Notes

Description: We give some definitions and properties of inaccessible cardinals. (Proofs of the results given below can be found in Drake [1974].)

Content:

We give some definitions and properties of inaccessible cardinals. (Proofs of the results given below can be found in Drake [1974].)

Definition: A cardinal \(\kappa\) is (strongly) inaccessible if \(\kappa\) is regular and for all \(\mu < \kappa\),\(2^{\mu} < \kappa\). (An inaccessible cardinal is also called a strong limit cardinal.)

Lemma: No successor cardinal, \(\aleph_{\alpha+1}\), is inaccessible. (This is [0 AJ].)

Definition: \(\mu\) is called a 2-valued,\(\kappa\)-additive (complete)  measure on \(x\) if \(\mu\) is a function from \(x\) to \(\{0,1\}\) such that

  1. \(\mu(x)=1\)
  2. If \(y=\bigcup_{\alpha<\kappa}x_{\alpha}\subseteq x\) is adisjoint union, then \(\mu(y)=\sum_{\alpha\in\kappa}\mu(x_{\alpha})\).

(The measure  is called non-trivial if the measure of a one-element set is 0.)

Definition: A cardinal \(\kappa\) is said to bemeasurable if \(\kappa > \aleph_0\) and \(\kappa\) has a 2-valued \(\kappa\)-additive non-trivial measure.

Theorem: (\(ZFC\)) Every measurable cardinal is inaccessible. (Form 319.)

Definition: A filter is \(\kappa\)-additive (complete)if the intersection of \(<\kappa\) members of the filter is in the filter.

Theorem: A cardinal \(\kappa\) is measurable iff there is a non-principal ultrafilter on \(\kappa\) which is\(\kappa\)-additive.

Definition: A measure \(\mu\) on \(\kappa\) is normal if \(\mu\) is a non-trivial \(\kappa\)-additive measure on \(\kappa\)such that for any function \(f: \kappa\to\kappa\) that is regressive (i.e., such that \(f(\alpha)<\alpha\) for all \(\alpha<\kappa\),\(\alpha\ne 0\)) there is a \(\beta<\kappa\) such that \(\mu(\{\alpha<\kappa: f(\alpha)=\beta\})=1\) (i.e., every regressive function on \(\kappa\) is constant on a set of measure 1).

Definition: If \(X= \langle X_{\alpha}:\alpha <\kappa\rangle\) is\(\kappa\)-sequence of subsets of \(\kappa\), the diagonal intersection, \(D(X)=\bigcap_{\alpha<\kappa}(X_{\alpha}\cup\alpha+1)=\{\beta<\kappa:\forall\alpha<\beta(\beta\in X_{\alpha}\}\).

Lemma: \(\mu\) is normal iff the diagonal intersection of any \(\kappa\)-sequence of sets of measure 1 is also of measure 1.

Theorem: (\(ZFC\)) If \(\kappa\) is measurable then \(\kappa\)has a normal measure.

Definition: If \(\kappa\) and \(\lambda\) are cardinals,\(\cal L_{\kappa,\lambda}\) is the language obtained by adding infinite conjunctions and disjunctions of cardinality less than\(\kappa\) and infinite quantifications of blocks of variables of cardinal less than \(\lambda\). (The ordinary first-order language is equivalent to \(\cal L_{\omega,\omega}\).)

Definition:

  1. A cardinal \(\kappa > \aleph_0\) is weakly compact if the language \(\cal L_{\kappa,\kappa}\) is (\(\kappa,\kappa\))-compact,i.e., if \(\Gamma\) is a set  of sentences of \(\cal L_{\kappa,\kappa}\)with cardinality \(\le\kappa\) and every subset of \(\Gamma\) of cardinality \(< \kappa\) has a model, then \(\Gamma\) has a model.
  2. A cardinal \(\kappa > \aleph_0\) is strongly compact if \(\cal L_{\kappa,\kappa}\) is compact, i.e., if \(\Gamma\) is any set of formulas of \(\cal L_{\kappa,\kappa}\) such that every subset of \(\Gamma\) of cardinality \(< \kappa\) has a model, then \(\Gamma\) has a model.
Definition:
  1. For \(n\in\omega\), \([X]^n\)  is the set of all \(n\)-element subsets of \(X\), and \([X]^{<\omega}\), is the set of all finite subsets of \(X\).
  2. The partition relation \(\kappa\to (\kappa)^n\) holds iff for every function \(f: [\kappa]^n\to 2\) there is an unbounded subset \(X\) of \(\kappa\) such that \(f/[X]^2\) is constant. (The set \(X\) is called homogeneous for \(f\).)
  3. The partition relation  \(\kappa\to (\kappa)^{<\omega}\) holdsiff for every function \(f: [\kappa]^{<\omega}\to 2\) there is an unbounded subset \(X\) of \(\kappa\) such that \(f\) is constant on \([X]^n\) for every \(n\in\omega\).

Theorem: (\(ZFC\)) If \(\kappa\) is inaccessible, then \(\kappa\)is weakly compact iff \(\kappa\to (\kappa)^2\).

Lemma: (\(ZFC\)) Every measurable cardinal is weakly compact.

Bull [1978] defines a measurable cardinal as acardinal that has a normal measure. With this definition, \(ZF\) \(\vdash\) Every measurable cardinal is weakly compact.)

Theorem: (\(ZFC\)) If \(\kappa\) is a  regular infinite cardinal,\(\kappa\) is strongly compact iff for any \(\kappa\)-complete filter \(J\)on a set \(X\), \(J\) can be extended to a \(\kappa\)-complete ultrafilteron \(X\).

