Since the stone-ages of Cornelius Van Til until modern times, many have claimed that Karl Barth's theology necessarily concludes universalism or else it is incoherent, as recently exemplified by Oliver Crisp's cavalier statement in his Deviant Calvinism, "that the scope of human salvation envisioned in the theology of Karl Barth either is a species of universalism or comprises several distinct, incompatible strands of doctrine that he does not finally resolve. " Contrary to these Barthian despisers, I do not understand how such a statement maybe justified, considering Barth's plain and understandable explanation at the end of the Church Dogmatics, Vol IV/3.1, especially since this quotation (quoted a length below) is consistent with everything Barth had written since the infancy of the Church Dogmatics. I'm tempted to respond with an equally cavalier (and humorous) statement that this animus towards Barth may be caused by a demon that has gone out of Van Til, wandering the pages of the Church Dogmatics, seeking an affirmation of universalism or evidence of incoherence in the Barth's Doctrine of Election (CD II/2) or elsewhere, but it finds none, so it takes up dwelling in these Barthian Despisers!
Was Karl Barth a Universalist? Nein!
Was Karl Barth a Universalist? The short answer is No. Barth did not self-identify as a universalist, and this is an important point that is frequently ignored. Barth once told a universalist preacher seeking consolation that "I do not believe in universalism, but I do believe in Jesus Christ, reconciler of all." Barth thought that it was nonsensical to believe in "universalism" because our only object of belief is in Jesus Christ, but does not mean that he was merely equivocating with universalism (i.e. affirming it in a hidden way) because Barth explicitly rejected Apocatastasis (or Universal Reconciliation).
The early Church Father, Origen (c. 185—254 AD) developed a form of Universalism known as Apocatastasis that teaches that God must necessarily save all people at the end of the age, and he based this doctrine on the Greek word apokatastasis from Acts 3:21 (often translated as the "restoration of all things" or "universal reconciliation"). Three centuries after Origen had died, the Fifth Ecumenical Council (553 AD) issued Fifteen Anathamas Against Origen for his doctrine of Apocatastasis, and henceforth, Origen had been branded a heretic. Some Origen scholars believe that these anathemas do not apply to what Origen actually taught, and suggest that his posthumous condemnation was unfair and inaccurate, especially since Origen lived long before any ecumenical council ever commenced.
Additionally, Apocatastasis was not rejected because it taught Universal Reconciliation, it was rejected because it taught the Platonic pre-existence of soul, as exemplified by the first anathema: "#1. If anyone asserts the fabulous pre-existence of souls, and shall assert the monstrous restoration which follows from it: let him be anathema." Other theologians who affirmed Universal Reconciliation but not the pre-existence of the souls were never condemned for their universalism; the most notable examples are Jerome (347—420 AD) and St. Gregory of Nyssa (335—395 AD). So the only form of universalism that was officially declared heretical by the ecumenical councils is the specific form of Apocatastasis that affirms both Universal Reconciliation and the pre-existence of the human soul. There are many forms of Universalism that do not fall within the rubric of Apocatastasis, especially in the Reformed Tradition, such as Amyrauldism and various hopeful universalists that have remained orthodox throughout their lifetime.
Why did Barth reject Universalism?
In the following quotation from CD IV/3.1, Barth explains why he rejects Universal Reconciliation (or Apocatastasis). His reasons are as follows:
1) the teaching that god necessarily must save all individuals undermines the biblical warnings about sin, and nullifies the biblical threats towards sin;
2) God is not obligated to save any sinner, and it is only by his grace that anyone is saved;
3) we do not have assurance or a promise that all people will be saved at the end of the age, so it may not be asserted as such;
4) although universalism may not be affirm, the bible provides us hope that all people will be finally be delivered until salvation at the end of the age.
Karl Barth's Rejection of Universalism (Apocatastasis)
At the end of the Church Dogmatics: The Doctrine of Reconciliation, Vol. IV/3.1 §70, Barth explains his rejection of universalism as follows:
First, if this is not the case, it can only be a matter of the unexpected work of grace and its revelation on which we cannot count but for which we can only hope as an undeserved and inconceivable overflowing of the significant, operation and outreach of the reality of God and man in Jesus Christ. To the man who persistently tries to change the truth into untruth, God does not owe eternal patience and therefore deliverance any more than He does those provisional manifestations. We should be denying or disarming that evil attempt and our own participation in it if, in relation to ourselves or others or all men, we were to permit ourselves to postulate a withdrawal of that threat and in this sense to expect or maintain an apokatastasis or universal reconciliation as the goal and end of all things. No such postulate can be made even though we appeal to the cross and resurrection of Jesus Christ. Even though theological consistency might seem to lead our thoughts and utterances most clearly in this direction, we must not arrogate to ourselves that which can be given and received only as a free gift.
Secondly, there is no good reason why we should forbid ourselves, or be forbidden, openness to the possibility that in the reality of God and man in Jesus Christ there is contained much more than we might expect and therefore the supremely unexpected withdrawal of that final threat, i.e., that in the truth of this reality there might be contained the superabundant promise of the final deliverance of all men. To be more explicit, there is no good reason why we should not be open to this possibility. If for a moment we accept the unfalsified truth of the reality which even now so forcefully limits the perverted human situation, does it not point plainly in the direction of the work of a truly eternal divine patience and deliverance and therefore of an apokatastasis or universal reconciliation? If we are certainly forbidden to count on this as though we had a claim to it, as though it were not supremely the work of God to which man can have no possible claim, we are surely commanded the more definitely to hope and pray for it as we may do already on this side of this final possibility, i.e., to hope and pray cautiously and yet distinctly that, in spite of everything which may seem quite conclusively to proclaim the opposite, His compassion should not fail, and that in accordance with His mercy which is "new every morning" He "will not cast off for ever" (Lam 3:22f,31). 
Barth did not identify as a Universalist, and he rejected universalism as an object of belief, and specifically rejected the doctrine of Apocatastasis that taught that God must necessarily save all people in the end of the age, because this nullifies the threat of sin and God's grace in election. However, Barth had hope that God may freely choose to deliver all people in accordance with His mercy at the end of the age.
[^Image Background] By Gustave Doré - Alighieri, Dante; Cary, Henry Francis (ed) (1892) "Canto XXXI" in The Divine Comedy by Dante, Illustrated, Complete, London, Paris & Melbourne: Cassell & Company Retrieved on 13 July 2009., Public Domain, https://commons.wikimedia.org/w/index.php?curid=93403
[^1] Crisp, Oliver. Deviant Calvinism: Broadening Reformed Theology. MN: Fortress, 2014. 155. Print.
[^2]Busch, Eberhard. Karl Barth: His Life from Letters and Autobiographical Texts. Trans. John Bowden. Eugene, OR: Wipf & Stock, 2015. 394. Print.
[^3] "NPNF2-14. The Seven Ecumenical Councils." Christian Classics Ethereal Library. Christian Classics Ethereal Library, n.d. Web. 18 Aug. 2016. <http://www.ccel.org/ccel/schaff/npnf214.xii.ix.html>.
[^4] Barth, Karl. Church Dogmatics: The Doctrine of Reconciliation, Vol. IV/3.2. Vol. 28. London: T & T Clark, 2010. 105-6. Print. Study Edition. [Original ET is CD IV.3.1, pages 477-8]