Election and the divine will
Election, hardening and the vessels of wrath beside God's will that all be saved (1 Timothy 2:4; 2 Peter 3:9; Matthew 23:37) — Augustine, Pelagius, and the Reformed and Arminian traditions after them.
Romans 9:14-24
14What shall we say then? Is there unrighteousness with God? May it never be!
15For he said to Moses, “I will have mercy on whom I have mercy, and I will have compassion on whom I have compassion.”
16So then it is not of him who wills, nor of him who runs, but of God who has mercy.
17For the Scripture says to Pharaoh, “For this very purpose I caused you to be raised up, that I might show in you my power, and that my name might be proclaimed in all the earth.”
18So then, he has mercy on whom he desires, and he hardens whom he desires.
19You will say then to me, “Why does he still find fault? For who withstands his will?”
20But indeed, O man, who are you to reply against God? Will the thing formed ask him who formed it, “Why did you make me like this?”
21Or hasn’t the potter a right over the clay, from the same lump to make one part a vessel for honor, and another for dishonor?
22What if God, willing to show his wrath and to make his power known, endured with much patience vessels of wrath prepared for destruction,
23and that he might make known the riches of his glory on vessels of mercy, which he prepared beforehand for glory,
24us, whom he also called, not from the Jews only, but also from the Gentiles?
1 Timothy 2:3-4
3For this is good and acceptable in the sight of God our Savior,
4who desires all people to be saved and come to full knowledge of the truth.
What each tradition says
Rabbinic / Midrashic
The Pharisees, fully aware that predestination precludes free will, adopted a middle view: man is subject to predestination in material life but completely free in spiritual life. R. Akiba expressed this: “All is foreseen, yet freedom is granted.” R. Ḥanina similarly said, “All is in the power of God, except the fear of God.”
The Jewish Encyclopedia (Funk & Wagnalls, 1905); article signed 'K. I. Br.', The Jewish Encyclopedia, vol. 10, s.v. 'Predestination' — Section on the Pharisees' middle view and R. Akiba (Pirke Abot iii. 15)
Patristic
Augustine says Paul explained why God does not teach all men: because God, willing to show His wrath and exhibit His power, endured with much patience the vessels of wrath perfected for destruction. He did so to make known the riches of His glory on the vessels of mercy He prepared for glory.
Augustine of Hippo (354-430), A Treatise on the Predestination of the Saints, Book I, in NPNF ser. 1, vol. 5: St. Augustin: Anti-Pelagian Writings (ed. Philip Schaff, 1887) — Book I, chapter 14, on Romans 9:18-22 (the vessels of wrath and of mercy)
Patristic
Chrysostom says this is not said to do away with free-will, but to show how far we ought to obey God. In calling God to account, we should be as little disposed to it as the clay is.
John Chrysostom (c. 349-407), Homilies on the Epistle to the Romans, Homily XVI (trans. J. B. Morris, rev. W. H. Simcox; Oxford 1876), in NPNF ser. 1, vol. 11 (ed. Philip Schaff, 1889) — Homily XVI, on Romans 9:20-21 (the potter and the clay)
Catholic
Thomas Aquinas holds that God antecedently wills all men to be saved. Consequently, God wills some to be damned, as His justice exacts.
Thomas Aquinas (c. 1225-1274), Summa Theologiae, Prima Pars (I), q. 19 (The Will of God) — I q. 19, art. 6, reply to objection 1 (on 1 Timothy 2:4)
Orthodox
The Synod of Jerusalem teaches that God eternally predestined the chosen to glory and the rejected to condemnation, but not so as to justify one and condemn the other without cause. That would contradict God's nature as common Father of all, no respecter of persons, who would have all men saved and come to the knowledge of the truth.
Synod of Jerusalem (1672), under Patriarch Dositheus; trans. J. N. W. B. Robertson (1899), The Acts and Decrees of the Synod of Jerusalem (London, 1899), 'Confession of Dositheus' — Decree III (on predestination)
Reformed / Protestant
Calvin says that with regard to the elect, God cannot be charged with unrighteousness, because he favors them with mercy according to his good pleasure. Yet the flesh murmurs, since it cannot concede God the right to show favor to one and not another unless the cause is made evident.
John Calvin (1509-1564), Commentaries on the Epistle of Paul the Apostle to the Romans (trans. John Owen, Calvin Translation Society, 1849) — Commentary on Romans 9:15
Evangelical
Spurgeon states that election is absolute: it does not depend on what we are. He cites the text, “God hath from the beginning chosen us unto salvation,” and contrasts this with opponents who say God chooses people because they are good or because of works they have done.
Charles Haddon Spurgeon (1834-1892), Election, Sermon No. 41-42, New Park Street Pulpit, vol. 1 (preached September 2, 1855) — Section II, 'election is absolute'
Evangelical
Wesley takes “all men” to mean not a part only, much less the smallest part. God seriously wills all to be saved eternally, and in order thereto to come; they are not compelled.
John Wesley (1703-1791), Explanatory Notes upon the New Testament (1755); note on 1 Timothy 2:4 — Note on 1 Tim. 2:4 (with the note on 2:3)