The fired Google engineer wrote his memo after he went to a 'shaming' programme

I studied mathematical logic quite extensively, and I'm pretty sure this person doesn't know what he's talking about. Godel's incompleteness theorem is about facts which are unprovable in a certain logical framework. Any sufficiently complex consistent system A has a statement G which cannot be either proven or disproven in that system. That by itself is not a paradox, there are plenty of unsolved mathematical problems that might not be solvable using the axioms we have. For example "A is/is not consistent" is such a statement.

On the contrary, any statement that can be derived within the framework A cannot contradict A because that would prove that A is inconsistent! Of course with a real sufficiently complex logical system we cannot know whether it is consistent or not, but that doesn't mean it is possible to prove a paradox in it.

I think the reason Damore mentioned those jobs is because they, like software engineering (can be), are high pay/high stress BUT are considered undesirable, UNLIKE software engineering, because you die a lot less on the job as a software engineer.

In Google's ideal world where ideal means all jobs are distributed evenly matching the population of men and women, there would be an equal push for women taking those undesirable jobs. However, there isn't, and everyone is fine with that (I personally haven't heard any rallying cries to get more women into garbage collection, at least). What this translates to (in my interpretation of what I believe Damore's point was, as a disclaimer) is "men take dangerous, stressful jobs because women don't want them, even when they are high paying, due to differing pressures on the sexes."

The problem is when you remove the "dangerous" part (and software engineering one could argue is also not as stressful as these other jobs, if only for its lack of danger. The worst thing that can happen to you on the job is to be fired), women supposedly want the jobs, and so there's an artificial push for them to be employed by employers like Google. So now the market is smaller for qualified men to take these jobs, and theoretically they are more incentivized to take lower paying/more stressful/more dangerous (worse) jobs that may also be male dominated, but where there are no pushes for artificially increased diversity.

The result is a net loss for men overall, and not necessarily a net gain for women, when you consider his other point that most women may not actually want software engineering jobs (this is a more controversial statement in my opinion, but not the one you chose, and I still believe it deserves to be responded to respectfully instead of with ridicule and outrage). Additionally, this could also lead to a reduced average level of talent if less qualified women were being hired over qualified men, but I won't argue that and Google is an elite enough company that everyone they hire is likely supremely qualified.

So to me it seems like a reasonable (or at least not inflammatory for the sake of being inflammatory) assertion to make. I don't think that specific passage was there to provoke (although his original intent with the entire memo was to spark dialogue, which I suppose you could call 'sparking controversy' but I wouldn't). I believe you missed the point.

He's looking for a big debate. I don't know if it's fair to call it trolling. That implies his primary goal is to make other people upset. I think he actually wants to change people's minds.

I think he cited dangerous jobs like coal mining because he is trying to build a case that certain personalities lead genders, on average, to different professions.

He does not support this opinion with any research, and that's where his argument falls apart. He ignores societal factors and assumes biological factors explain gender employment gaps.