3.2 Is It Possible to Cross an Infinite Past?

3.2 Is It Possible to Cross an Infinite Past?

Another argument for a beginning of time and the universe rests on a mathematical intuition: the past cannot be infinite, because it would be impossible to cross such an infinity and arrive at the present moment.

Potential Infinity vs. Actual Infinity #

It is important to distinguish two fundamental notions of infinity:

  • Potential infinity: a quantity that can increase without bound but is always finite at each stage. For example, counting 1, 2, 3… without ever reaching an “infinite number.” In mathematics this corresponds to the concept of a limit, where a value tends toward infinity without ever attaining it.

  • Actual infinity: a completed infinite totality, existing all at once, such as the set of all natural numbers. This kind of infinity can contain a proper subset of the same size (for example, the natural numbers and the even numbers can be matched one-to-one). Unlike potential infinity, an actual infinity is not built by successive additions but is given in its entirety.

This distinction is crucial for thinking about an infinite past: is it a potential infinity (the past extends without limit but is never complete), or an actual infinity (the past is a completed, infinite sequence of events)?

Why an Actual Infinite Past Cannot Be Crossed #

The argument can be framed as follows:

  • If time extended infinitely into the past, then the past would consist of an infinite series of events.
  • Such a series would be actually infinite, not merely potentially infinite.
  • But a series formed by successive addition cannot be actually infinite.
  • The temporal sequence of past events has in fact been formed by successive addition, event after event.
  • Therefore, the past cannot be an actual infinity stretching infinitely backward.

Conclusions #

If we adopt a dynamic view of time — in which time genuinely “flows” — then an infinite past is impossible. While an actual infinity might be conceivable in a static view (where all events are laid out simultaneously, as in a “block universe”), it becomes incoherent when we consider the real, successive passage of time.

Even if actual infinities exist in some sense, the dynamic nature of time prevents an infinite sequence of past events from being crossed to reach the present.

In short: the real passage of time rules out an infinite past, since such an infinity would make the present unattainable. The logical conclusion is that time had an absolute beginning.

Further Reading #

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