I know that early axiomatizations of real arithmetic (in the first half of the nineteenth century) were often inadequate. For example, the earliest axiomatizations did not include a completeness axiom. (For example, there is no completeness axiom in Cauchy's *Cours d'Analyse*).

I also know that mathematicians in the early nineteenth century had some false beliefs about real analysis which were later uprooted as the process of rigorization continued. For example, it was widely believed that a continuous real function must be differentiable except at isolated points - a claim which Weierstrass refuted in the 1870s by defining a function that is continuous everywhere but differentiable nowhere. (http://en.wikipedia.org/wiki/Weierstrass_function).

What I'd like to know is whether at any point an "axiom" was proposed for real arithmetic, which subsequently turned out to be false.

I'd also like to hear about such cases from other branches of mathematics.

However, I'm not so interested in cases of inconsistent sets of axioms (e.g. Gottlob Frege's *Grundgesetze*, or Church's first formulation of the lambda calculus).

The best example I've found so far is Leibniz's "principle of continuity", according to which "what is true *up to the limit* is true *at the limit*". Apparently this was sometimes called an "axiom" although it is obviously not true in general. (I'm getting this from Lakatos's *Proofs and Refutations*, pg. 128). I'm not entirely happy with this example, because it's so obviously false that I can't believe that it was accepted, except as a heuristic.

Thanks in advance!