Is there a way to prove that God’s foreknowledge is incompatible with genuine human freedom?

Logic requires definitions. In particular, does "omniscient" mean "knows everything" or "is capable of knowing everything"?

If you want the former, you might also wonder why, if God is omnipotent, he hasn't done everything. The obvious answer is that omnipotence must mean the latter kind of definition.

Similarly, God's ability to know everything doesn't mean that he does know everything.

I live near Maple Hills Creek. I know that its water will flow into the Grand River, and thence into Lake Ontario, and eventually into the Atlantic Ocean.

I know this in general, but am incapable of knowing it with absolute certainty (perhaps Yellowstone will explode tonight and mess everything up), nor am I capable of knowing whether any specific individual water molecule will evaporate before it reaches the ocean.

But most of the time, knowing the big picture is enough, the details usually don't matter.

God knows with absolute certainty how history will flow. But, with a few exceptions, he has no need to know how an individual person will behave within that flow of history. Even with specific individuals that are predestined to play a role in his plan, it's sufficient to occasionally nudge them back onto the right track without knowing every little detail of what they are going to do.

God might be capable of knowing everything, but why would he bother?

As another analogy, suppose you record a video of a child playing. Later, when you view it, you will know for sure what will happen, but that in no way means that the child didn't use free will to decide what to do; you simply know what that decision will be.

Open Theists say that God's knowledge of the future is incompatible with libertarian notion of human freedom. Their solution is to limit God's omniscience so God does not know the future.

Their position is discussed in the IEP entry Open Theism, as well as in a 2001 article by theologian John Frame Open Theism and Divine Foreknowledge. Wikipedia also has a good article on Open Theism which includes:

• Comparison with Reformed theism
• Varieties of open theists
• A large list of scholars since 1980
• A nice table of representative books by both sides

In this scheme, your 2 axiomatic statements can remain true (highlighted in the quote from John Frame's article below) and you will get your libertarian style free will:

A free act in the libertarian sense⁸ is an act that is utterly uncaused, undetermined. It is not caused by God, nor by anything in creation, nor even by the desires and dispositions of the one who performs the act. Such causes may “influence” or “incline” us to a certain choice, but they never determine a choice, if that choice is free in the libertarian sense. At the moment of choice, on this view, we are always equally able to choose or not to choose a particular alternative.⁹ For this reason, libertarian freedom is sometimes called “liberty of indifference,” for up to the very moment of choice nothing is settled; the will is indifferent.¹⁰

Now if people are free in the libertarian sense, then human decisions are radically unpredictable. Even God cannot know them in advance. If in 1930 God knew that I would be writing this article in 2000, then I would not be writing it freely. I could not avoid writing it. So if my writing is a free choice in the libertarian sense, even God cannot have been certain of it in advance. Libertarian freedom excludes the classical view of God’s foreknowledge.¹¹

On this view, the future is of such a nature that it cannot be known exhaustively. So open theists claim that on their view God is indeed omniscient, in the sense that he knows everything that can be known. That he lacks exhaustive knowledge of the future is no more of a limitation than his inability to make a square circle. Just as his omnipotence enables him to do everything that can be done, so his omniscience enables him to know everything that can be known. That includes knowledge of the past and present, but not the future, so open theists name their view presentism.¹²

For open theists, therefore, libertarian freedom is a fundamental premise, a standard by which all other theological statements are judged. Typically, open theists do not argue the case (such as there is) for libertarian freedom; rather, they assume it. ¹³ It is their presupposition. So God cannot have exhaustive knowledge of the future. Pinnock says,

However, omniscience need not mean exhaustive foreknowledge of all future events. If that were its meaning, the future would be fixed and determined, much as is the past. Total knowledge of the future would imply a fixity of events. Nothing in the future would need to be decided. It also would imply that human freedom is an illusion, that we make no difference and are not responsible. ¹⁴

He is saying that God cannot know the future exhaustively, because if he did we would not have libertarian freedom.

All 3 articles discuss scriptural support cited by their proponents, but

• there is big cost: i.e. no guarantee that everything will work out as God wants in the end, God's glory is diminished (see section 4 of the IEP article)
• there is incoherency when taking the full scriptural evidence into account (see the last 2 sections of John Frame's article: 'Divine Ignorance in Scripture?' and 'God’s Exhaustive Knowledge of the Future').