With the exception of an image demonstrating an argument so utterly and awfully illogical, it deserves public shaming, this post largely works with abstract cases as examples in the hope of not upsetting quite as many people as it otherwise might.
We start with the background stuff …
Kurt Gödel, in 1931, dropped a bit of a bombshell on a mathematical and logical world (that was quietly believing the opposite) by showing that there are things that can’t be proven or disproven. In other words, in all ‘vaguely normal’ systems, there are propositions that can be either true or false and it doesn’t really matter. ‘Mathematics is incomplete‘. In 1936, Alan Turing proved that there are problems that can’t be computed/solved and the rest of the computer science research community spent the next few decades realising that these were kind of the same thing.
Pretty disastrous, huh?
Well, no, not really. The mathematical and computer logic world dusted itself off and got on with it. And anyway, other branches of science – and beyond – had similar problems. In physics, for example, there’s a limit to how closely you can measure something before you change what you’re trying to measure. Turns out, in one form or another, ‘incompleteness’ is pretty normal in life.
So, no, incompleteness, isn’t that much of an issue. (it just means we don’t know everything – in fact, can’t know everything: there’s some bits of science, philosophy, etc., we can’t do from our little three-and-a-half-dimensions backwater of the universe; or we’re not God, if you like.)
However, rather than ‘incompleteness’, something called ‘inconsistency’ would be a problem. Why? And what does that look like?
Turing's Radiator 