Cap and trade is not the same as carbon offsetting at all. You are mixing up things. The idea behind cap and trade is that there is a limit to the total amount

Cap and trade is not the same as carbon offsetting at all. You are mixing up things. The idea behind cap and trade is that there is a limit to the total amount of emissions and you need a certificate from the government to pollute. The trading part just moves certificates around with a monetary incentive. Carbon offsetting is the exact opposite of cap and trade, since it raises the cap by issuing more certificates.