That was my (charitable) interpretation too, until, to my dismay, Eliezer confirmed (at a meetup) that he had "leanings" in the direction of constructivism/intuitionism -- apparently not quite aware of the discredited status of such views in mathematics.

And indeed, when I asked Eliezer where he thinks the standard proof of infinite sets goes wrong, he pointed to the law of the excluded middle.

His idol E.T. Jaynes may be to blame, who in PTLS explicitly allied himself with Kronecker, Brouwer, and Poincaré as opposed to Cantor, Hilbert, and Bourbaki -- once again apparently not understanding the settled status of that debate on the side of Cantor et al. One is inclined to suspect this is where Eliezer picked such attitudes up.