Arminians will often say that God wishes to save everyone. He has done, and is doing, all that He can to save them because of this, but it is their free will that gets in the way. God has so chosen to allow a man to choose whatever he wishes without restraining him one way or another.
However, we don't choose in a vacuum. We choose what we have been convinced is the most desirable and pleasant good. We then love and desire what we have been persuaded to love and desire.
Saying that God does all that He can to save everyone, but will not interfere with our free will misses the point. Leave the argument about the will behind for a moment. Let's just assume we have a completely unshackled free will. As I said before, we do what we are influenced to believe is the most desirable in the moment. These influences come from our environment, the devil, and our sinful tendencies. What this says, therefore, is that these influences, whether individually or collectively as a group working together are more powerful and have a greater ability to persuade us of the desirable good of a deception (i.e., they persuade us to love and choose self exaltation) than God has to convince of the desirable good of a truth (i.e., to persuade us to love Him).
In other words, forget the debate about free will. Let's talk about whether God has the ability to convince a person to love and desire Him above all other things from the outside, without affecting the will from the inside. Does God have the ability to influence a person and sway him to look upon Him as the greatest and most desirable good without changing his will?
If He does, and He wants everyone saved, why doesn't He do it?
What you have in Arminian theology is a God who is either not as powerful to convince as lesser influences, or a God who isn't doing all that he can to save people. Which is it?
Either way, you end up with a contradiction in Arminian theology that claims that God is more powerful than all other existing things (it does not matter whether he is almighty here, only that he is more powerful than everything else individually or collectively) and desires all people to be saved; but does not save everyone. Either he isn't trying his hardest or he doesn't have the ability to persuade men to see him as the most desirable good because the world, the devil, and the self (i.e., created things) have a greater ability than God has to convince others that evil and self rule is the most desirable good of the moment.
But if this is all true then not only is salvation not something one should pray about as an Arminian, since God is either already doing all that he can to save people and there is nothing else he can do, and/or he is not capable of answering that prayer, since he can only try to save people, which he is already doing, but can't, but he cannot ask God to persuade anyone of anything where our nature and nurture have a foothold (which is pretty much everything under the sun).
In fact, in this scheme, no one should be saved in the end, since God does not have a greater ability to convince anyone against those entities that influence all people toward the love of self worship.
Furthermore, he doesn't have the ability to affect the will from the outside then, since those other entities have greater ability to do so than he does. If that is true, then praying for someone to be filled with true knowledge and understanding when the adverse enemies of mankind effectively prevent that person from being filled with true knowledge and understanding would be worthless. Praying that he be strengthened with all power that would enable him to be steadfast in his faith and patience becomes just a wish, not something God can actually do. The influence of those other forces are too powerful. Hence, prayer becomes like one poor buddy saying to another poor buddy, "I sure wish I could fly to Mars." The friend could simply nod in agreement, and maybe go to work to try to give him that wish, but in the end, he is simply powerless to accomplish it. It's just the expression of a nice fantasy between friends, but nothing more.
So we are really left with one question, If this is the case, that God cannot effectually convince a person that He is the most desirable good against the persuasion of the world, the devil, and the person's sinful inclinations that convince him otherwise without altering the person's will from the inside, but only seeking to convince from without, then why is anyone saved at all?
This question must be answered in one of two ways by the Arminian: (1) It's a mystery. In other words, it's a complete and utter contradiction and destroys our entire system, but we don't want to be Calvinists so we're going to appeal to mystery. (2) People are saved because there was something better about them. They were able to use what God gave them (prevenient grace, knowledge, revelation, etc.) to convince themselves that God is the greatest and most desirable good, and hence, they choose Him. In other words, people are saved because they're not only stronger than all of the influences that would have deceived them, but because they are stronger than God as well, as they were able to overcome the deception of those influences where God was not able to do so. Surely, they were given power by God to overcome, but the point is that God was not able to use his supposedly superior power to do what they did with less power than He has. Hence, their salvation is a result of their being better than those who are damned and more powerful than God Himself.
Now, no Arminian is going to say the second, so it's either going to be the first or they're just going to deny that there is a contradiction here. But please note. The free will debate apologetic is ineffective here. I'm speaking about what God can and cannot do without affecting a man's will from the inside.
If, however, an Arminian wishes to acknowledge the contradiction and remain orthodox, he must then admit that God is all-powerful, and therefore, capable of convincing/persuading all of mankind that He is the most desirable good (i.e., to persuade them to love Him), but therefore must not be doing all that He can to save everyone, since not everyone (and indeed most people so it seems) are not saved.
He can then consistently pray for God to do more than what He is doing, since He is not doing everything He can; but if He really wanted everyone saved, why wasn't He doing everything He could beforehand?
What a tangled web we weave when first we exalt philosophies that deceive.