TITLE: Statements which were given as axioms, which later turned out to be false.
QUESTION [16 upvotes]: 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!

REPLY [5 votes]: Perhaps one of the earliest examples would be with the Pythagoreans, who held that any two magnitudes were commensurable, measured as integer multiples of a smaller common unit, a belief that was connected with their mystical religious views and also with their mathematical theory of musical harmony. The Pythagoreans were shocked by the discovery of incommensurable numbers, such as $\sqrt{2}$.
But it may be anachronistic to refer to the fundamental Pythagorean beliefs or principles as "axioms".