I think AJAX is a concept frequently misunderstood.
AJAX is just a new name to speak about rich web interfaces that use Java Script (to execute things), Dynamic HTML (to render things) and XML (to pass information between clients and server). The element that makes AJAX so "cool" is the Asynchronous messages, that allow the browser to make new requests without reloading the entire page.
PHP is a technology to program mainly the server-side of web applications. Access to databases, mailboxes or whatever should be done on the server, has to be programmed using server-side languages like PHP, Perl, JAVA, .net.
Using AJAX technologies is not possible to contact a database, for example, since only the HTTP protocol is supported, but PHP scripts can invoked to make what is necessary on the server.
One of the downside of AJAX is that Java Script is not a very friendly language for developers and is not as easy to develop and debug complex applications as with other languages. That is why a lot of AJAX frameworks exist for other languages. For example, a library called XAJAX allows you to develop your code in PHP and it translates it to the appropriate Java Script; that way you only have to easily call the generated Java Script functions.
It could be possible to write applications using only AJAX, but surely they would be very boring since they have to be run inside a browser (in a sandbox) and can't save or read local files or do many things other than what is allowed to do with DHTML.