The Theology of Sets

“It’s turtles all the way down.” That’s how one audience member explained the universe to a shame faced Stephen Hawking after Hawking had spent a lifetime searching fruitlessly for a unified theory to explain the phenomena of our world. What a waste! Someone had already come up with the solution: “It’s turtles!” 

And if not turtles, then what? The 20th century grounded logic and mathematics in Set Theory (ZFC). So perhaps it’s sets all the way down, not turtles.

The intellectual history of the West can be seen as one massive attempt to define the ‘fundamental substance’. The Ionians narrowed it down to earth, fire, air, and water. The Eleatics called it Aletheia (Truth), the Christians, Agape (Love), the Positivists, Aether, the Quantum Mechanics, Fields. Pop culture calls it Energy. I guess Sets is just as good…or Turtles. 

Set theory captures the scale invariance of Universe: again, turtles all the way down; but better than turtles, sets make a stab at modeling the interactivity of things. Every set is a collection of elements (or members), which are themselves sets (collections of elements) and so on, ad infinitum (sets, all the way down). An element of one set can be an element of another set; one set can be an element of another set. Sets of sets!

There are sets with just one member; there’s even a set with no members (called the ‘null set’ and commonly represented by the symbol, Ø). Counter intuitively, Ø is a subset of every other set! Everything that is is defined in part by what is not.

So, according to this theory, nothing is ‘something’…sort of. Even though the null set contributes no members to the Great Loom of Being, it contributes its own emptiness. That emptiness (what is not) helps define what is. 

First, think of the universe as a hierarchy of Russian Nesting Dolls. But things get a great deal more interesting when we look at overlapping (intersecting) sets, i.e., sets with some, but not all, of their members in common. Such sets (i.e., sets with common members) give the world its solidarity. They provide the scaffolding for the various phenomena we call Universe.  

Parenthetically, this accounts in part for the non-linearity of Being (why there are lumps in your  mashed potatoes). The becoming of X is accelerated by its not being Y. Everything is what it is in part because of what it is not. 

Is there such a thing as a set of all sets? No – at least not according to the dominant model (ZFC)! Bertrand Russell showed that allowing a ‘set of all sets’ leads to paradoxes, even contradictions, and mathematics abhors contradictions. But ZFC is ‘limiting’ (if not strictly speaking ‘wrong’). Contradictions be damned! “A foolish consistency is the hobgoblin of little minds…” (Emerson) Cosmology demands that there be a Set of All Sets, even when Logic says that there can’t be. But news flash: it is not the job of cosmology to conform to the strictures of logic; it is the job of logic to account for the phenomena of cosmology. 

Sidebar: Since the ‘dawn’ of the so-called Enlightenment (not), intellectual history has pursued a more or less uninterrupted search for a God-free universe. Frankly, it came as a breath of fresh air after the ‘revelation bound’ cosmologies of the Middle Ages. It triggered major conceptual advances such as Evolutionary Science and Big Bang Theory; but fundamentally, it’s wrong!

Excluding by postulating the possibility of a ‘set of all sets’ goes a long way toward excluding by postulating the existence of God. How 20th century of you! Any interesting notion of God, IMO, includes the idea that all beings roll-up to one Being (aka, God). Genesis describes the Alpha point of this process; Paul defines the Omega point in his first Letter to Corinthians: “Then comes the end…when everything is subjected to him (Father)…so that God may be all in all.” (15: 24 – 28)

For purposes of this article, let’s throw orthodoxy to the winds and consider the possible characteristics of the forbidden uber-set, U, the set of all sets. For one thing, the set of all sets would have to be a member of itself. Such sets are relatively rare, but they do exist, for example: The set of all mathematical objects is itself a mathematical object.

It seems arbitrary to say that the set of all mathematical objects is a mathematical object while saying that the set of all sets is not a set. If we decide to allow the “set of all sets”, then we are immediately confronted with the phenomenon of Incarnation. Now, the set of all sets, U, must be a member of itself by definition. 

In fact, theology modeled on this expanded theory of sets is inherently Trinitarian: (1) The set of all sets, U, is a member of the set of all sets; (2) the set of all sets, U, is the aggregate of its members, including itself; (3) the set of all sets, U, transcends as unity what would otherwise exist only as multiplicity (i.e., the aggregation of all sets). 

Mapping the above onto traditional Christian Trinitarianism, (1) evokes ‘the Son’, (2) ‘the Spirit’, (3) ‘the Father’. Set theory can be mapped as ‘a set’ of vectors: (1) ↓, (2) ↔, (3) ↑, each suggestive of the unique identity of a member of the Holy Trinity.

So, set theory (ZFC plus U) is theology: and cosmology? It’s God, all the way down!

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