> In article <firstname.lastname@example.org>, > Aatu Koskensilta <email@example.com> wrote: > >> Peter Percival <firstname.lastname@example.org> writes: >> >> > I think that in a theory of sets without the axiom of choice there can >> > be non-comparable cardinalities. >> >> Sure. But even in such theories the inequality |N| < |R| will hold, >> witnessed by the injection taking the natural n to the real n. > > That would certainly justify |N| <= |R|, but why would it necessitate > strict inequality?
By the uncountability of the reals. In any case, this more or less trivial injection -- the identity map, morally speaking -- suffices to establish comparability.
-- Aatu Koskensilta (email@example.com)
"Wovon man nicht sprechen kann, darüber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus