That inconsistency was subject to an immense amount of scrutiny by thousands of scientists for over a hundred years. Albert Einstein received a Nobel Prize for resolving the inconsistency. The theory that ultimately resolved it is called quantum mechanics. That's no longer an inconsistency.
Better examples would be the inconsistencies between general relativity, which demands curved spacetime, and quantum mechanics, which prohibits curved space. This is a very real inconsistency at the heart of modern physics. Many, many people are actively working on solutions for it. Search for Grand Unified Theory or Theory of Everything for more information. Candidate theories include string theory and its derivatives, loop quantum gravity, etc. There are plenty more.
The problem is that designing experiments for these theories is Hard. Big-O Capital H Hard. My flight's about to board, maybe I can expand on it during my layover.
Dark matter is another example. Observations of the speeds and orbits of stars in galaxies and galaxies in galactic clusters are not consistent with our measurements of their masses. Plenty of candidates for dark matter have been and are continuing to be tested.
Candidates include MOND, which supposes that our theories of gravity need to be modified when acceleration is astronomically low. We have designed experiments to support or disprove these theories and most of the results have landed on the side of disproving them. (search for "bullet cluster" or "dark matterless galaxies")
Another candidate is MACHOs. ("MAssive Compact Halo Object") Basically the universe is teeming with small black holes, brown dwarfs, loose planets unassociated with any star, basically a lot of stuff we can't see. We have designed and executed experiments to search for these, but the results have concluded that there are insufficient such objects to explain the inconsistency.
The third candidate with traction is WIMPs, or "weakly interacting massive particle". Basically theorizing that there are other types of unobserved particles that have mass but do not interact via the electromagnetic force, which makes them very difficult to observe. There was hope that neutrinos could explain all these, especially when it was demonstrated that neutrinos have mass. However, experiments trying to bound the mass of the neutrino have shown they are not nearly massive enough to explain the observations. Experiments are ongoing to find these particles, but have yet not discovered anything we can't already explain. However, there's a problem: maybe they're just too difficult for any experiment to observe. In that case, we might be SOL.
These are just two examples. All of science are all the other examples.
The idea that this person is the only person probing inconsistencies is pure hubris. It's not a useful starting point for a conversation, unless the point of the conversation is to talk about how awesome you are and how much everyone else sucks. Which is all I got from this article.