That God's foreknowledge plays a role in predestination is as undeniable as predestination itself. As Romans 8:29 tells us, "For whom he did foreknow, he also did predestinate to be conformed to the image of his Son, that he might be the firstborn among many brethren." (Emphasis mine). This does not mean, as some claim, that foreknowledge is synonymous with predestination. The two are distinct, however it is undeniable that foreknowledge and predestination are also inextricably linked.
To the Calvinist, however, this is not the case. I even heard one account of a Calvinist professor reading this verse aloud before telling his class, with zero justification "foreknowledge has nothing to do with predestination". But this very verse explicitly states the opposite.
The fact is, God knows all things. He knows every possible, probable, and actual reality. We actually see this throughout Scripture, with God describing alternate realities that never came to pass. All of this makes perfect sense. It allows God to remain fully Holy, fully coherent, and fully sovereign over all things, including salvation, while nevertheless giving mankind free will.
God, in His infinite wisdom, knows every possible outcome of every possible scenario. He even knows what a free willed being would do in a given circumstance. When Joseph, for example, says "...ye thought evil against me; but God meant it unto good..." (Genesis 50:20), we see God using sins freely committed by sinful human beings to bring about a good end. But directly causing them to commit the sin of selling their own brother into slavery and telling his father he'd been eaten by wild beats is beyond God's abilities (James 1:13-14).
Nevertheless, He knew every Holy step He could take to exploit the sinful nature of human beings in order to bring about this circumstance. He knew what would happen if He had made Joseph the oldest or youngest brother. He knew what would happen if He had never given Joseph any brothers at all. He knew what would happen if Joseph was the one who didn't exist, and He instead selected Reuben. God, being a master of cause and effect, knows exactly how to bring about His desired end without ever once affecting man's free will.
So what of the hardening of hearts thing? For this, we can go to the Bible's own example: Pharaoh. It is undeniable here that God did harden Pharaoh's heart, for He Himself says so (Exodus 4:21). But we must ask the question, how? If God cannot tempt a man to sin (James 1:13-14), it cannot be the case that He not only tempted, but actually forced him to sin against the Lord. But it can be the case that God provided the circumstances that would cause such hardening. In Exodus 8:15, for example, Pharaoh is said to have hardened his own heart. Why? Because he saw that there was respite. In other words, God relented of the disaster He had brought upon Egypt, and so Pharaoh felt safe continuing in his rebellion.
As I have often suggested, God is like a master Chess player. A good Chess player considers not only his own moves, but his opponents' as well. A great Chess player not only considers them, but may even manipulate them. Controlling your opponent's pieces directly is cheating, but controlling them through various strategies is fair game. In much the same way, God is perfectly capable of providing the circumstances required to bring about His purpose, particularly by controlling the practically infinite number of causes to bring about His desired effect.
Whom God foreknew, those He predestined. If you're a Christian reading this, God foreknew you. He knew what circumstances would bring you to faith, when they would do it, and what effect that would have. Had He desired, He could have brought you to faith earlier, later, or even not at all. It would, after all, be perfectly just for Him to condemn you for the sins you committed. Just as He has done with you, so also is He able to do with every individual who has ever lived, from the mighty Solomon to one of Adam and Eve's many children whose names have faded into the obscurity of history.
Over the last 6,000 years, God has been sovereign, man has been free, and God has known how to use both facts perfectly. That His foreknowledge plays a role in Predestination is obvious from Scripture, but He has certainly not just placed a speck of dust out in space and left things to develop while He slept soundly. He has been working to achieve His purpose every second of every day. Therefore, there is no contradiction between the sovereignty of God and the responsibility of man in salvation. Both Calvinism, which overemphasises God's sovereignty, and Arminianism, which overemphasises man's responsibility, are wrong, and contrary to Scripture.