No, simply because it is not in Europe. If we started accepting countries where people can claim European descent, why not accept all American countries, South Africa, Australia and NZ ? That just wouldn't be a European Union anymore. I am not saying a greater union won't happen in the (not so near) future, but that's another debate.
Exactly, why not? I am an American of European descent. My grandmother came from Ireland, and my grandfather from Greece. Though I have American nationality, I feel European. I share European values. I identify with the culture, and I have more in common with Europeans than with other people.
It would still be "a European Union", as if you have European decent you are "European".
My point is that why not consider the Anglophone countries of the world for EU membership. Why not consider Hispanophone countries like Argentina? This countries would help the EU expand, but would also complement the EU; due to their shared culture.
The precedent has already been set with Cyprus. Though in Asia, it is European culturally. Countries such as Canada are European culturally. Look at New Zealand, their government is similar to Scandanavian countries, they speak a European language (English), and share a head of state with a Europeam country (The UK). Same with countries such as Australia, and Canada (though their governments are much more similar to the British government than Scandanavian governments).
And regarding Israel. I believe that once they obey international law, give the Palestinians back their rightful land and their rights, and reform their own country, that; they should then consider EU membership. And if they did that, I would support their membership.
As to what another poster said regarding Turkey and Lebanon; Turkey is partially in Europe, and genetically; Turks are Europeans. I think that it is likely that Turkey will join the EU.
Lebanon has no way in hell. Maybe in 50 years or so. But I highly doubt it.