Notes

Description: Form 200 is studied in Dawson/Howard [1976]

Content:

Form 200 is studied in Dawson/Howard [1976] where  it is shown that

  1. Each of
    1. \(\exists\) infinite \(x\) such that \(2^{x} < x!\)
    2. \(\exists\) infinite \(x\) such that \(2^{x} > x!\)  and
    3. \(\exists\) infinite \(x\)  such  that \(2^{x}\)  and \(x!\)  are incomparable
    are consistent with \(ZF\). (\(x!\)  is the cardinality of the set of all one to one functions from \(x\) onto \(x\).).
    2. for all \(x\), \(2x = x\) implies \(2^{x} \le  x!\) .

Pincus [1978] a model of \(ZF\) is constructed in which \(AC\) (Form 1) is false, \(\forall\) infinite \(x\), \(2x = x\) (Form 3) is false and \(\forall\) infinite \(x\), \(2^{x} = x!\) is true.  Also true in this model are \(\forall\) infinite \(x\), \(2^{x} = | x^{x}| \) and \(\forall\) infinite \(x\),\(2^{x} = |x^{x} - x!|\).

Howard-Rubin number: 64

Type: Summary of results

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