Mathematical Analogies: God's Ordinal Infinitude

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An advantage of mathematical thought is that exposure to advanced mathematical concepts can provide a strong, philosophically imaginative mental framework for other kinds of complex ideas.  This is especially the case with theological ideas relating to God.  Given God's metaphysically unique status as Being in itself, we can only imagine him analogically.  Any attempt to fully capture him falls short.  If we think we've understood Him, then our conception is not God.  And yet, this does not mean that our conception is totally unrelated to God.  Since we are created in His image and likeness, our intellectual relationship with God is something real and true, and our conceptions can be a true "projection," so to speak, of who He is. 

Here, we give a way to consider God's infinitude, using the mathematical study of infinities found in set theory.  One thing about this analogy, as we shall see, that sets it apart from others is that it contains within itself an additional analogy of its limitations.  Typically, we have to supply the limitations of an analogy in a manner exterior to it.

So what do we mean when we say that God is infinite?  We mean that there is no limit to him: i.e., that he is not finite.  He does not have "boundaries" like we do. Now this means two things in terms of creation: the first is that God permeates creation.  This doesn't mean that we are all a part of God (given his simplicity), but rather that he is present in all creation.  As the Psalmist says:

Where shall I go from your Spirit?
Or where shall I flee from your presence?
If I ascend to heaven, you are there!
If I make my bed in Sheol, you are there!
If I take the wings of the morning
and dwell in the uttermost parts of the sea,
even there your hand shall lead me,
and your right hand shall hold me.
If I say, “Surely the darkness shall cover me,
and the light about me be night,”
even the darkness is not dark to you;
the night is bright as the day,
for darkness is as light with you.

 

The second is that God is not bound by creation itself, but is beyond it.   This contrary to the notion of certain forms of Pantheism which identify God with the collection of creation itself.  As you shall see, this one will be a bit harder to grasp analogically.  But let's see how we can start.

Analogy 1.0: The natural numbers

The natural numbers have a set theoretic representation. Zero corresponds to the empty set $\{ \}$ , $1 = \{ \{ \} \}$, $2 = \{0, 1 \} = \{ \{\}, \{ \{ \} \} \}$, etc. In general, assuming that $0, 1, ..., n$ are represented as sets, we can just let $n+1 = \{0,1,2,..., n\}$. Now this is intuitively nice for us for two reasons: 

    1. The number of elements in the set "$n+1$" is $n+1$ itself, and 

    2. we tend to think of the natural numbers as comprising the numbers under it. 

So let's make a first attempt at an analogy: God isn't finite, but he contains within himself all finite things. In some sense, then, God is like the set of natural numbers itself. Call this set $\omega = \{0, 1,2, 3, ... \}$.   This analogy does not quite capture, as much as we would like, however, God's infinitude being totally "other", as we can in some sense comprehend the natural numbers as an increasing union of the finite sets.

Now this seems more like in the spirit of what you are trying to get at, so if we stop here with the recognition of the imperfection of analogies, we will be ok. 

Analogy 2.0:  Ordinal numbers and countability

But what if we could mathematically improve the analogy? Let's see how we can do this. The reason I don't like the natural numbers as an end point is that we human beings also contain them within our minds. In other words, "God's infinitude = ω" is too human for me. Note, for one, that we easily consider ω to be another set (in set theory, this assumption is the typical instantiation of what is called the "axiom of infinity", i.e., there exist infinite sets). That, and note that ω itself is comprised of the natural numbers. In some sense, ω behaves a lot like another number, just not a finite one. Then we can extend "numbers" beyond the natural numbers, for example: $\omega+ 1 = \{0,1,2,..., \omega\}$, $\omega+2 = \{0,1,2,..., \omega, \omega+1\}$, or something like $2\omega = \omega \cup \{\omega,\omega+1, \omega+2,...\}$.

