There may not be another topic in Christian theology on which people have stronger opinions than reformed soteriology. The doctrines of Calvinism (though I myself prefer to employ 'doctrines of grace') are found by some to be wondrous truths yet others find them to be sickening. Some say that they necessarily contradict the concept of God as all-loving. To that end, opponents of the doctrines of grace will typically employ John 3:16 as the catch-all proof text that the entire system of reformed soteriology is a sick myth.
The responses from those of the reformed conviction can sometimes be puzzling. I've often heard reformed folks argue that the phrase "For God so loved the world" is more accurately rendered as "For God so loved the believing [elect]". Though the Greek phrase (Οὕτως γὰρ ἠγάπησεν ὁ θεὸς τὸν κόσμον) does not in any way signify that. The object of the love is κόσμον (world). Of the 185 usages of κόσμον in the NT, "world" is almost always the obvious rendering. And so, to say that τὸν κόσμον would be more accurately translated as 'the believing' or 'the elect' is definitely an agenda-driven rendering. However, I myself have reformed convictions. If I was to agree with the non-reformed that this is a terrible argument, what grounds then do I have to say that John 3:16 does not contradict reformed soteriology? Well, this is where we have to dig a bit deeper into what John means by the term "world".
It is evident that John has different conceptions of the term. It is employed heavily in Jesus' high priestly prayer such as when he says "the world has hated them because they are not of the world" (John 17:4) which seems to refer to a particular sort of people that the disciples are not a part of just as Jesus mentioned that he was praying for his disciples and not the world (John 17:9). This world would eventually no longer see Jesus but his disciples still would (John 14:19). Yet the usage of κόσμον in the high priestly prayer elsewhere seems to imply a category that would include the disciples such as Jesus' statement, "so that the world may believe that you have sent me" (John 17:21). And so, it must be clear that John does not use κόσμον in the same sense each time. Then question then is, how does he use it in the 3:16 text? In order to know that, we much appreciate the greater context in which the phrase is found.
Jesus' statement "For God so loved the world..." is presented as an answer to a Pharisee. Nicodemus, who is specified as "a ruler of the Jews", calls Jesus Rabbi and states that he knows Jesus is from God. Jesus responds by telling Nicodemus what must take place before one can "see the kingdom of God". Though Nicodemus, a Pharisee and ruler of the Jews, has a very low and limited view of the kingdom of God, Jesus has a greater one. One that is qualified not by background but by being born again. Jesus elaborates further on this in verses 5-8 saying that whatever is born of the flesh is flesh but a Spirit birth is the qualifier of the kingdom. Nicodemus objects by saying, "How can these things be?". Jesus' response, then, is to present just how radically wide his view of the atonement and the kingdom is. He does this in verses 14-15 by illustrating that partaking of his atoning work and being a part of the kingdom requires faith and faith alone. Greek readers will note the presence of ἵνα in verse 15 which renders "in order that". Thus, the atoning work is directed towards the believing ones.
The kingdom is not a particular ethnic group (by which one could say being born of the flesh would be the qualifier) but rather it is those whom have been transformed by the power of the Spirit. Thus enters the phrase enjoyed by reformed theologians, John's usage of "world" is not all without exception (by which we could say Jesus came for all people) but rather all without distinction (by which we can say there is no social boundary between the people for whom Jesus came). This makes the transition to the "whoever believes" statement at the end of the verse more naturally understood. The kingdom will not be designated by physical birthright but by a spiritual one.
Though this argument would no doubt be enjoyed by reformed folks, one should acknowledge the hesitancy that some have toward reformed theology. We should continue the conversation but do so with dignity and love.