Implications

We have the following indirect implication of form equivalence classes:

14 \(\Rightarrow\) 221
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
14 \(\Rightarrow\) 52 On the application of Tychonoff's theorem in mathematical proofs, L o's, J. 1951, Fund. Math.
Two applications of the method of construction by ultrapowers to analysis, Luxemburg, W.A.J. 1970, Proc. Symp. Pure. Math.
Applications of Model Theory to Algebra, Analysis and Probability, Luxemburg, 1969, 123-137
52 \(\Rightarrow\) 221 clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
14:

BPI: Every Boolean algebra has a prime ideal.

52:

Hahn-Banach Theorem:  If \(V\) is a real vector space and \(p: V \rightarrow {\Bbb R}\) satisfies \(p(x+y) \le p(x) + p(y)\) and \((\forall t > 0)( p(tx) = tp(x) )\) and \(S\) is a subspace of \(V\) and \(f:S \rightarrow {\Bbb R}\) is linear and satisfies \((\forall  x \in S)( f(x) \le  p(x) )\) then \(f\) can be extended to \(f^{*} : V \rightarrow {\Bbb R}\) such that \(f^{*}\) is linear and \((\forall x \in V)(f^{*}(x) \le p(x))\).

221:

For all infinite \(X\), there is a non-principal measure on \(\cal P(X)\).

Comment:

Back