How can Paul quote from Psalms and say things such as, ‘There is none that seeks after God’ (see Romans 3:11) when we know that some do, such as Cornelius in Acts?

This text is from a transcript of a talk by David Gooding, entitled ‘God’s Power for Salvation’ (2005).

That is very perceptive of you, because I lay some emphasis on the fact that God recognizes those that seek him, even though they are not yet Christians. You mention Cornelius who sought after God (Acts 10). You could also quote Lydia, if you like, who Paul found at a prayer meeting in Philippi when he went there (Acts 16:11–15). She was at the hour of prayer. She was a pagan woman but a woman who sought God and joined with the Jewish women, perhaps, in Philippi. There weren't enough of them to make a synagogue, but they had a regular prayer meeting, and they sought for God. How do we reconcile that with the psalmist who is quoted by Paul, saying, 'There is none that seeks after God?'

The best explanation that I know of, the best way of understanding it is, of course, what is called by the theologians 'total depravity.' It can be a doubtful doctrine anyway, but it is not meant to infer that we are every bit totally corrupt but that no one part of us is perfect. So in the absolute sense there is none that seeks after God. If you want evidence of that, ask the most holy man or woman you know. There goes a believer, and you say that she is a holy woman. She seeks after God. And you ask her, 'My dear, good, lady, do you mind my asking you? You strike me as very holy; you seek the Lord daily. Do you seek him with one hundred percent of your heart?' What do you suppose she would answer?

'Oh, I come far short of it. There's much selfishness still within me.' And so forth.

In the absolute sense, there is none that seeks after God one hundred percent as we ought to, and Paul is stating in Romans 3 the extreme, the absolute. Judged by God's law, we all come short. That doesn't mean that no one seeks God at all. And, rightly, you point out Cornelius and others who, to the best they know how, seek after God.