Lemma: (\(ZFC\)) Every strongly compact cardinal is measurable.

Definition:

  1. \(\cal P_{\kappa}(\lambda)\) is the set of all subsets of\(\lambda\) of cardinality \(<\kappa\).
  2. Let \(X=\cal P_{\kappa}(\lambda)\),
    1. For each \(\xi<\lambda\), let \(C_{\xi}=\{x\in X:\xi\in x\}\). An ultrafilter extending the filter \(\{C_{\xi}:\xi < \lambda\}\) is called a regular ultrafilter. (The measure corresponding to the regular ultrafilter is called a regular measure.)
    2. A measure \(\mu\) on \(X\) is called normal if it is \(\kappa\)-additive, regular and such that if \(f(x)\in x\) for each \(x\in X\), then for some \(\xi <\lambda\), \(\mu(\{x\in X: f(x)=\xi\})=1\).

Definition:

  1. If \(\kappa\) and \(\lambda\) are cardinals such that \(\aleph_0<\kappa\le\lambda\), \(\kappa\) is \(\lambda\)-supercompact if there is a normal measure on \(\cal P_{\kappa}(\lambda)\).
  2. If \(\kappa>\aleph_0\), \(\kappa\) is supercompact if \(\kappa\) is \(\lambda\)-supercompact for every \(\lambda\ge\kappa\).

Lemma: \((ZFC)\) Every supercompact cardinal is strongly compact.

Lemma: Form 319 implies Form 320. (Use [0 AJ].)

Definition: \(\kappa\) is a Ramsey cardinal iff \(\kappa\to(\kappa)^{<\omega}\).

Lemma: Every measurable cardinal is a Ramsey cardinal.

Definition: A cardinal \(\kappa\) is called a Rowbottom cardinal iff, whenever
\(\frak A =\) \(\langle A, U,\cdots\rangle\) is a structure of countable length, where \(|A|=\kappa\), \(U\subset A\),and \(|U|<\kappa\), then \(\frak A\) has an elementary substructure \(\frak B = \langle B, U',\cdots\rangle\) with\(|\frak B|=\kappa\) and \(|U'|\le\aleph_0\).

Lemma: Every Ramsey cardinal is a Rowbottom cardinal.

Definition: A cardinal \(\kappa\) is said to be almost huge iff there is an \(n\in\omega\) and a cardinal \(\lambda>\kappa\), a \(\kappa\)-additive normal ultrafilter \(U\) on\(2^{\lambda}\), and a sequence \(\kappa=\lambda_0<\lambda_1<\cdots<\lambda_n=\lambda\) such that for each \(i \le n\), \(\{x\subseteq \lambda:x\cap\lambda_{i+1}=\lambda_i\}\in U\).

Lemma: Every almost huge cardinal is supercompact.

In a series of papers (Apter [1985b], Apter [1988], Apter [1990], Apter [1992] and Apter/Henle [1991]), various consistency results about large cardinals are obtained. We give some examples below.

  1. \(Con(ZFC +\) There exists cardinals \(\kappa<\lambda\) such\(\kappa\) is \(\lambda\)-supercompact and \(\lambda\) is measurable)\(\to\) \(Con(ZF +\) cof(\(\kappa\))=\(\omega\) \(+\) \(\kappa\) is a Rowbottomcardinal \(+\) \(\kappa^+\) is a measurable cardinal).
  2. (\(Con(ZFC +\) There exists an almost huge cardinal) \(\to Con(ZF + \aleph_1\) is measurable \(+\) All uncountable cardinals are singular).
  3. (\(Con(ZFC +\) There exists an almost huge cardinal) \(\to Con(ZF + \neg C(\aleph_0,\infty)\) \(+\) An uncountable cardinalis measurable iff it is the successor of a singular cardinal).
  4. (\(Con(ZFC +\) There exists an almost huge cardinal) \(\to Con(ZF + \neg C(\aleph_0,\infty)\) \(+\) \(\forall\alpha\) \(\aleph_\alpha\) or \(\aleph_{\alpha+1}\) is measurable).

In Bull [1978], besides Form 321, there are the following consistency results.

Let \(S\) and \(S'\) be the following statements:

  • \(S\):  There are cardinals \(\aleph_\alpha < \aleph_\beta <\aleph_\gamma\) such that \(\aleph_\alpha\) is \(\aleph_\gamma\)-supercompact; \(\aleph_\beta\) is the first measurable cardinal greater than \(\aleph_\alpha\); and \(\aleph_\gamma = |2^{\aleph_\beta}|\).
  • \(S'\):  There are cardinals \(\aleph_\alpha < \aleph_\beta\) such that \(\aleph_\beta\) is the first weakly compact cardinal greater than \(\aleph_\alpha\)  and  \(\aleph_\alpha\) is \(\aleph_\beta\)-supercompact.
Then we have
  1. \(Con(ZFC +\) \(S\)) \(\to Con(ZF + \)Form 43 \(+ \aleph_1\) is weakly compact \(+ \aleph_2\) is measurable).
  2. \(Con(ZFC +\) \(S'\)) \(\to Con(ZF + C(\aleph_\alpha,\infty) + \aleph_\alpha\) and \(\aleph_{\alpha+1}\) are both weakly compact. (In fact, 2 may be extended  to a finite sequence of weakly compact cardinals.)

Howard-Rubin number: 20

Type: Proofs and statements

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