Casper FFG
Casper FFG (the Friendly Finality Gadget), designed by Vitalik Buterin and Virgil Griffith, is the part of Ethereum's proof of stake that turns an otherwise probabilistic chain into one with provable, economically enforced finality. Rather than reaching consensus on every block, it operates on checkpoints — the first block of each epoch — and overlays a voting process that periodically declares some checkpoints irreversible. It is a gadget because it sits on top of an underlying block-proposal mechanism rather than replacing it.
Validators cast attestations that include an FFG vote: a link from a source checkpoint to a target checkpoint, effectively saying 'I see the chain extending from here to there.' When links carrying more than two-thirds of total staked ETH point to a target, that target becomes justified. Finality is a two-step ratchet: when a justified checkpoint's immediate successor checkpoint also gets justified by a supermajority link, the earlier one becomes finalized, meaning reverting it would require an attacker to destroy at least one-third of all staked ETH.
That economic guarantee is enforced by two slashing conditions a validator must never violate: do not cast two distinct attestations for the same target epoch (no double voting), and do not cast an attestation whose source-target span surrounds, or is surrounded by, one of your earlier attestations (no surround voting). Together these make it mathematically impossible to finalize two conflicting checkpoints unless a third of the stake signs slashable, self-incriminating evidence — the property called accountable safety. In Ethereum, Casper FFG provides finality while LMD-GHOST provides the live fork choice, and the combination is named Gasper.
Checkpoint C(n) is justified by a supermajority link. In the next epoch a supermajority link justifies C(n+1) with source C(n). That second link finalizes C(n) — no honest validator will ever build a chain that excludes it.
Finality here is economic, not absolute: a finalized checkpoint can in theory be reverted, but only by an attacker willing to have at least one-third of all staked ETH slashed — currently many billions of dollars — which is why it is treated as irreversible.