The existing email system will not survive. Too many people are receiving too much unwanted email. Too many people are making lots of money sending email to too many people. Let's assume that the current system is going to die. What will replace it?
Instead of writing email, people will publish web pages. Instead of the recipient paying for the resources to hold the email, the sender will pay for the resources to publish the web page. There's basically two types of email: mailing lists, and email addressed to one or more individuals. Let's address the latter first.
Right now, when you send email to someone, you send a message from your client machine to their server. The email is stored on the recipient's machine until they read the email. Under the new system, you would create a web page, and store it on a server. The URL would be something unguessable like http://example.com/~nelson/827134282173614682732.html. If I wanted to send that email to someone, I would point them to that URL. If I wanted to send it to many someones, I would point a program to that URL, and it would create an index of available messages.
I'm glossing over some details. In particular, the notification mechanism. At first, the notification mechanism would be ordinary email. When you published a message (in essence, like hitting the 'send' button), your publishing software would send a notification to the recipient(s) using the new protocol. If their computer didn't understand the notification, then your publishing software would email a URL to them.
I'm also glossing over privacy. Some people would want to send encrypted emails. In order to do that, you would have to have a copy of the recipient's public key signed by yourself. It would work in a similar method to PGP's web of trust, except that everybody would have a published public key. Casual encryption is built-in.
Why is this going to help with spam? Because it shifts the cost of receiving email from the recipient to the sender. Let's say that somebody tries to notify the entire Internet of their message. Some people would keep track of how many people are notified. If too many people are notified about the same message, or about the same server, then it gets marked as a possible source of unwanted messages. In order for their spam to be successful, they would have to remain as a sitting duck the whole time, accessible to blocking.