By all means, take Mosul, and continue on until ISIS is no more.
But ISIS, also known as Islamic State, is and has been the wrong focus. Were it not holding hostage the scattered populations it controls in urban areas, a properly directed military coalition of two or three Western powers, or the United States alone, could roll it up in a week. Even as things are, and despite the chaos and cross-loyalties in the present theater of war, with competent diplomacy and military force ISIS could be crushed in a matter of months. The key is NATO’s activation under Article 5 in behalf of alliance member Turkey, which, if only technically, has nonetheless come sufficiently under attack to do so.
With air support from American and French carriers in the Mediterranean, the U.S. Air Force at Incirlik and Gulf bases, and the Turkish, Saudi, and Gulf States air forces, in very short order Turkish divisions from the north could link up with Saudi, Jordanian and an Egyptian expeditionary force from the south, stiffened by American, British, and other NATO units where needed, to cut Syria in half. With Kurds and Iraqis closing from the east, this would simultaneously surround ISIS and confine the Syrian regime in a truncated enclave shielded by its Russian patrons.
The primary purpose of such action, however, would not be to defeat ISIS. Though at the moment ISIS is undeniably the most publicity-rich and barbaric of the jihadist movements, in relation to its structure and resources its ambition to unify the Islamic world has—as in the case of bin Laden, Nasser, and the Mahdi of the Sudan—doomed it from the start. While much has been made of its links to other jihadists in Africa and elsewhere, these alliances have little practical effect, being little more than the distant salutes from one group of psychotics to another. That ISIS has survived for years is less a testament to it than an indictment of the quaking West.