Some objects have no mana cost. This represents an unpayable cost. An ability can also have an unpayable cost if its cost is based on the mana cost of an object with no mana cost. Attempting to cast a spell or activate an ability that has an unpayable cost is a legal action. However, attempting to pay an unpayable cost is an illegal action.
July 16, 2010 · Magic 2011
If a player has 0 or less life, that player loses the game as a state-based action. See rule 704.
February 1, 2010 · Worldwake
Damage marked on a creature remains until the cleanup step, even if that permanent stops being a creature. If the total damage marked on a creature is greater than or equal to its toughness, that creature has been dealt lethal damage and is destroyed as a state-based action (see rule 704). All damage marked on a permanent is removed when it regenerates (see rule 701.11, "Regenerate") and during the cleanup step (see rule 514.2).