Jonluw said:
Your friend then goes ahead and presses the "random" button. Furious, you decide to chase after them (they owe you money) by the only means you know: pressing the random button (let's say you can bring the time machine with you when you travel so you can try as many times as you like, but your friend didn't for some reason).
Of course; an infinite number of timelines spring from the point your friend has travelled to. This means that your friend is currently in an ifinite number of timelines. There is also the infinite number of timelines springing from the line in which they didn't press the button in the first place.
Or rather, there exist infinitely many possible events in which your friend spontaneously existed where he hadn't before, as a result of his having travelled from another world. We might suppose this infinity for a given timeline is the cardinality of the time line, in that if it's really totally random, they could quite literally appear at any point whatsoever. Time is often understood to be a continuum, and hence there are not only infinitely many such possible events, but also
uncountably infinitely many such events.
This only determines the number of possible events within a timeline. It is a conceptual leap to go from "Possible event" to "genuine alternative timeline" (even though the various modalities of quantum mechanics form some kind of difference in possibility, it's not apparent that this means that all possible differences amount to physically possible alternatives). If the "random" button is based on an actual, metaphysical randomness, then yeah, you guarantee that the mere pressing of the button reveals uncountably many alternative timelines. But this only exponentiates with the number of other such "truly random" events that the universe encounters.
Suppose that we live in an causally deterministic universe with the single exception of the button. What the button does to our physical model is reveal that continuum many other worlds exist. Since these alternative worlds are all physically possible, they too are bound by the physical laws, so are entirely deterministic unless they, too, contain buttons. How many such worlds contain buttons? This is a question for the physics of building the button, rather than the metaphysics of the many-worlds framework.
Jonluw said:
Problem is, there is also an infinite number of timelines in which your friend is not.
Again, this is a question for the final grand unified physical theory of everything rather than for the metaphysics of time travel. How much does genuine alternative history vary other than in respect to this incident of time travel? Without saying something fairly substantial about the degree of genuine randomness inherent in fundamental physics, you haven't given us enough to go on to presume that there actually are infinitely many such timelines. The consistent suggestion I've made is that the button is the only truly random thing there is, and that only one world (the actual world) contains it. Therefore, the timelines that you can state to be distinct prior to your pressing the button yourself would be limited to "the actual world" and "worlds reached by my friend travelling with the button".
In all of those timelines, some version of your friend exists. You are guaranteed that the world you arrive at has at some point had your friend on it.