Description:
For research on set theory
without \(AC\)
Content:
For research on set theory
without \(AC\) the following references may be consulted:
- Results on Dedekind finite, infinite fields in set theory
without \(AC\) see Hickman [1978a], Hickman [1980a], and Hickman [1982]
- The Dedekind cardinals without \(AC\) see Ellentuck [1970],
Ellentuck [1973],Ellentuck [1976], and Monro [1975].
- Order types of linearly ordered Dedekind finite sets without \(AC\) see Hickman [1979a]
- Ordered sets satisfying ``there is no infinite descending
sequence'', see Hickman [1975].
- Possible properties of rings see Hodges [1974] where the
negation of Form 278 is shown to be consistent with \(ZF\).
- Estimates of the sizes of \(\aleph (x)\) and \(\aleph ^{*}(x)\) in set
theory without AC. \(\aleph (x)\) is the smallest ordinal \(\alpha \) such
that \(\neg (|\alpha| \le|x|)\) and \(\aleph^{*}(x)\) is the smallest
ordinal \(\alpha \) such that \(\neg (|\alpha|\le * |x|)\) where
\(\le *\) is the surjective cardinal ordering. See Kruse [1974]
- Possible cardinality of the set of Dedekind finite cardinals
in a Fraenkel-Mostowski model see A. Rubin/J. Rubin [1974].
- H. Amorphous sets \(X\) for which \(\cal P \cal P \cal P(X)\) is large see Monro [1973c]. (\(X\) is amorphous if \(X\) is infinite and
\(\forall Y\subseteq X\), \(Y\) is finite or \(X-Y\) is finite.)
- Topology without the axiom of choice see Chandler [1972].
- Possible order types of convex sets of cardinal numbers see Truss [1973e]
- Large cardinals in set theory without the axiom of choice see
Spector [1991] and Apter [1982]
- Logic without \(AC\) see Malitz [1992]
- Posets and lattices without \(AC\) see Hickman [1977a]
Howard-Rubin number:
45
Type:
Reference summary
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