Examples of two inequivalent definitions which were once thought to be equivalent

Question

I once thought two definitions that I made in proof theory were equivalent, but then I discovered a counterexample. So I am now led to wonder, have there ever been cases in the history of mathematics when people made two inequivalent definitions and thought they were equivalent, but later a counterexample was discovered?

Answer 1

It was once thought that the solution set of $x^2 + 1 = 0$ was $\emptyset$, until the notion of the complex numbers became widespread. Or is that not the kind of thing you want?

In a similar vein, nobody knew for a good long while how to define infinitesimals properly; the best thing we had until Robinson's nonstandard analysis in the 1960's was simply that infinitesimals are zero, end of story.

There are any number of "equivalent" statements in elementary analysis, which are actually inequivalent when you relax Choice to a greater or lesser extent, or you go through the various stages in reverse mathematics to discover exactly which ones are equivalent assuming an extremely weak background logic. For example, "a continuous function on a closed bounded interval is bounded" is equivalent to WKL0 (see https://en.wikipedia.org/wiki/Reverse_mathematics), so it's not equivalent to "every nonempty bounded set of reals has a least upper bound" which is equivalent instead to ACA0. There's also a truly wonderful paper by Propp which examines just what properties of the reals do indeed characterise the reals.

Answer 2

The answer depends on what exactly is meant by "inequivalent definition" and "counterexample". For example, if we have the system of equations $$a = u+v,\; b = u-v $$ then is this equivalent to the system $$ (a+b)/2 = u,\; (a-b)/2 = v? $$ This used to be accepted without question until fields with characteristic $2$ were introduced. In those fields, division by $2$ is not possible and the second system does not even make sense.