Wednesday, March 23, 2022

An analogy for divine infinity

Here’s an analogy I’ve been thinking about. God’s value is related to other infinities like (except with a reversal of order) zero is related other infinitesimals. Just as zero is infinitely many times smaller than any other infinitesimal (technically, zero is an infinitesimal—an infinitesimal being a quantity x such that |x| < 1/n for every natural number n), and in an important sense is radically different from them, so too the infinity of God’s value is infinitely many times greater than any other infinity, and in an important sense is radically different from them.

Suppose we think with the medievals that value and being are correlative. Then zero value corresponds to complete non-being. There isn’t anything that has that. Between ordinary non-divine things like people and oak trees and non-being we have a radical ontological difference: there are people and oak trees, but there is no non-being. Suppose we push the analogy on the side of God. Then between ordinary non-divine things like people and oak trees and God we will have a radical ontological difference, too. Some theologians have infamously tried to mark this difference by saying that people and oak trees are but God is not. That way of marking the difference is misleading by making God seem like non-being instead of like its opposite. A better way to mark the difference is to say that in an important sense God is and people and oak trees are not (compare what Jesus is said to have have said to St Catherine of Siena: “I am who I am and you are she who is not”). In any case, the gap between God’s “is” and our “is” is at least as radical as the gap between our “is” and the “is not” of non-being.

In fact, I think the gap is more radical: we and all other creatures are closer to non-being than to God. So the analogy I’ve been thinking about, that God’s value is related to other infinities like zero to other infinitesimals (but in reverse order) is misleading: God’s value is in a sense further from other infinities than zero is from other infinitesimals. (And not just because all infinitesimals are infinitesimally close to zero. The relevant scale should not be arithmetic but logarithmic, so that the gap between zero and anything—even an infinitesimal—bigger than zero is in an important sense infinite.)

Don’t take this too seriously. Remember this.


Wesley C. said...

1) Is this a potential answer to how God may relate to large cardinal axioms? Whether or not the LCA hierarchy is infinite (no largest infinity, even the universe V of all sets can be taken as a sort of hyperinfinite number that can be expanded on forever with greater models, and if you complain that there is no set X such that X = V, just remember the finitists also say the same thing about "w") or if it has an upper ceiling beyond which contradictions arise and so there is a greatest LCA, God is uniquely absolutely infinite in a sense beyond which it is impossible to go, just like you can't go beneath zero...or can you?

Couldn't someone come up with small cardinal axioms and make it such that they are smaller than zero? Or even to take the existence of negative numbers and say there is something truly less than zero, so there may be something greater than God?

2) I'd just disagree with the idea that God's infinity means everything else is just worthless or literally nothingness - there is a sort of ontological gradation or intermediary when it comes to being, in such a way that God's radical difference is preserved yet without making created beings illusory.

Alexander R Pruss said...


1. Yeah, I was initially going to write it as a response to that comment of yours (though I didn't think up the idea in way of response to that). Of course, you can talk about negative numbers. But there is no such thing as "negative being". And on a privation theory of evil, there is no possibility of negative value.

2. I would say that *compared to God* everything else is nothing. The move in being from us to God is greater than the move from nothing to us.

BTW, I love the Jewish joke on this stuff. The rabbi in a sermon exclaims in ecstasy "I am nothing!" The cantor looks raptured, too, and calls out "I am nothing!" Then Shmul the begger calls out "I am nothing!" And the people around Shmul say: "Look who thinks he's nothing!"

I certainly don't want to say that we are literally nothing. But we are more like nothing than like God.

Alexander R Pruss said...

Related to all this is also Cantor's idea that there is an absolute infinity above all the cardinals, an absolute infinity that somehow escapes Cantor's theorem (which says that for every set there is a larger set). I have heard that Cantor wanted somehow to identify the absolute infinity with God. I think this is a mistake: if there were such a mathematical object as absolute infinity, God would be beyond that, too.

Wesley C. said...

1) So I guess this solves the concern I've had for years as to how God's infinity compares to LCA's. There's still one more site which I find interesting where a person tries to show that you can always expand in math in such a way that there is no highest infinity or model-universe:

"Those same adepts have enriched Cantor's paradise with a great bestiary of enormous cardinals, inaccessibles, Mahlo, Vopenka, Woodin cardinals, etc. Big fellows, no doubt. Yet... In comparison with the size of V they are puny, nil in fact, no more no less as Graham number, or Friedman's TREE(3) stand in comparison to (for finitists) almighty ω0."

