monadic theories.