Implications

We have the following indirect implication of form equivalence classes:

264 \(\Rightarrow\) 74
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
264 \(\Rightarrow\) 202 Variations of Zorn's lemma, principles of cofinality, and Hausdorff's maximal principle, Part I and II, Harper, J. 1976, Notre Dame J. Formal Logic
202 \(\Rightarrow\) 40 clear
40 \(\Rightarrow\) 39 clear
39 \(\Rightarrow\) 8 clear
8 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 74 note-10

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

Howard-Rubin Number Statement
264:

\(H(C,P)\): Every connected relation \((X,R)\) contains a \(\subseteq\)-maximal partially ordered set.

202:

\(C(LO,\infty)\): Every linearly ordered family of non-empty sets has  a choice function.

40:

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

39:

\(C(\aleph_{1},\infty)\): Every set \(A\) of non-empty sets such that \(\vert A\vert = \aleph_{1}\) has a choice function. Moore, G. [1982], p. 202.

8:

\(C(\aleph_{0},\infty)\):

94:

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

74:

For every \(A\subseteq\Bbb R\) the following are equivalent:

  1. \(A\) is closed and bounded.
  2. Every sequence \(\{x_{n}\}\subseteq A\) has a convergent subsequence with limit in A.
Jech [1973b], p 21.

Comment:

Back