Implications

We have the following indirect implication of form equivalence classes:

302 \(\Rightarrow\) 19
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
302 \(\Rightarrow\) 392 clear
392 \(\Rightarrow\) 394 clear
394 \(\Rightarrow\) 337 clear
337 \(\Rightarrow\) 92 clear
92 \(\Rightarrow\) 94 clear
94 \(\Rightarrow\) 34 Non-constructive properties of the real numbers, Howard, P. 2001, Math. Logic Quart.
34 \(\Rightarrow\) 19 Sur les fonctions representables analytiquement, Lebesgue, H. 1905, J. Math. Pures Appl.

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

Howard-Rubin Number Statement
302:

Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain.

392:

\(C(LO,LO)\): Every linearly ordered set of linearly orderable sets has a choice function.

394:

\(C(WO,LO)\): Every well ordered set of non-empty linearly orderable sets has a choice function.

337:

\(C(WO\), uniformly linearly ordered):  If \(X\) is a well ordered collection of non-empty sets and there is a function \(f\) defined on \(X\) such that for every \(x\in X\), \(f(x)\) is a linear ordering of \(x\), then there is a choice function for \(X\).

92:

\(C(WO,{\Bbb R})\):  Every well ordered family of non-empty subsets of \({\Bbb R}\) has a choice function.

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.

34:

\(\aleph_{1}\) is regular.

19:

A real function is analytically representable if and only if it is in Baire's classification. G.Moore [1982], equation (2.3.1).

Comment:

Back