Bitcoin is created for decentralization of funds, anonymity features, fast transactions and all those good stuffs but never to take down Paypal. There's no competition going on. The bitcoin network is not managed by a single company. It never intended to fight against other payment methods that exists. If Paypal starts accepting bitcoin, then it's okay. Bitcoin's not going to mock Paypal for doing that.
The competition between PayPal and bitcoin system has started when bitcoin provides a solution for online payments.
Bitcoin has all the features above mentioned, but all the features are possible because it is existing online whereas PayPal is doing business mainly targeting online users.
Maybe bitcoin was not invented to compete PayPal but it will happen because only fittest will dominate and will survive.