This is one of those questions that comes up often in theological circles: is salvation for all persons, or only for an elect subset of all persons? Augustinians and Calvinists and other Reformed types typically affirm the latter position, whereas Arminians and the Orthodox and others typically affirm the former.
I want to consider this question informed by one particularly important passage in the Old Testament. I am referring specifically to one of the servant songs in Second Isaiah:
"It is too light a thing that you should be my servant
to raise up the tribes of Jacob
and to restore the survivors of Israel;
I will give you as a light to the nations,
that my salvation may reach to the end of the earth" (Isa 49.6).
Here we see that the servant of God was chosen in order to bring Jacob back to [God], and that Israel might be gathered to him (v. 5). The servant's work, in other words, is the restoration of the chosen people of God to their creator and redeemer. Yet God tells him that this work, the restoration only of God's chosen people, is yet too light a thing. On the contrary, his work is much larger than this: now he must be a light to the nations, because God's salvation has to reach the very ends of the earth.
What is important about this? God's salvation is now no longer explicitly being limited to the people of God, or in other words, God's chosen elect. Now, those persons who are considered non-elect are objects of God's salvific will. Those persons who were normally outside of God's purview and excluded from the community of the elect are now being sent a light, so that they too might know the salvation of God.
Of course, we know that Christ is the true Servant of God. And his salvific work was for the whole world, not merely for the chosen people of Israel (cf. 1 John 2.2). Yet might not there still be a group of the chosen elect, for whom alone he is concerned? This is certainly logically possible and perhaps compatible with the strict letter of what is written. But it seems to me that the message and general tenor of this passage doesn't allow that. For within the Isaian context, Israel is the chosen and the nations are not. God's message to the servant is that his salvation now is aimed at the whole world, and not merely the company of the chosen. In the same way, I think, we should understand Christ's salvific work as aimed at the salvation of all persons, not merely a group of the elect but for all people.
Now this doesn't, by itself, entail that all people will be saved. But it is important to affirm that God wants everyone to be saved and to come to the knowledge of the truth (1 Tim 2.4). God's intention and desire is that everyone be saved, not only some who are chosen to this end. In a way, we can say that God chooses everyone to be saved.