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Him hath God exalted with his right hand to be a Prince and a Saviour, for to give repentance to Israel, and forgiveness of sins.
Arminian view: This shows that God gives opportunities for repentance on the basis of Christ's work. Otherwise, this means that all of Israel must have been saved.
Calvinist view: The point here is that both repentance and forgiveness are gifts, and if we did have the ability to repent on our own, why would that need to be a gift?
Our finding here is that both sides of this issue are in error -- because they have understood "forgiveness" and "repentance" in terms of the 21st century, Western understanding of those words, and also misunderstood this passage in terms of personal salvation.
Malina and Rohrbaugh note in Social-Science Commentary on the Synoptic Gospels (303-4) that within the context of the ancient world, the "introspective, guilt-oriented outlook of industrialized societies did not exist...." "Guilt" referred not to personal feelings, but to liability and responsibility for an offense. "Forgiveness" meant, in turn, "being divinely restored to one's position" and being "freed from fear of loss at the hands of God." Forgiveness was restoration to a previous position and not an emotion.
Thus, both Arminian and Calvinist views are misplaced, for the issue here is not personal salvation, but the restoration of corporate Israel and freedom from the bondage of their "exile" at the hands of the Romans. (Note that the same word is used in Luke 4:18, "The Spirit of the Lord is upon me, because he hath anointed me to preach the gospel to the poor; he hath sent me to heal the brokenhearted, to preach deliverance to the captives...")
God exalted Jesus "to be a Prince and a Saviour" -- both a ruler and a deliverer -- "for to give repentance to Israel" -- since there is no "guilt" in the modern sense in the ancient mind, the "repentance" here is a turning around, or reformation (the word here is used not just of repentance from evil, but also of any change of plans), of corporate Israel, with Christ at its head -- "and forgiveness of sins" -- in other words, a restoration of corporate Israel to its position.
As Christ came into Jerusalem on a donkey, thereby fulfilling the prophecy of the coming Messianic king and expresses an intention (which was rejected), so this passage speaks of his intent to restore corporate Israel (which was also rejected) through a change in plan offered by God. In other words, God is "repenting" as in changing His offer in the time-space continuum (not in the sense of guilt or sorrow -- note our reply to that typical Skeptical objection in the link below).
Neither Calvinists nor Arminians should be making use of this passage.