Implications

We have the following indirect implication of form equivalence classes:

333 \(\Rightarrow\) 366
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
333 \(\Rightarrow\) 67 clear
67 \(\Rightarrow\) 89 On cardinals and their successors, Jech, T. 1966a, Bull. Acad. Polon. Sci. S'er. Sci. Math. Astronom. Phys.
89 \(\Rightarrow\) 90 The Axiom of Choice, Jech, 1973b, page 133
90 \(\Rightarrow\) 91 The Axiom of Choice, Jech, 1973b, page 133
91 \(\Rightarrow\) 79 clear
79 \(\Rightarrow\) 367 Eine Basis aller Zahlen und die unstetigen Losungen der Functionalgleichung: \(f(x+y) = f(x) + f(y)\), Hamel, G. 1905, Math. Ann.
367 \(\Rightarrow\) 366 Eine Basis aller Zahlen und die unstetigen Losungen der Functionalgleichung: \(f(x+y) = f(x) + f(y)\), Hamel, G. 1905, Math. Ann.

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

Howard-Rubin Number Statement
333:

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.

67:

\(MC(\infty,\infty)\) \((MC)\), The Axiom of Multiple Choice: For every set \(M\) of non-empty sets there is a function \(f\) such that \((\forall x\in M)(\emptyset\neq f(x)\subseteq x\) and \(f(x)\) is finite).

89:

Antichain Principle:  Every partially ordered set has a maximal antichain. Jech [1973b], p 133.

90:

\(LW\):  Every linearly ordered set can be well ordered. Jech [1973b], p 133.

91:

\(PW\):  The power set of a well ordered set can be well ordered.

79:

\({\Bbb R}\) can be well ordered.  Hilbert [1900], p 263.

367:

There is a Hamel basis for \(\Bbb R\) as a vector space over \(\Bbb Q\).

366:

There is a discontinuous function \(f: \Bbb R \to\Bbb R\) such that for all real \(x\) and \(y\), \(f(x+y)=f(x)+f(y)\).

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