“For He was foreknown before the foundation of the world, but has appeared in these last times for the sake of you” I Peter 1:20
Foreknowledge shows God’s omniscience even over the dimension of time. He knows all before it occurs to man - because to God He sees all of time at the same time. God does not base His actions on what He sees ‘in the future’, for to Him He is not bound to travel through time. For Him, there is no mystery or surprise. Nor must He mandate or decide all events to know them.
Predestination deals with God’s omnipotence. The word itself means to “to mark out beforehand’; to pre-establish limits and boundaries. Specifically, this word references how God set limits/boundaries/laws upon everything before creation. He set the laws of physics, placed the boundaries of the sea, determined the eternal plan by which mankind would be saved, etc (Prov 8:22-31, Eph 1:3-10, Eph 3:10-11, Job 38:33, Rom 8:29, etc).
In the plan of salvation, God also predestined it to include the gentiles, not just the Jews (Eph 3:2-6, Rom 3:21-31, Rom 9:1-26, Rom 15:5-13, John 1:11-13, Isa 45:9-10, Rom 9:11-16, etc).
Predestination does not mean God decided the specific movement of every person and molecule, but rather that He in His power and wisdom set the rules and limits by which space, time, and people are bound and governed. In regards to man, God set laws regarding sin and righteousness, and the consequence of death for sin. In regards to salvation, God predestined that only Christ was the Way, the Truth, and the Life. He predestined that Christ would need to die to cover man’s sin. He predestined that all in Christ, those who believe, would be adopted as sons and made holy (Eph 1; ) etc.
Further details and scriptures regarding predestination and God’s sovereign rule can be found here: https://ebible.com/questions/265-what-is-predestination-is-predestination-biblical#answer-15328 https://ebible.com/questions/3181-is-god-sovereign-or-do-we-have-a-free-will#answer-14667
Elect means chosen/select, such as how God chose Israel among all the nations of the world to be His people. In like manner, God also chose all who believed in the promise of the savior; both Jew and Gentile; to be His daughters and sons. Predestination is a related concept, as God predestined the way of salvation by which we become His chosen children. Fundamental to the concept of election is that it is by God’s good pleasure that a people is chosen, not by works or the wisdom of man. Israel had nothing to recommend her, nor was Israel even a nation, when God chose her (Ezek 16:1-14, Deut 7:6) to be His treasured possession. God not only favored Israel as His people, but predestined the savior to come through Israel (Is 9:6-7, Jer 33:20-21). He chose Israel to redeem a people for Himself and to magnify His name (II Sam 7:23). Likewise, it pleased God to choose/elect His people not by anything works we have done or any favor with men, but rather to redeem us as a people through the blood of Christ alone (Eph 1:3-14).
This way, it is by God’s mercy and not by man’s might or wisdom (I Cor 1:20-21, I Cor 2:5, Titus 3:4-6). By coming to God by the way He predestined for us - faith in Christ alone, we are redeemed by God as His people for the praise of His name (I Pet 1:1-10, Eph 1:3-14, Luke 12:27-33).
Some key verses where these terms are used in tandem are I Pet 1:1-2, Rom 8:28-20, Eph 1:4-5, and Eph 1:11:
"Peter, an apostle of Jesus Christ, To the elect, exiles of the Dispersion throughout Pontus, Galatia, Cappadocia, Asia, and Bithynia, chosen by the foreknowledge of God the Father and sanctified by the Spirit for obedience to Jesus Christ and sprinkling by His blood..." I Pet 1:1-2
The elect are the people of God. God chose His people by His foreknowledge - not just those who were ethnically Jewish, but those who would have faith in Christ. Specifically here in context, Peter is speaking of the 'remnant' of ethnic Israel who would believe in the Messiah, as described in Rom 11. "And we know that in all things God works for the good of those who love him, who have been called according to his purpose. For those God foreknew he also predestined to be conformed to the image of his Son, that he might be the firstborn among many brothers and sisters. And those he predestined, he also called; those he called, he also justified; those he justified, he also glorified." Rom 8:28-20
It is important to note in context of the chapter that all these people Paul is addressing love the Lord and have His Spirit - they are already in Christ. It is these believers whom God foreknew, predestined to be conformed, called, justified, glorified, etc.
Also in context, this particular calling includes Gentile believers as well, as explained in Rom 9:22-26: "What if God, although choosing to show his wrath and make his power known, bore with great patience the objects of his wrath—prepared for destruction? What if he did this to make the riches of his glory known to the objects of his mercy, whom he prepared in advance for glory— even us, whom he also called, not only from the Jews but also from the Gentiles? As he says in Hosea:
It is important to note in context of the chapter that all these people Paul is addressing love the Lord and have His Spirit - they are already in Christ. It is these believers whom God foreknew, predestined to be conformed, called, justified, glorified, etc.
and I will call her ‘my loved one’ who is not my loved one,”
and, “In the very place where it was said to them,
‘You are not my people,’
there they will be called ‘children of the living God."
(This form of calling, then, is for believers, for it is being called the children of God. This is not referring to the general call of unbelievers to repentance and faith.)
Eph 1:4-5 & 11-13 "For he chose us in him before the creation of the world to be holy and blameless in his sight. In love he predestined us for adoption to sonship through Jesus Christ, in accordance with his pleasure and will—In him we were also chosen, having been predestined according to the plan of him who works out everything in conformity with the purpose of his will, in order that we, who were the first to put our hope in Christ, might be for the praise of his glory. And you also were included in Christ when you heard the message of truth, the gospel of your salvation. When you believed, you were marked in him with a seal, the promised Holy Spirit,"
Eph 1:4-5 & 11-13 "For he chose us in him before the creation of the world to be holy and blameless in his sight. In love he predestined us for adoption to sonship through Jesus Christ, in accordance with his pleasure and will—In him we were also chosen, having been predestined according to the plan of him who works out everything in conformity with the purpose of his will, in order that we, who were the first to put our hope in Christ, might be for the praise of his glory. And you also were included in Christ when you heard the message of truth, the gospel of your salvation. When you believed, you were marked in him with a seal, the promised Holy Spirit,"
Eph 1 is an incredibly important chapter for understanding how foreknowledge, predestination, and election relate. Paul is addressing a mixed group of believers, Jew and Gentile. He begins by pointing out that God had chose before creation that those Jews 'in Him' (faith in Christ) would be made holy and blameless in His sight. He predestined that these believing Jews would be adopted as sons, according to His plan, so that these Jewish believers would be for the praise of His glory. Paul then expands the argument to include the Gentile believers. They, too, were included 'in Christ' when they believed! As such, they too have all the blessings predestined for Jewish believers - holiness, adoption to Sonship, etc.
Hopefully this helps explain these often-debated terms, and how when taken together they are a wonderful picture of how God planned before creation that anyone who placed their faith in Christ would be included 'in Him' and receive all the blessings of adoption as sons of God such as justification, holiness, and eternal life.