Catálogo de publicaciones - libros

This book is devoted to a particular recursive method of constructing mathematical structures that live on a given ordinal θ , using a single transformation ξ ↦ C _ξ which assigns to every ordinal ξ < θ a set C _ξ of smaller ordinals that is closed and unbounded in the set of ordinals < ξ . The transfinite sequence $$ C_\xi \left( {\xi < \theta } \right) $$ which we call a ‘ C -sequence’ and on which we base our recursive constructions may have a number of ‘coherence properties’ and we shall give a detailed study of them and the way they influence these constructions. Here, ‘coherence’ usually means that the C _ξ’s are chosen in some canonical way, beyond the already mentioned and natural requirement that C _ξ is closed and unbounded in ξ for all ξ . For example, choosing a canonical ‘fundamental sequence’ of sets C _ξ ⊆ ξ for ξ < ε _0, relying on the specific properties of the Cantor normal form for ordinals below the first ordinal satisfying the equation x = ω ^x, is a basis for a number of important results in proof theory. In set theory, one is interested in longer sequences as well and usually has a different perspective in applications, so one is naturally led to use some other tools besides the Cantor normal form. It turns out that the sets C _ξ can not only be used as ‘ladders’ for climbing up in recursive constructions but also as tools for ‘walking’ from an ordinal β to a smaller one α , β = β _0 > β _1 > ... > β _ n −1 > β _n = α where the ‘step’ β _i → β _ i +1 is defined by letting β _ i +1 be the minimal point of C _βi that is bigger than or equal to α .

The space ω _1 of countable ordinals is by far the most interesting space considered in this book. There are many mathematical problems whose combinatorial essence can be reformulated as problems about ω _1, which is in some sense the smallest uncountable structure. What we mean by ‘structure’ is ω _1 together with a system C _α ( α < ω _1) of fundamental sequences, i.e., a system with the following two properties: (a) C _ α +1 = { α }, (b) C _α is an unbounded subset of α of order-type ω , whenever α is a countable limit ordinal > 0.

In this section we study a characteristic of the minimal walk that satisfies certain triangle inequalities reminiscent of those found in an ultra-metric space. Some applications of the corresponding metric-like theory of ω _1 will appear already in this section and some of them will later on get separate treatments.

In this and the following section we present a general study of the following notion already encountered at several places above.

Recall that a walk from a countable ordinal β to a smaller ordinal α along the fixed C -sequence C _ξ ( ξ < ω _1) is a finite decreasing sequence $$ \beta = \beta _0 > \beta _1 > \cdots \beta _n = \alpha , $$ where β _ i +1 = min( $$ C_{\beta _i } $$ \ α ) for all i < n . Recall also the notion of the upper trace of the minimal walk, $$ Tr\left( {\alpha ,\beta } \right) = \{ \beta _0 ,\beta _1 , \ldots ,\beta _n \} , $$ the finite set of places visited in the minimal walk from β to α . The following simple fact about the upper trace lies at the heart of all known definitions of square-bracket operations, not only on ω _1 but also at higher cardinalities.

One of the most basic questions frequently asked about set-theoretical trees is the question whether they contain any cofinal branch , a branch that intersects each level of the tree. The fundamental importance of this question has already been realized in the work of Kurepa [65] and then later in the works of Erdős and Tarski [32] in their respective attempts to develop the theory of partition calculus and large cardinals. A tree T of height equal to some regular cardinal θ may not have a cofinal branch for a very special reason as the following definition indicates.

The purpose of this section is to study walks along C -sequences that have the following pleasant coherence property.

In what follows, θ will be a fixed regular infinite cardinal. 8.1.1 $$ osc:\mathcal{P}\left( \theta \right)^2 \to Card $$ is defined by 8.1.2 $$ osc\left( {x,y} \right) = \left| {x\backslash \left( {\sup \left( {x \cap y} \right) + 1} \right)/ \sim } \right|, $$ where ∼ is the equivalence relation on x \ (sup( x ⋂ y )+1) defined by letting α ∼ β iff the closed interval determined by α and β contains no point from y . Hence, osc( x, y ) is simply the number of convex pieces the set x \ (sup( x ⋂ y )+1) is split by the set y (see Figure 8.1). Note that this is slightly different from the way we have defined the oscillation between two subsets x and y of ω _1 in Section 2.3 above, where osc( x, y ) was the number of convex pieces the set x is split by into the set y \ x . Since the variation is rather minor, we keep the same old notation as there is no danger of confusion. The oscillation mapping has proven to be a useful device in various schemes for coding information. Its usefulness in a given context depends very much on the corresponding ‘oscillation theory’, a set of definitions and lemmas that disclose when it is possible to achieve a given number as oscillation between two sets x and y in a given family X . The following definition reveals the notion of largeness relevant to the oscillation theory that we develop in this section.

In this section κ is a regular cardinal and C _α ( α < κ ^+) is a fixed C -sequence with the property that tp( C _α) ≤ κ for all α < κ ^+. When the C -sequence is necessarily coherent, then it is natural to define the corresponding mapping 9.1.1 $$ \rho :[\kappa ^ + ]^2 \to \kappa $$ as follows: 9.1.2 $$ \rho (\alpha ,\beta ) = \sup \{ tp(C_\beta \cap \alpha ),\rho (\alpha ,\min (C_\beta \backslash \alpha )),\rho (\xi ,\alpha ): \xi \in C_\beta \cap \alpha \} , $$ with the boundary value ρ ( α, α ) = 0 for all α < κ ^+, a definition that is slightly different from the one given above in (7.3.2) above. Clearly, 9.1.3 $$ \rho (\alpha ,\beta ) \geqslant \rho _1 (\alpha ,\beta ) for all \alpha < \beta < \kappa ^ + , $$ and so, using Lemma 6.2.1, we have the following fact.

The reader must have already noticed that in this book so far, we have only considered functions of the form f : [ θ ]^2 → I or equivalently sequences $$ f_\alpha :\alpha \to I\left( {\alpha < \theta } \right) $$ of one-place functions. To obtain analogous results about functions defined on higher-dimensional cubes [ θ ]^n, one usually develops some form of stepping-up procedure that lifts a function of the form f : [ θ ]^n → I to a function of the form g : [ θ ^+]^ n +1 → I . The basic idea seems quite simple. One starts with a coherent sequence e _α: α → θ ( α < θ ^+) of one-to-one mappings and wishes to define g : [ θ ^+]^ n +1 → I as follows: 10.1.1 $$ g\left( {\alpha _0 ,\alpha _1 , \ldots ,\alpha _n } \right) = f\left( {e\left( {\alpha _0 ,\alpha _n } \right), \ldots ,e\left( {\alpha _{n - 1} ,\alpha _n } \right)} \right). $$