Why does an infinite regress have to be terminated?

I’m guessing that what you mean by your question is what is so bad about a regress that is NOT terminated, and hence infinite.

An infinite regress which is considered bad is sometimes called a ‘vicious regress’.

An example of an argument that uses the idea of an infinite regress is the Cosmological argument for the existence of God. If we trace chains of cause and effect back far enough we either get to a Big Bang (the beginning of the universe) or the causes and effects go back for ever.

If the series terminates with the Big Bang, then that would be an event without a cause, contrary to the belief that every event has a cause. If the series doesn’t terminate but just keeps going back further and further in time, that would be an infinite regress.

But why is the regress vicious?

First, I need to explain the difference between an infinite regress — for example, a regress of causes and effects, or of explanations — and an infinite series.

In mathematics, there are examples of an infinite series that is terminated at one end and and not the other, e.g. the series 0, 1, 2, 3… which terminates at 0, or the series 0, -1, -2, -3… which also terminates at 0. Or you can have a series that is infinite in both directions, e.g. … -3, -2, -1, 0, 1, 2, 3… . It doesn’t terminate anywhere. Nothing bad or vicious about that at all.

What is infinity? In mathematical terms,

“An infinite set is a set whose members can be put into a 1-1 correlation with a proper subset of itself.”

What that means in layman’s terms is that if you remove half of an infinite series, you still have the same ‘number’ of items. E.g. 0, 1, 2, 3… can paired off with 0, 2, 4, 6… That’s just one of the weird properties of the infinite.

In mathematician David Hilbert’s story of the Grand Hotel (cited by George Gamov in his book One Two Three Infinity, 1947 — the inspiration for 123infinity.com) there is still room for an infinite number of new guests even though each one of the infinitely many rooms is occupied — the old guests are simply asked to move to the room which was double the number of the room they were in before, leaving an infinite number of empty rooms. Job done.

Back to the cosmological argument. We have a problem — or so it is alleged. Identifying the cause of an event is supposed to supply a reason or explanation why that event happened. The event occurred because of the cause. Going back in time, that cause occurred because of another cause and so on.

Here’s an example that illustrates the problem. In an empty field, you discover a magnificent chandelier on a chain going up into the sky and through the clouds. Each link in the chain holds the link immediately below it. Then you are told that there’s no ‘ceiling’ the chain is attached to, no first link. The chain just goes up for ever to infinity. But if there isn’t a first link then what’s holding the entire chain up?!

The problem of a vicious regress, in short, is a problem of dependency. A required explanation is indefinitely deferred. An indefinitely deferred explanation is no explanation at all.

Theists hold that God from a vantage point outside time is responsible for the entire physical series of causes and effects — the entire ‘chain’ — which is either finite or infinite, depending on whether or not the Big Bang theory is true. God is the ultimate, non-temporal ’cause’. If the series of ‘links in the chain’ is infinite, God can still be there outside of space and time to ‘hold it up’.

Is that a good argument? If you don’t believe in God, then the alternative is to say that there is no ‘ultimate cause’. Things just are the way they are rather than some other way. The existence of the universe is contingent, but it is not contingent ON anything. It just IS.

Say that if you like, but there is still a question, a problem. Maybe a singular event without a cause, or alternatively an infinite regress of effects and prior causes is bearable, but there is some discomfort there that indicates that there is something that we do not fully understand about the nature of our world.

That’s not a problem for physics but for metaphysics.