Axiom Of Choice

We have the following indirect implication of form equivalence classes:

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

Implication	Reference
292 \(\Rightarrow\) 90	The axiom of choice and linearly ordered sets, Howard, P. 1977, Fund. Math.
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
292:	\(MC(LO,\infty)\): For each linearly ordered family of non-empty sets \(X\), there is a function \(f\) such that for all \(x\in X\) \(f(x)\) is non-empty, finite subset of \(x\).
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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