Since that is the clear statement of this sentence, the question is, "What does the Bible mean when it talks about the 'ungodly'?"
The term "ungodly" could simply mean to be unlike God. I suppose all of us, in our better moments, would be willing to confess that is true of us. After all, God is infinite and we are finite. God is all-powerful and we are weak. God is all wise and we are limited. Most of us would be willing to admit that in these ways we are unlike God. But this word is far more severe than that. It not only means that we are unlike God, but in the context of Romans, it also means that down deep inside we are opposed to God.
It's very much like that word, "un-American." When you say that someone is "un-American," you do not mean he lives in Canada or Mexico or Great Britain or Russia, but that the individual supports an ideology opposed to the principles on which the government of the United States was founded. That is the sense in which this word "ungodly" is used in the New Testament.
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Not only are we different from God, but we are opposed to God and to God's will in our lives. Again and again, we say "yes" when God says "no," or we say "no" when God says "yes." For example, we know what we ought to do and yet we do things we know we ought not do. We know that we ought to be kind, but often we are unkind. We know we should be honest, but we are dishonest. We know we should be pure, but often we are impure. We know we should honor our parents, but often we dishonor them. Again and again, by deliberate choice, we have said "no" to God's "yes" and "yes" to God's "no."
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Romans, chapter 3, Paul, quoting the Old Testament, said that God looked down among the children of men and declared that "there is none righteous, no, not one. There is none that seeks after God."
So you see, some time in your life, if you are going to be made right with God, you must admit that you are wrong with Him. If you are ever going to get to heaven, you must admit that you are ungodly -- not because it's the religious thing to do, or out of modesty, but because you recognize that it is true of you.
Every year the American Cancer Society spends thousands of dollars telling us about cancer's seven danger signs. By means of advertisements in the newspaper, motion pictures, through articles in magazines, the American public has become aware of the symptoms of cancer. Yet, every year thousands of Americans die of cancer who have recognized the evidence in their bodies. Because of fear of the physician, or a fear of spending money, or worse, the fear of finding out they have the disease, they do not seek out a doctor. As a result, they die of cancer.
In the same way, if we are ever to gain health with God, we must admit that the Bible speaks the truth about us. We have gone our own way. We have rebelled against what we know we ought to do. We are ungodly people. Of course, since the Bible says this is true of all of us, it places all of us on exactly the same level before God. The prostitute and the preacher, the lawyer and the lawless, the gunman and the governor, the sophisticate and the savage, the doctor and the dunce -- all stand before God as ungodly men and women in desperate need of God's salvation.