Baloney, Cornelius is a perfect example of an unregenerate, unsaved man who feared God, prayed always, and did "righteous works" according to Peter himself.
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Was Cornelius saved? NO, because the angel told him to send for Peter whereby he would hear words and be saved. (Acts 11:13-14)
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It is indisputable, Cornelius was not saved, and yet he feared God, and did righteous works.
Did Cornelius have the Holy Spirit before Peter preached to him? NO.
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So there is absolutely evidence that unregenerate man can seek God in the scriptures. It may be exceptional, but it is there.
You are attempting to prove a point that is not proved by the Scriptures you used.
Cornelius is just the same as many in whom God has selected for salvation. He has already placed in their heart a new will to seek the things of God. They are under Godly conviction, and merely needing to be given the intellectual insight in which their heart has already been prepared by God.
If anything, your use of Cornelius shows all the more how God appoints a person to salvation and moves a believer into the relationship to bring the Word.
It shows all the more the Calvinist thinking is basically correct.
However, YOUR view obliges that this be done for all humankind - or it is unfair.
That is just not the historical way God has conducted business, irregardless of your own desires.
One other point:
You desire to use the evidence of the Holy Spirit falling upon the house of Cornelius as it did in Pentecost as proof. However, Peter accepted this as a sign (not as proof of salvation) to show that for certain the Gentiles were to be saved just as the Jews at Pentecost. Later, this would prove vital when Paul was sent to Jerusalem for discussions as to how Jewish the Gentiles had to be to be saved. Peter was the one who made the final decision, and (in my opinion) it was because he witnessed first hand the events at Cornelius' home.
Another example of an unregenerate man who sought God is the Philipian jailer;
Acts 16:29 Then he called for a light, and sprang in, and came trembling, and fell down before Paul and Silas,
30 And brought them out, and said, Sirs, what must I do to be saved?
31 And they said, Believe on the Lord Jesus Christ, and thou shalt be saved, and thy house.
The Philipian jailer could not have been spiritually alive when he burst in and asked Paul and Silas what he must do to be saved, because he had not yet believed on Jesus. And until you believe on Jesus you are DEAD in trespasses and sins, yet he was aware that he was lost, and he sincerely desired to be saved.
And Paul and Silas's answer tells us he was not regenerated, because they told him he must believe on Jesus to be "saved".
So, here are TWO examples of unregenerate men who sincerely sought God in the scriptures.
Do you not see the intention of God that a specific jailor was in charge when Paul and Silas were thrown into the prison?
You make the mistake of assuming that a person is "dead" until they believe on Jesus.
Dead people do not have the ability to believe.
Unless the person is made alive (as was Cornelius, as was the jailor) they remain unresponsive to the claim of God on the life.
Expressions of belief come from a heart that has already believed.
The heart that has already believed is a heart that has already been awakened not only to the need, but has actually been changed. It is made alive in Christ.
BOTH Cornelius and the Jailor's show evidence their heart was already changed, and that the intellect had to be "upgraded."
Do not neglect: "For with the heart man believes ... with the mouth ..."
Just for those who mistake this verse as meaning someone has to proclaim belief to receive belief, I suppose I should clarify one word of the verse.
The "unto" of Roman's 10 is not a statement of "this has to be done in order for this to happen." One does not have to have the mouth moving in order that salvation be attained.
The "unto" is a word of response (movement) to a condition (belief).
The condition of the heart causes the response of the mouth.
Just as at the condition of birth causes the new born to cry out (and all with ears to hear do hear) the birth of the new believer causes them to cry out.
Those who were schooled in the thinking that god/God (as reflected in idol worship, or OT sacrifices in which something had to be done to appease a god or God) would most certainly cry out "What must I DO to be saved." For intellectually that is all they know. And the instructor must guide them into understanding.
The crying out is an expression that salvation is already at work in the heart.