What is Paul trying to prove in Romans 1:18-3:20? Here are a few, non-exhaustive, suggestions:
1) He is trying to close “every mouth” and demonstrate that “all the world” is “accountable to God,” and guilty before Him. It is sometimes said in recent Pauline scholarship that this is not a central thrust in Rom 1-3, but that seems to be belied by the concluding statements of 3:19-20. True, Paul could assume from the OT the universality of human sin. However redundant, it does seem to be a crucial purpose for Paul to establish what should be an axiom. But the rhetoric of the passage indicates that his main purpose in this is to convict the JEWS of guilt before God. That is the main theme of chapter 2, where he charges that the Jews who judge the Gentile world are not for that reason exempt from God’s indictment against the “unrighteousness and ungodliness of man.” Jews have failed in their performance of the law, and therefore before the righteous Judge, they stand guilty. Gentiles who have been circumcised by the Spirit show up the sinfulness of Israel by fulfilling the law that they do not possess by nature. That, as much as anything, is a bitter indictment of the Jews who possess and boast in their possession of the “embodiment of knowledge and of the truth” in the Torah (2:20).
2) But this is NOT the largest purpose that Paul has in this passage. The indictment of all humanity, Jews and Gentiles, as sinners serves the larger purpose of expounding on the gospel of God’s righteousness. The sinfulness of Jews, of Israel, after all, is not merely a matter of “universality of sin.” Israel was supposed to be Yahweh’s SOLUTION to the problem of sin, and therefore Israel’s failure puts a question mark over Yahweh’s program. Is God’s purpose in history going to fail? Is sin going to be dealt with? Is Babel going to be reversed through the seed of Abraham?
3) #2, in turn, raises questions of theology proper. If God’s purpose for the Gentiles and the world through Israel has come to nothing, what does that say about God? Can He be trusted? Is He really committed to doing right in the world, to bringing about righteousness, justice, and peace in His creation? Or is He the kind of God who makes a world and then leaves it to its own sorry devices? Does the Judge of all the earth in fact do right? That is the kind of question Paul is dealing with in 3:1-8.
4) Finally, Paul is setting up for 3:25-26 by expounding on the crucial problem of “wrath.” One could say that the whole of 1:18-3:20 is under the shadow of the “wrath of God revealed from heaven,” for the entire passage has to do with the “ungodliness and unrighteousness of men” against whom God is wrathful. Wrath, further, is described in 1:18-32 as the CAUSE of the disorder and disorientation of human society. If justice and peace are going to reign, then something has to be done about wrath, which means, of course, that something has to be done about sin. What can be done? Is Torah the solution to wrath? No: because those who received Torah did not keep it, and because no flesh can be justified by the works of the law. Is Israel the solution to wrath? No: Israel is under sin (and therefore under wrath) with the Gentiles. Might God deal with wrath by just giving up, ignoring sin? No: Because that would leave His righteousness in question. Only the God of wrath can deal with wrath. And He has, by displaying Jesus as a “propitiation” in blood.