Implications

We have the following indirect implication of form equivalence classes:

100 \(\Rightarrow\) 106
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
100 \(\Rightarrow\) 347 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
347 \(\Rightarrow\) 40 Partition principles and infinite sums of cardinal numbers, Higasikawa, M. 1995, Notre Dame J. Formal Logic
40 \(\Rightarrow\) 43 Consistency results for $ZF$, Jensen, R.B. 1967, Notices Amer. Math. Soc.
On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
43 \(\Rightarrow\) 106 Injectivity, projectivity and the axiom of choice, Blass, A. 1979, Trans. Amer. Math. Soc.
On the role of the Baire category theorem and dependent choice in the foundations of logic, Goldblatt, R. 1985, J. Symbolic Logic

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

Howard-Rubin Number Statement
100:

Weak Partition Principle:  For all sets \(x\) and \(y\), if \(x\precsim^* y\), then it is not the case that \(y\prec x\).

347:

Idemmultiple Partition Principle: If \(y\) is idemmultiple (\(2\times y\approx y\)) and \(x\precsim ^* y\), then \(x\precsim y\).

40:

\(C(WO,\infty)\):  Every well orderable set of non-empty sets has a choice function. Moore, G. [1982], p 325.

43:

\(DC(\omega)\) (DC), Principle of Dependent Choices: If \(S\)  is  a relation on a non-empty set \(A\) and \((\forall x\in A) (\exists y\in A)(x S y)\)  then there is a sequence \(a(0), a(1), a(2), \ldots\) of elements of \(A\) such that \((\forall n\in\omega)(a(n)\mathrel S a(n+1))\).  See Tarski [1948], p 96, Levy [1964], p. 136.

106:

Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.

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