Cain
Cain (compare (Genesis 4:1) a type of the religious natural man, who believes in a God, and in "religion," but after his own will, and who rejects redemption by blood. Compelled as a teacher of religion to explain atonement, the apostate teacher explains it away.
error of Balaam
Balaam, the "error" of Balaam must be distinguished from his "way"
( See Scofield) - (2 Peter 2:15)
and his "doctrine"
( See Scofield) - (Revelation 2:14).
The "error" of Balaam was that, reasoning from natural morality and seeing the evil in Israel, he supposed a righteous God must curse them. He was blind to the higher morality of the Cross, through which God maintains and enforces the authority and awful sanctions of His law, so that He can be just and the justifier of a believing sinner. The "reward" of (Jude 1:11) may not be money, but popularity, or applause.
Core
See Numbers 16. The sin of Korah was denial of the authority of Moses as God's chosen spokesman, and intrusion into the priest's office.
gainsaying
(Greek, "antilogia", means "against the Word").