Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
109 \(\Rightarrow\) 66 clear
66 \(\Rightarrow\) 367 clear
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
109:

Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.

66:

Every vector space over a field has a basis.

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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