What is the "world" God so loved that he gave His life for it " (true agape love )
kosmos {kos'-mos}
1) an apt and harmonious arrangement or constitution, order, government
2) ornament, decoration, adornment, i.e. the arrangement of the stars,
'the heavenly hosts', as the ornament of the heavens. 1 Pet. 3:3 3) the world, the universe
4) the circle of the earth, the earth
5) the inhabitants of the earth, men, the human family
6) the ungodly multitude; the whole mass of men alienated from God, and therefore hostile to the cause of Christ
7) world affairs, the aggregate of things earthly
a) the whole circle of earthly goods, endowments riches, advantages, pleasures, etc, which although hollow and frail and fleeting, stir desire, seduce from God and are obstacles to the cause of Christ
8) any aggregate or general collection of particulars of any sort a) the Gentiles as contrasted to the Jews (Rom. 11:12 etc) b) of believers only, John 1:29; 3:16; 3:17; 6:33; 12:47 1 Cor. 4:9; 2 Cor.5:19
When the jews spoke of the "world" they were usually speaking of The gentile nations around them. So the words that the Messiah was for "the world" was truly shocking to them. this was after all the JEWISH messiah they were looking for.
So the question is NOT the meaning of the love of Christ for the elect, that surely is agape love, it is for whom did Christ hold that love? . If Christ holds agape love for those he will condemn for an eternity of fire one might really be concerned what kind of love that is.
If Christ had agape love for all men and died for all men then God is a liar and he is demanding a double payment for what was purchased at the cross.
You make the cross of questionable effect
When the jews spoke of the "world" they were usually speaking of The gentile nations around them.
That's certainly not true. When they spoke of "the nations," that is true, but "world" was not used to show the non-Jew. kosmox was used in the LXX to represent "universe," "inhabited earth," and "humanity," 3 TDNT 867 at 882.
If Christ holds agape love for those he will condemn for an eternity of fire one might really be concerned what kind of love that is.
It might be theologically inconvenient, and not very neat, but it is what the text says, and as Protestants, we're supposed to subjugate our theology to the text, not vice versa. kosmos is never used to describe just the elect in any other verse; why would it be here?