"Now, let us be brave and say: what about breaking through into the trans-transfinite? What about , for instance, starting from V itself and state that its size is some hyperinfinte number, say ℵ0,1 ?"

"(SIDE NOTE ON NOTATION: The standard aleph series would now be ℵ0,0 , ℵ1,0, .... The second subindex controls the degree of hyperfiniteness, much like degrees of unsolvability. I could have put it on top, but then it would cause troubles with cardinal exponentiations )."

"Wait, I hear you say loud and clear. Are you crazy? Don't you know that there is NO SET X such that X=V? Don't you know that there is no max ordinal? Yes, ladies and gentlemen, I do know it. But I do reply: and so what? The objection is exactly the same as the one of the finitists vis-a'-vis ω. Someone has broken through the finite, so why not the transfinite? There is no set, but who said that it must be a set?"

"In fact, start with a pairs of transitive countable models of ZFC, M0 and M1, with M0≤M1, of different tallness (the ordinal height of the first being strictly smaller than the height of the second). From the point of view of M0, IT is the full universe of sets, and the ideal ordinals of M1 some unimaginable higher level of infinity. Of course, say you, M0 does not see M1. True, but we do. And -I think- nothing prevents us from formalizing their reciprocal relation as some new theory of sets (the elements of M0) and classes (the elements of M1). Note that here all sets are classes, but not viceversa."

"Also, being more reckless, we could generalize the above by stipulating an entire chain of ascending hyper-infinities, and perhaps enrich ZFC with an axiom that says that for each model there is a cofinal (in V) ascending chain of taller models, the Cofinal Tallness Axiom...."

What do you think of the above proposal? Would it's implications have any impact on God's infinity?

Wesley C. said...

2) Agree about Cantor's absolute infinity not being God but a mathematical object. In fact, I think I saw somewhere that it's been proven that either the Absolute Infinity can't exist due to coherency reasons, or that what Cantor's intuitions were actually pointing towards is the universe V, which doesn't have the other attributes Cantor wanted.

3) As for comparisons to God, I think there are some noteworthy cautions about this, as God even in His divine nature can be said to enjoy creation and appreciate it, and even became a human.The whole relation between creation and God is in some senses akin to marriage, as the bridegroom (God) has reasons to appreciate and value the bride (creation); this is also how parents broadly relate to children. The active cherishes the passive which is dependent on the active.

Another thing coming to mind is a video by Jonathan Pageau who points out the differences between Christian philosophy and other eastern / New Age philosophies whereby created things are seen as illusions or even as bad nothings because God is the absolute, as if the absolute somehow devalues the finite by comparison in such a way. Couldn't find it earlier, so if I do I'll post it here.

Walter Van den Acker said...

There no non-being,so it doesn't make Sense to say we are 'likr' something that is impossible.
We are not more like square circles than like God.

Andrew Dabrowski said...

"I have heard that Cantor wanted somehow to identify the absolute infinity with God. I think this is a mistake: if there were such a mathematical object as absolute infinity, God would be beyond that, too. "

Clearly you can't identify absolute infinity with God. But as Wesley C. pointed out, the intention may be rather to analogize God to V, in that the order of V is ON.

In a similar way, one might think about applying an ordinal to every object according to the "day" of creation on which became possible for it to be created.

RunDec said...

I think I agree with Walter on this one, at least to an extent.
The difference between literally *nothing* and being strikes me as, in a way, greater and more radical than that between us and God, or maybe anything. Just like the difference between a square circle and an actual square.
Nothing is literally no-thing. In some way, creatures and God share an analogy of being. There is, however, no such analogy between NOTHING and creatures, or finite being.
Sure, the difference between us and God is radical. But not greater than between us and... nothing. We have an analogy with God, and a real relation. There's no such a thing with nothingness and anything else. And as Walter puts it, it makes no sense to say we are more like nothing.

I think, Pruss, that your point would make more sense if you spoke of *possibilia* or *potentialities* instead of nothingness. You could say that we are more like possibilities than like God. A possible, non-actual human might at least some kind of potentiality or idea. We are radically different from mere potentialities. But we might be more like them than we are like God (indeed, I think Aquinas would probably think so, given that we are all potentialities-being-actualized, whereas God just is pure actuality)