No, you cannot play PS3 games on a PS4. PS4 has no backward compatibility with PS3, as they use different kinds of processors. However, PS4 will be able to play PS3 game titles through Cloud gaming, known as Gaikai, but it may take time for them to add in games to play via this Cloud gaming.
More discussion on how to play PS3 Games on PS4 here:
If you want to play PS3 games now, then is still a good idea to go for the PS3 console. It is now being released as a cheaper bundle, and it’s quite popular. Remember, you can always resell the PS3 console and games that you have when you feel ready to upgrade to the more expensive PS4. See our article Why get a PS3 even though PS4 has arrived?, which talks about the hot PS3 bundle and all the pros and cons.
On the other hand, there are some titles which are available on both consoles, but they are each in their respective console’s format and won’t be able to run on the other console. For example, a PS4 version of Battlefield 4 won’t play on a PS3 console, and a PS3 version of Watch Dogs won’t work in a PS4 console.
Some very good games that are available on both PS3 and PS4:
Finally, there is a misunderstanding among users that if they are playing online on the PS3 platform, they are able to play the same game with PS4 online users. This is wrong; they are different platforms and you cannot play the games across platforms. PS4 will have different servers of COD Ghosts than PS3, so they cannot be played with each other.