Kristol, born in Brooklyn to Orthodox Jews, was a Trotskyite at City College and an infantryman in World War II. When he came home, he edited Commentary, founded The Public Interest, and in the 1970s became the world's first Neoconservative.
Neoconservatives were, basically, former leftist intellectuals who decided they hated liberals, radicals, and Goldwater conservatives, and loved American moral superiority and, uh, tax cuts. It was much "sunnier" and nicer than regular conservatism. And they liked FDR. And Israel.
The big idea of Neoconservatism was, per 2003-era Irving, "cutting tax rates in order to stimulate steady economic growth." A revolutionary concept! Great for getting elected. And pretty good for getting reelected, until it stops working.
In fact, nearly every major tenet of his political philosophy, as he laid it out in 2003, informed the worst abuses of Bush administration. The entire section on foreign policy and American military might would be laughable if it weren't for the disastrous way in which his theories were tested on the ground.
He is survived by his son Bill Kristol—a remarkably embarrassing partisan party hack, which is at least not something Irving ever was—his wife Bea, and their daughter Elizabeth Nelson.