Form Classes

Statement:

Rado's Selection Lemma: Let \(\{K(\lambda): \lambda \in\Lambda\}\) be a family  of finite subsets (of \(X\)) and suppose for each finite \(S\subseteq\Lambda\) there is a function \(\gamma(S): S \rightarrow X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda)\in K(\lambda))\).  Then there is an \(f: \Lambda\rightarrow X\) such that for every finite \(S\subseteq\Lambda\) there is a finite \(T\) such that \(S\subseteq T\subseteq\Lambda\) and such that \(f\) and \(\gamma (T)\) agree on S.

Howard_Rubin_Number: 99

Parameter(s): This form does not depend on parameters

This form's transferability is: Unknown

This form's negation transferability is: Negation Transferable

Article Citations:
Rado-1949: Axiomatic treatment of rank in infinite sets

Book references

Note connections:

The following forms are listed as conclusions of this form class in rfb1: 99, 324, 14, 18, 45-n, 70, 80, 98, 128, 154, 198, 216, 225, 344, 358,

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Complete List of Equivalent Forms

Howard-Rubin Number Statement References
99 A

Weak Rado Lemma: Let \(\Lambda\) be a set. Suppose that \(\gamma\) is a function with domain all finite subsets of \(\Lambda\) such that for each finite \(S\subseteq\Lambda\), \(\gamma(S): S \rightarrow \{0,1\}\). Then there is a \(\phi\) defined on \(\Lambda\) with the property that for every finite \(S\subseteq\Lambda\) there is a finite \(T\subseteq \Lambda\) such that \(S\subseteq T\) and \(\gamma(T)\) and \(\phi\) agree on \(S\). \iput{Rado's selection lemma}

Howard [1993]

Book: Handbook of Analysis and its Applications
99 B

Rado's Lemma restricted to families of two element sets: Let \(\{k(\lambda): \lambda\in\Lambda\}\) be a family of 2 element subsets of a set \(X\) and suppose that for each finite \(S \subseteq\Lambda\), there is a function \(\gamma(S)\) from \(S\) into \(X\) such that \((\forall\lambda\in S)(\gamma(S)(\lambda) \in k(\lambda))\). Then there is a function from \(\Lambda\) into \(X\) such that for every finite \(S\subseteq\Lambda\), there is a finite \(T\) with \(S\subseteq T \subseteq\Lambda\) such that \(f\) and \(\gamma(T)\) agree on \(S\).

Rav [1977] Howard [1993]

99 C

Strong Rado Lemma 2: Let \(T\) be a family of finite sets and let \({\cal F}\) be a collection of finite subsets of \(T\) satisfying

  1. \((\forall K\in T)(\exists S\in {\cal F})(K\in S)\)
  2. \((\forall S_{1}, S_{2}\in{\cal F})(S_{1}\cup S_{2}\in {\cal F})\)
Assume there is a function \(\phi\) with domain \({\cal F}\) such that for each \(S\in{\cal F}\), \(\phi(S) = \phi_{S}\) is a function with domain \(S\) and satisfies \((\forall K\in S)(\phi_{S}(K)\in K)\). Then there is a function \(f\) with domain \(T\) and with the following property: For all \(S\in{\cal F}\) there is an \(S'\in{\cal F}\) such that \(S\subseteq S'\) and \(f\) and \(\phi_{S'}\) agree on \(S\).

Howard [1993]