"First of all, then, I urge that supplications, prayers, intercessions, and thanksgivings be made for all people, for kings and all who are in high positions, that we may lead a peaceful and quiet life, godly and dignified in every way. This is good, and it is pleasing in the sight of God our Savior, Who desires all people to be saved and to come to the knowledge of the truth. For there is one God, and there is one mediator between God and men, the man Christ Jesus, who gave Himself as a ransom for all, which is the testimony given at the proper time. For this I was appointed a preacher and an apostle (I am telling the truth, I am not lying), a teacher of the Gentiles in faith and truth."
This passage frequently comes up in debates about the extent of the atonement. With all Calvinists, I hold that Jesus died for all of the elect, and only the elect, fully redeeming them from the just judgment for their sins. In opposition to this view, Arminians hold that Christ died for every person in the world, throughout history, making their salvation possible, but not certain. Arminians claim verse four here, "Who desires all people to be saved," as supporting their view. But does it really?
First, I have just one simple, logical question for the Arminian: if God desires something, anything, who can refuse Him? Wouldn't your interpretation of this passage logically lead to the doctrine of universal salvation?
But to particulars: Let's look at the context. Verse one ends with that same phrase, "all people," but the sentence continues in verse two, referring to kings and others in authority. Then in verse seven, Paul refers to himself as called to be a teacher of the Gentiles. Thus, in context, "all people" here cannot refer to "every individual without distinction," but rather to "individuals of every class or ethnicity." God desires, effectually, for the Gospel to impact every level of society, and every nation. And in this desire, He attains that goal! The Apostle John envisions the success of the Gospel in this very fashion (Revelation 7:9): "After this I looked, and behold, a great multitude that no one could number, from every nation, from all tribes and peoples and languages, standing before the throne and before the Lamb, clothed in white robes, with palm branches in their hands."
I can't speak for anyone else, but I prefer this vision of the success of the Gospel, over the Arminian view of potential redemption!