If time is necessary, it's discrete

Say that a total order is locally finite provided that between any two elements there are at most finitely many elements. A locally finite total order is discrete: each non-extreme element has an immediate predecessor and successor.

Here’s a valid argument:

1. If time is necessary, the order structure of time cannot be problematically arbitrary.

2. The only possible order structures for time that are not problematically arbitrary are locally finite.

3. Thus, if time is necessary, it has to be locally finite, and in particular discrete.

Premise (1) is motivated by my and Josh Rasmussen’s research project on which fundamental and necessary things cannot be problematically arbitrary. But if time is necessary, it is very likely to be fundamental.

Why accept (2)? I am thinking like this. Suppose there are two times between which there are infinitely many times. The size of that infinity of times is arbitrary—is it countable, is it the continuum, is it something in between (if there is anything in between), is it something even bigger?

So, if time is necessary, the order of time is locally finite.

Up to isomorphism, the following are the only possible locally finite total orders:

1. ...,  − 3,  − 2,  − 1, 0, 1, 2, 3, ...: no beginning or end

2. ...,  − 3,  − 2,  − 1, 0: end but no beginning

3. 0, 1, 2, 3...: beginning but no end

4. 0, 1, 2, 3, ..., n (for some finite n): beginning and end.

Now a complex thesis:

4. If time is necessary, then either (i) necessarily time has structure A or B, or (ii) necessarily time has structure C or D.

Here’s why. If there were possible worlds where time has structure A or B and other possible worlds where time has structure C or D, it would be problematically arbitrary whether time starts off with an infinite past or a finite past.

Can we go further? Maybe, but I get less confident. Still, it seems like it would be weird that time could end at some point, but had to have started. Thus:

5. If time does not have to go on endlessly, there need not have been time at all.

It follows that:

6. If time is necessary, then it necessarily has structure A or it necessarily has structure C.

In other words, if time is necessary, it must be like the integers (option A) or like the natural numbers (option C).

Can we narrow it down further? Well, causal finitism suggests but does not prove that options with no beginning are problematic. That would get one down to option C. But this is a weak argument. Option A also has the disadvantage that it leaves unexplained why events in time didn’t all happen a day earlier than they did.

In any case, there is another move we can make at this point. It follows from 6 that:

7. If time is necessary, all the specifics of the order structure of time are necessary.

I am personally inclined to add:

8. The specifics of the order structure of time are not necessary.

9. So, time is not necessary.

But my intuitions for 8 aren’t that strong, so I don’t think this is a very strong argument for the contingency of time.