Notes

Description:

A general method for adding Dependent Choice (Form 43) to models of \(ZF\) is described

Content:

In Pincus [1977a], a general method for adding Dependent Choice (Form 43) to models of \(ZF\) is described and the following models of \(ZF\) are constructed with the properties indicated:

   
Model Properties
Model A
(\(\cal M43\))		 Form 62 (\(C(\infty ,<\aleph _{0}))\) and Form 43 \((DC)\) are true and Form 30 \((OP)\) is false.
Model B
(\(\cal M44\))		 Form30 and Form 43 are true and Form 15 (Kinna-Wagner Principle) is false.
Model C
(\(\cal M45\))		Form 30 and Form 43 are true and Form 49 (the order extension principle) is false.
Model D
(\(\cal M46(n,M)\))		 For each finite set \(Z\) of positive natural numbers and natural number \(n > 1\), for which the following is false
  • \(S(Z,n)\):  For every decomposition \(n = p_{1} + \ldots + p_{s}\) of \(n\) into a sum of primes at least one \(p_{i}\) divides an element  of  \(Z\),   (See Mostowski [1945]. Also see Note 15. Condition \(S\) implies condition \(D\) of Note 15.)
there is a model in which Form 46 (\(C(\infty ,Z)\)) and Form 43  are  true and Form 45 (\(C(\infty ,n)\)) is false.
Model E
(\(\cal M47(n,M)\))		For every finite set \(Z\) of positive integers and integer \(n > 1\), for which the following condition is false,
  • \(M(Z,n)\):  For any decomposition \(n = p_{1} + \ldots + p_{s}\)of \(n\) into a sum of primes there are non-negative \(q_{1}, \ldots, q_{s}\) such that \(q_{1}p_{1} + \ldots  + q_{s}p_{s} \in  Z\).
there is a model in which Form 46 (\(C(\infty ,Z)\)) and Form 43 are true andForm 47 (\(C(WO,n)\)) is false. (See Mostowski [1945].  Also, condition \(K\) of Note 15 implies condition \(M\).)

Howard-Rubin number: 73

Type: General Method Described

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