Non-calvinists believe in election, we simply see it differently than you. I believe that God can see through all time and those that he sees will believe are the elect.
Calvinists often use several verses from Romans 9 to prove their doctrine, but ignore the whole chapter which is contrasting faith to works. It is true that God said he will have mercy on whom he will show mercy and show wrath on whom he chooses, but if you read all the verses before this you will see he is speaking of showing mercy to those who answer the calling of faith and rejecting those who try to earn salvation by works.
Rom 9:10 And not only this; but when Rebecca also had conceived by one, even by our father Isaac;
11 (For the children being not yet born, neither having done any good or evil, that the purpose of God according to election might stand, not of works, but of him that calleth; )
12 It was said unto her, The elder shall serve the younger.
Here election is explained if you read it carefully. It shows that God knew before being born that Jacob would have faith, and Esau would not. It says the "purpose of God according to election" and then it explains what that is, "not of works, but of him that calleth".
So these verses are showing God's foreknowledge, he knowing that Jacob would have faith and Esau would not before they were born which it clearly explains, and therefore choosing Jacob over Esau.
You need to understand who Paul was preaching to in Romans 9. He was primarily preaching to Jews, although there were many Gentiles as well.
Acts 28:16 And when we came to Rome, the centurion delivered the prisoners to the captain of the guard: but Paul was suffered to dwell by himself with a soldier that kept him.
17 And it came to pass, that after three days Paul called the chief of the Jews together: and when they were come together, he said unto them, Men and brethren, though I have committed nothing against the people, or customs of our fathers, yet was I delivered prisoner from Jerusalem into the hands of the Romans.
18 Who, when they had examined me, would have let me go, because there was no cause of death in me.
19 But when the Jews spake against it, I was constrained to appeal unto Caesar; not that I had ought to accuse my nation of.
20 For this cause therefore have I called for you, to see you, and to speak with you: because that for the hope of Israel I am bound with this chain.
21 And they said unto him, We neither received letters out of Judaea concerning thee, neither any of the brethren that came shewed or spake any harm of thee.
22 But we desire to hear of thee what thou thinkest: for as concerning this sect, we know that every where it is spoken against.
23 And when they had appointed him a day, there came many to him into his lodging; to whom he expounded and testified the kingdom of God, persuading them concerning Jesus, both out of the law of Moses, and out of the prophets, from morning till evening.
24 And some believed the things which were spoken, and some believed not.
25 And when they agreed not among themselves, they departed, after that Paul had spoken one word, Well spake the Holy Ghost by Esaias the prophet unto our fathers,
26 Saying, Go unto this people, and say, Hearing ye shall hear, and shall not understand; and seeing ye shall see, and not perceive:
27 For the heart of this people is waxed gross, and their ears are dull of hearing, and their eyes have they closed; lest they should see with their eyes, and hear with their ears, and understand with their heart, and should be converted, and I should heal them.
28 Be it known therefore unto you, that the salvation of God is sent unto the Gentiles, and that they will hear it.
29 And when he had said these words, the Jews departed, and had great reasoning among themselves.
You might believe the book of Romans was written only to Gentiles, but it was not. Paul first gathered all the Jews in Rome and preached to them. And this is who I believe he is speaking to in Romans 9. And he is explaining to them that they are not saved simply because they are the physical descendents of Abraham. Look how Paul starts the chapter.
Rom 9:1 I say the truth in Christ, I lie not, my conscience also bearing me witness in the Holy Ghost,
2 That I have great heaviness and continual sorrow in my heart.
3 For I could wish that myself were accursed from Christ for my brethren, my kinsmen according to the flesh:
4 Who are Israelites; to whom pertaineth the adoption, and the glory, and the covenants, and the giving of the law, and the service of God, and the promises;
5 Whose are the fathers, and of whom as concerning the flesh Christ came, who is over all, God blessed for ever. Amen.
6 Not as though the word of God hath taken none effect. For they are not all Israel, which are of Israel:
7 Neither, because they are the seed of Abraham, are they all children: but, In Isaac shall thy seed be called.
8 That is, They which are the children of the flesh, these are not the children of God: but the children of the promise are counted for the seed.
So you see how Paul explains in verse 7 how that you are not saved because you are the physical seed of Abraham, but rather the children of "promise" are counted as Abraham's seed. These are those who believe and have faith in God's word.
This is why some objected. They had been raised all their life to believe they were the chosen or "elect" people. They believed they were saved simply because they descended from Abraham. And this is the very topic Paul is discussing in this chapter. It is after Paul explains this and some object that Paul says God will show mercy on whom he will, and wrath on others.
And right after the verses you pull out of context Paul goes right back to this subject and once again explains that he is contrasting works verses faith.
Rom 9:30 What shall we say then? That the Gentiles, which followed not after righteousness, have attained to righteousness, even the righteousness which is of faith.
31 But Israel, which followed after the law of righteousness, hath not attained to the law of righteousness.
32 Wherefore? Because they sought it not by faith, but as it were by the works of the law. For they stumbled at that stumblingstone;
33 As it is written, Behold, I lay in Sion a stumblingstone and rock of offence: and whosoever believeth on him shall not be ashamed.
Look how Paul starts this passage. He says, "What shall we say then?". He is summing up this chapter. And he clearly explains that the Jews who sought after God were rejected because they tried to attain righteousness through the works of the law. He also explains that the Gentiles who had not the law have attained righteousness by believeing in Christ.
So, you can't pull a few verses out of context and use it to prove your doctrine. Read these verses within the full context of the chapter and you will easily see that God has elected those who come by faith, and rejected those who try to attain righteousness through works.
And Paul continues this theme in the very next chapter.
Rom 10:1 Brethren, my heart's desire and prayer to God for Israel is, that they might be saved.
2 For I bear them record that they have a zeal of God, but not according to knowledge.
3 For they being ignorant of God's righteousness, and going about to establish their own righteousness, have not submitted themselves unto the righteousness of God.
4 For Christ is the end of the law for righteousness to every one that believeth.
Carico, you often criticize non-Cals like me, but in my opinion you do not know the scriptures. You cherry-pick verses to prove a doctrine taught by men and ignore scripture that contradicts it.