Just lately, for what ever reason, I've come across many people muttering that Gödel proved that time doesn't exist. Indeed there are several books on the subject and a nice pile of papers citing Gödel original paper, "An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation". The contents of Gödel's paper are well known, indeed you can find them in Hawking and Ellis (which your nearest universities library will almost certainly have).
Gödel presents an exact solution of Einstein's Field equations for a perfect fluid. That is he gives a model of the universe. It contains what are called closed timelike curves but not closed timelike geodesics (although there are closed lightlike curves). This means that time travel is possible, but that it requires acceleration to achieve it, i.e. a spaceship with an engine. Have a look at Pfarr's, "Closed Timelike Curves—Time and Again" for a simple derivation of this and a nice bibliography. In fact any two points in Gödel's model can be joined by a future directed timelike curve. That is a person, with a spaceship, can travel between any to points in this universe. Specifically a person could travel anywhere into their past.
The argument against the existence of time seems to go like this:
We are are for events on the world able to tell when one even happens at the same time as another. If I note the time when I have breakfast I can work out what time it would be in, say, England when I was having breakfast. Every event that happened at that time in England happened at the same moment as when I was having breakfast. All these events, and all the other events that happened at this time can be thought of as happening simultaneously. This gives us a very well defined idea of which events must have come before or after others. But this only works because, with respect to the earth, we're all pretty much moving at the same speed relative to the speed of light.
A person in a space ship moving much faster, relative to the earth, would see some of these simultaneous events happening before and after each other due to the persons speed and the distance between all the simultaneous events. The smaller the distance between the events the closer the events would occur to each other, with respect to the person in the spaceship. My point is only this: Even when gravity is not taken into account and even in very simple models of our universe, the concept of future/past is dependent on who is doing the measuring. Effectively any definition of simultaneity is arbitrary and depends on the distance between points. Our perception of time is built on this idea.
When points are close the difference in the simultaneity of points with respect to the different definitions of simultaneity will be small, when the points are far away the differences will can be large. Mathematically the definition of simultaneity is the same as building a coordinate system about your path in a space-time (a future directed timelike curve). This coordinate system will extend a short distance around yourself but, at some point, is very likely to fail (e.g. it'll assign the same event with multiple sets of coordinates). Generically coordinate systems about timelike curves only extend a very small distance, hence any idea of simultaneity and therefore of time will only be defined, with respect to yourself, only for a very small distance. It just so happens that in the case of the Earth, there exists a coordinate system which extends over the whole planet, this coordinate system is the one we use to tell what's happening in England when I have breakfast.
Hence any definition of simultaneity and therefore of time can only ever be done for events that are close. That is for events that are local. In fact, regardless of what has ever happened in your individual past of future you can always define a set of coordinates. Hence you can always, locally define time.
In this way, I think, the argument about the existence of time is correct: There is no such thing as "global" time. In this way, I think, the argument about the existence of time is incorrect: Even if you've travelled in time you will be able define time locally.
Anyway... for having made it this far I've included some interesting facts about time travel in Gödel's universe.
In "Minimal acceleration requirements for "time travel" in Gödel space‐time" Malament shows that a spacecraft must lose at least