Actual Infinity in Reality?

[polymath257]

At least as far as I understand it, Einstein's views were not *exactly* in line with the B theory of time. But they certainly were not at all in line with the A theory. Let me explain.

The B theory seems to suggest that time is a universal, objective thing in a way that contradicts general relativity.  For example, if I measure the time between two events to be 5 seconds, so will everyone else. But that is NOT the case in relativity! It is quite possible in relativity for one observer to measure 5 seconds between events and another to measure 3 seconds between those same two events. From what I can see, the B theory isn't very comfortable with this aspect of reality.

General relativity, on the other hand, considers time and space together to be a single entity: spacetime, which has an objective reality. So, while distances in space and durations between events are not separately well-defined (they depends on the motion and position of the observers), a combination of them *is* objective and is agreed upon by all observers. In this system, time does NOT 'flow'. ALL of space and ALL of time are, together, the geometry of spacetime. Time, like space, simply exists. Neither is well-defined in and of itself. BOTH are required for a consistent system.

One of the issues here is that it is possible that one observer will see event A as happening before event B and another may see it the other way around. So the idea of causality needs to be somewhat modified: instead of causes only having to be in the past (which is observer dependent), they have to be in the *past light cone* (which all observers agree to). The past light cone is the collection of events in spacetime from which light could reach the event in question. Nothing outside of this light cone can be a cause for an event.

With this modification, however, causality *is* a central aspect of general relativity. But it isn't in the sense that time flows from one set of events to another. It is that the particular distribution of particles in spacetime has to obey certain rules that reflect this causality. The spacetime is *still* one complete entity, but the 'pictures' on that manifold are not completely random: they reflect constraints on the solutions of the basic dynamical equations.

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OK, we have a good question here:

Is {1,2,3,4,5....} an actual infinity or a potential infinity?

Steve claims it is a potential infinity. I claim it is an actual infinity. The difference is that Steve seems to think of it as a process of creating that set and I see the set as simply there and our attempts to *list* it are doomed to failure.

But I can tell what is in and what is not in that set with perfect precision. For example, the number 59749873 is in that set and the number 82.389587 is not. That is *all* that is required to completely define the set: to know what is in and what is not. You don't have to list the elements, just know how to tell when something *is* an element.

So the set {1,2,3,4,5....} is an actually infinite set. It is the *complete* collection of counting numbers. Nothing is being added or taken away from it.

And we are NOT taking the succession of finite sets
{1}, {1,2}, {1,2,3}, {1,2,3,4},....
NONE of those is the set above. THAT I certainly agree with. At no finite stage in this progression do you get the set above. I agree with that also.

But that isn't the same question. The question is whether the completed set
{1,2,3,4,.....}
is an actually infinite set or not. Since it is well defined and since it is none of the finite sets above, it is.