This question already has an answer here:
Sounds crazy i know, but hear me out there are place's all over the universe with ice glaciers much bigger then what we can even imagine on earth.
So what if he build a Tower in the middle of Antartica where temperatures would never raise higher then 28 degrees (freezing point of salt water) we could make it any thickness and the most of materials would be much cheaper then anythings.
if anything happened it would be in the middle of no where and wouldn't be able to hurt anyone. also since no one owns Antartica it would be a mutual interest for ever country to invest into.
with all of that it may also help with countries that have raising sea level problems and would lower the sea around the globe.