Actual and potential infinities

I've spent much time on the ontological argument for God, but only ever invested one post, years ago, on the cosmological argument.  Looking back, my essay is a jumble of too many objections in too small a space.  I also pushed some of the most interesting objections aside because they were not particularly important.  Therefore, I am declaring a new blogging series: not "Why the cosmological argument is wrong," but "Here are a few things that are wrong about the cosmological argument."

I will place emphasis on math and physics, and deemphasize the cosmological argument itself.  I trust this post will demonstrate what I mean.

Actual and Potential Infinities meet Mathematics

In William Lane Craig's version of the cosmological argument, he makes a distinction between "potential" and "actual" infinities.  William Lane Craig (henceforth WLC) contends that potential infinities can exist in the real world, but actual infinities cannot.¹

The thing is, in mathematics, there is no distinction between actual and potential infinities.  At least not one I've heard of.  Luckily, WLC explains.  He identifies actual infinity with the cardinality of natural numbers, ℵ₀ ("Aleph-nought").  As for potential infinity...

Crudely put, a potential infinite is a collection which is increasing toward infinity as a limit, but never gets there. Such a collection is really indefinite, not infinite. The sign of this sort of infinity, which is used in calculus, is ∞.

From a mathematician's perspective, WLC's definition of actual infinity is perfectly well-defined, but his definition of potential infinity is poorly-defined.  In calculus, ∞ doesn't actually have any meaning on its own, but when inserted into mathematical expressions it gives the expressions new meaning.

These are two possible contexts in which ∞ can be used, and it technically has a different meaning in each context.²  WLC appears to be using ∞ in yet a third context where the meaning is unclear.  I can guess fairly well what he's trying to say, and the charitable thing to do would be to simply give potential infinity a precise and appropriate definition, even though WLC could only be bothered to give a crude definition.  But if I did that, WLC's supporters would likely claim that I've defined it incorrectly.

But let's try it anyway.

Any particular set of objects has an exact "size", called its cardinality.  The cardinality may either be a finite number (eg 0, 1, 2, 3), or an infinite cardinality (ℵ₀ or larger).  We say that the set of objects is actually infinite if its cardinality is ℵ₀ or larger.

Potential infinity is not a cardinality, and does not refer to any particular set of objects.  If we say that the set of all apples in the world is potentially infinite, we are really talking about the set of all possible sets of apples.  And if we say that distance between two stars is potentially infinite, we are really talking about the set of all possible distances between two stars.

We say that a set of objects is potentially infinite if for every finite number M, there exists some possible world where that set of objects has cardinality greater than M.  Similarly, we say that a number X is potentially infinite, if for every finite number M, there exists some possible world where X is greater than M.

Notes on my definition:

1. Even if the set of all apples in the world is potentially infinite, this does not necessarily imply that there is some possible world in which the set of apples is actually infinite.  It just means that there is no maximum number of apples in the world.
2. However, if the set of all apples in the world is potentially infinite, this does imply that the set of all possible worlds is actually infinite.  I think this is okay with WLC, because he would not consider possible worlds to be "existing" objects.
3. Mathematically speaking, this is still poorly-defined because the set of all possible worlds is poorly-defined in mathematics.  However, I think it will suffice for my purposes.

I sincerely hope that my definition is satisfactory to WLC's supporters, but I couldn't know for sure.  It stands to reason that they should be able to tell me whether the definition is satisfactory before I apply the definition.  Therefore, I offer a pause here for objections (though it's a symbolic pause, since realistically I don't expect any supporters to pay attention to a little blog).

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1. This is relevant to the cosmological argument because WLC contends that a universe without beginning is an actual infinity, and I suppose he contends that a god is not.  But let's not get sidetracked.

2. Equation (1) means that for any positive number ε, there exists some number M such that for all x greater than M, 1/x is between -ε and ε.  Equation (2) means that for any number M, there exists some positive number ε such that if x is between 0 and ε, then 1/x is greater than M.

"A few things wrong about the cosmological argument"
1. Actual and potential infinities
2. Actual infinities in physics
3. What is real?
4. The "absurdity" of Hilbert's Hotel 
5. Interlude: God is infinite 
6. Forming Infinity, one by one 
7. Uncertain beginnings
8. Entropy: The unsolved problem
9. Kalam as an inductive argument
10. Getting from First Cause to God