We call these numbers "ordinals" in set theory. Using techniques for expanding ordinals (i.e., taking increasing unions like we did to get ω, or getting successors like ω+1), we can give constructions for things like $2\omega$ or $\omega^2$ , or $\omega^\omega$, $\omega^{\omega^\omega}$, and so on. There is also something funny about these ordinals though... it's that you're not really increasing the "size" of the set. By this, I mean that I can create a one-to-one correspondence between ω and, for example, ω+1 (by mapping 0 to ω and n > 0 to n - 1), or mapping ω to 2ω (for example, map even numbers 2n to n, and odd numbers 2n+1 to ω+n). In math jargon, this means that these sets all have the same "cardinality," and having a one-to-one correspondence with ω, or some subset of it, is called being "countable." Intuitively, this means that I can enumerate the elements sequentially as a human,  and assuming I had enough time could eventually reach any number inside my ordinal set that I wanted. So whatever God is in our improved analogy, he isn't countable, because he is beyond our own "infinity". It turns out that in set theory, we have ordinals that are uncountable sets, and we have several sizes of infinities, all surpassed by God.  For instance, the set $P(\omega)$, which is the set of all subsets of $\omega$, is not a countable set.

Analogy 3.0: Sets vs Proper Classes

But of course, we don't want to stop at any such ordinal, because we can increase that ordinal like we did with $\omega$. So how do we address this? It turns out that we can still envisage this incomprehensibility of God's infinity by just letting God's infinitude correspond to the collection of ALL ordinals in our set theoretic "universe". But wouldn't this just be another set subject to the same problem? Here's where it gets interesting. Such a collection is actually not a set, but rather what we call a "proper class". When Cantor was developing set theory, he thought sets were simply collections of things, but this led to some contradictions (see "Russell's paradox"). 

  So naive set theory had to be tweaked by adding something called the Axiom of Foundation.  Essentially, what this means is that you cannot have an infinite chain of relation of reverse inclusion.  Using technical terms, you can't have an infinite sequence $a_1, a_2, a_3, ...$ of sets such that for each natural number $n$, $a_{n+1} \in a_n$.  How does this imply that the ordinals are not a set?  This is proved using two facts: the first is that we can assume the axiom of foundation.  The second is that assuming this axiom we can actually determine that a set $S$ is an ordinal if it satisfies these conditions:

1.  It is transitive: for any elements $a,b,c \in S$ with $a\in b$ and $b\in c$, $a\in c $ as well.
2.  It is has the following trichotomy: for all $a,b \in S$ either $a\in b$, or $b\in a$, or $a = b$.
3. It is well-founded: each non-empty $A\subset S$ has a "smallest" element:  meaning, an element $x$ in $A$ such that every other $y$ in $A$ contains $x$.
   

Note that foundation implies that you can't have an alternating loop of inclusion with $a \in b$ and $b\in a$.  Then you would have an infinite sequence $a,b,a,b,a,b...$ that contradicts foundation.

 It's too complicated to explain how here, but basically, the collections of elements in our set theoretic universe have to be "small" in some sense. You can't, for example, have a "set of all sets." It takes some argumentation, but you also can't have an "ordinal of all ordinals" either.  Essentially, if there were such a set of all ordinals, let's call it $O$, then with some work, you can show that the set $O' = \{O\} \cup O$ is also an ordinal, so $O' \in O$.  This violates the axiom of Foundation, as now $O \in O' \in O$.

Analogy 4.0+: Expanding the set theoretic universe

 So now we have a mathematically cool/ sophisticated way of describing God's infinitude. God's infinitude transcends all categories of human understanding as the class of ordinals transcends the standard set theoretic universe. But I'm going to crash the party one more time to show, also mathematically, why this isn't enough. Set theorists like to expand our typical set theoretic universe by throwing in certain "large cardinals" whose existence can't be proved or disproved using our standard set theoretic axioms. Under such an extended universe, this class of ordinals can become a set. But then we are capturing God in our universe, which means that what we have analogized as God's infinitude ceases to be so.  While in this expanded universe our current class of ordinals is now a set, there class of ordinals in that universe is not a set in that universe.

Analogy ω.0: The futility is the analogy

The moral of the story is that the stronger the analogy becomes, the more elusive his infinitude turns out to be. The cool thing about this though, is that with set theory, we can actually experience for ourselves how this is, this coming closer to God and recognition of our own limitations at the same time.