Pretty much what fink says.
The guy can do what he want, but if he kills the tree he is on the hook. Old trees are worth a crap ton, since they are irreplaceable 99 times out of 100 due to any potential tree that could be moved there would be insanely expensive to move (tree + Tree spade + potential utility rerouting to make it happen) and would not likely live since a large portion of its root system would be damaged.
If you want to keep the tree as is, I would find a reputable arborist to take a look at it and tell you if the branches removed as he wants would pose a serious risk to the tree. Assuming you can find one who would do that, then find a quote for someone selling and installing ANY tree even close to that caliper, and show your neighbour the potential bill if the tree dies. Also be sure to remind him that your tree is bigger than the one that would be installed by that quote, so finding an equivalent tree would be exponentially expensive for the reasons I listed above.
My bet is he backs down.
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