Hereditary Cardinality

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The collection of sets of hereditary cardinality $<\lambda$, usually denoted $H_\lambda$, play a fundamental role in many set-theoretic constructions. Because of their strong closure properties (and the fact that they are transitive), they are typically used in model-theoretic arguments, various forcing constructions and even Shelah's pcf theory to carry out various computations in the cardinal arithmetic of singular cardinals. Their importance stems from the fact that they are natural models for strong fragments of ZFC and even theories extending ZFC.

Definition and Basic Absoluteness Properties

The set $H_\lambda$ is informally defined as the collection of all sets whose members have size smaller than $\lambda$, each of which themselves contain only members of size smaller than $\lambda$, each of which themselves contain only members of size smaller than $\lambda$, etc. Formally, $H_\lambda = \{x: |trcl(x)|<\lambda\}$ where $trcl(x)$ denotes the transitive closure of $x$ and is defined inductively as \begin{eqnarray}cl(x,0)=x \\cl(x,n+1)=cl(x,n)\cup\bigcup cl(x,n)\\ trcl(x)=\bigcup_{n<\omega} cl(x,n).\end{eqnarray} The notion of transitive closure of a set is important and interesting in its own right.

For large regular $\lambda$, the set $H_\lambda$ exhibits many absoluteness properties for formulas that aren't typically absolute for transitive models. For example, $H_\lambda$ correctly interprets or computes various facts about its members: "$B$ is the powerset of $A$", "$B$ is an ultrafilter over $A$", "$A$ is the collection of functions from $B$ to $C$", "$\kappa$ is cardinal", "$\kappa$ is regular cardinal" and "$\kappa$ is the successor of $\gamma$", etc. Because these properties are absolute for $H_\lambda$ when $\lambda$ is large, much of Shelah's pcf theory can be implemented inside these models.

When $\lambda$ is uncountable, $H_\lambda\subseteq V_\lambda$ but in general, the set $H_\lambda=V_\lambda$ only when $\lambda$ is inaccessible or $\omega$.

Hereditarily Finite Sets

The collection of hereditarily finite sets is the model $\langle H_{\aleph_0}, \in\rangle$ and is isomorphic to $\langle V_\omega, \in \rangle$. This model is, in essence, the standard model of the natural numbers and satisfies the axioms of Peano Arithmetic. In the language of set theory, this model satisfies ZFC without the axiom of infinity (in fact, it models the formal negation of the axiom of infinity) and in particular witnesses that the axiom of infinity is proof-theoretically independent of the other axioms of ZFC.

Hereditarily Countable Sets

Let $ZF^-$ denote the theory ZFC without the powerset axiom. Then $\langle H_{\aleph_1}, \in\rangle \models ZF^-$ (in fact, this model also satisfies the formal negation of the powerset axiom). In particular, this set is witnesses that the powerset axiom is proof-theoretically independent of the other axioms of ZFC. In general, $\langle V_\alpha, \in\rangle\not\models ZF^-$. More generally, if $\lambda$ is regular and uncountable then $\langle H_\lambda, \in\rangle |\models ZF^-$

Role in Elementary Embeddings

If $\kappa$ is $\lambda$-supercompact then there is some elementary embedding $j:V\to M$ with $M$ a transitive class and closed under arbitrary $\lambda$-sequences from $M$ and this implies that $M$ contains the model $H_{\lambda^+}$. This is also true of stronger large cardinals which are rank into rank types of embeddings. This latter fact plays a role in Woodin's proofs related to the HOD Conjecture.

References

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