Form Classes

Statement:

\(CT_{\hbox{fin}}\): The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.

Howard_Rubin_Number: 270

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

This form's transferability is: Unknown

This form's negation transferability is: Unknown

Article Citations:
Kolany-Wojtylak-1991: Restricted versions of the compactness theorem

Book references

Note connections:

The following forms are listed as conclusions of this form class in rfb1: 271-n, 62,

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Complete List of Equivalent Forms

Howard-Rubin Number Statement References
270 A

\(C(\infty,<\aleph_{0})\wedge CT_{2}\) (\(CT_{2}\) is the compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most two formulas. See Form 271(\(n\)).)

Kolany-Wojtylak-1991

270 B

\(F_{\mathrm{fin}}\): For any family \({\cal F}\) of pairwise disjoint, non-empty, finite sets if \(R\) is a symmetric binary relation on \(\bigcup {\cal F}\) which satisfies

  1. the set \(\{b\in\bigcup {\cal F}: \neg a\mathrel R b\}\) is finite for every \(a\in\bigcup{\cal F}\)
    and
  2. every finite subfamily of \({\cal F}\) has an \(R\)-consistent choice function,
then \({\cal F}\) has an \(R\)-consistent choice function.

Kolany-Wojtylak-1991

270 C-n

\(F^{\mathrm{fin}}_n\) (\(n\ge 3\) a natural number): For any family \({\cal F}\) of pairwise disjoint, \(n\) element sets if \(R\) is a symmetric binary relation on \(\bigcup {\cal F}\) which satisfies

  1. the set \(\{ b\in \bigcup {\cal F} : \neg a\mathrel R b\}\) is finite for every \(a \in \bigcup {\cal F}\), and
  2. every finite subfamily of \({\cal F}\) has an \(R\)-consistent choice function,
then \({\cal F}\) has an \(R\)-consistent choice function.

Kolany [1992]