World + God = God: A Mathematical Analogy

 The Big Henry offers the following comment on my post, World + God = God?

"World + God = God" is (mathematically) analogous to "number + infinity = infinity", where "number" is finite. If God embodies all existence, then God is "existential infinity", and, therefore, no amount of existence can be added to or subtracted from God's totality.

 

The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction. Similarly, if God is presumed to be the embodiment of all existence, He does not comply with the rules of arithmetic addition or subtraction.

To supply an example that supports Big Henry's point, 8 + = .  (aleph-nought, aleph-zero, aleph-null) is the first transfinite cardinal.  A cardinal number answers the How many? question.  Thus the cardinal number of the set {Manny, Moe, Jack} is 3, and the cardinal number of {1, 3, 5, 7} is 4.  Cardinality is a measure of a set's size. What about the infinite set of natural numbers {0, 1, 2, 3, 4 . . . n, n + 1, . . .}?  How many?  .  And as was known long before Georg Cantor, it is possible to have two infinite sets, call them E and N such that E is a proper subset of N, but both E and N have the same size or cardinality.  Thus the evens are a proper subset of the naturals, but there are just as many of the former as there are of the latter, namely, .  How can this be?  Well, EACH element of the evens can be put into 1-1 correspondence with an element of the naturals.

 

So far the analogy holds.  But I think Big Henry has overlooked the transfinite ordinals.  The first transfinite ordinal, denoted , is the order type of the set of nonnegative integers.  (See here.)  You could think of as the successor of the natural numbers.  It is the first number following the entire infinite sequence of natural numbers. (Dauben, 97)  The successor of    is  + 1.  These two numbers are therefore different.  Here the analogy breaks down.  God + Socrates = God.  + 1 is not equal to .

 

Moreover, it is not true to say that "The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction."  This ignores the rules of transfinite cardinal arithmetic and those of transfinite ordinal arithmetic.  Big Henry seems to be operating with a pre-Cantorian notion of infinity.  Since Cantor we have an exact mathematics of infinity.

 

In any case,  I rather doubt that mathematical infinity provides a good analogy for the divine infinity.  God is not a set!

Comments

[Kevin Kim]

I don't know what Henry had in mind when he made his analogy, but it sounds at least somewhat consistent with something I'd read, once, in a textbook on philosophy of religion. In the textbook, the metaphor of an infinite bookshelf is used: imagine a bookshelf that stretches forever to your left and to your right. On this shelf is an infinity of books that alternate between red and black covers, i.e., every other book is black, and every other book is red. That's the setup for the thought experiment.

If I pluck a single book, of either color, from this shelf, how many books remain on the shelf? An infinity of books! In that sense, perhaps we can say that "infinity minus one is still infinity." If I were somehow able to remove all the red books from the shelf, how many books would I remove? Why, an infinity of books. And how many books would still be on the shelf? An infinity! So it may be legitimate to say that, at least in this case, "infinity minus infinity equals infinity." Were I to add an infinity of green books to this shelf, such that the books now alternated "black-red-green, black-red-green, etc.," how many books would be on the shelf? An infinity! So perhaps "infinity plus infinity equals infinity," too.

I don't know, but perhaps it's to this additive oddness that Henry is referring, and as for "God is not a set," well... no matter what analogy we try to use when talking about God, it's a safe bet that, in any "God:X" analogy, God is not an X.

[TheBigHenry]

"God is not a set!"
___

Bill,

I am reminded of Niels Bohr's famous reply (when Big Al Einstein said, "God does not play dice."): "Einstein, stop telling God what to do".

https://en.wikipedia.org/wiki/Solvay_Conference#Fifth_Conference

[TheBigHenry]

Kevin,

That is essentially the infinity I had in mind when I offered my analogy for Bill's consideration. And, indeed, Bill is correct in noting that I had not considered Cantor's work on the concept of infinity. Bill's claim that, "Since Cantor we have an exact mathematics of infinity", however, is questionable. As the greatest mathematician of all time (Johann Carl Friedrich Gauss) noted, "Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics".

The analogy I suggested was not meant to imply that God is humanly conceivable by any analogy. From the only philosophy course I ever took as an undergrad (philosophy of religion) I remember that God's attributes are deemed to be "wholly other". That is to say (to use another mathematical analogy) if human attributes are real numbers, then God's are complex numbers.

Finally, in the spirit of Bohr's response to Einstein, if God wants to be a "set" or any "X" you care to mention, God can be whatever He wants to be.

[Kevin Kim]

Henry,

"God can be whatever He wants to be."

Fascinating theological claim. My inner Sunday-school student wants to accept this unquestioningly because, after all, God can do anything.

But can He really?

What if God says to Himself, in the spirit of Rachel Dolezal's claiming to be black, "You know... I want to be a deer. And you know what else? I think I am a deer! So be it!" So God transforms Himself into a deer, and does so in such a thorough, complete way that He is now wholly a deer, i.e., an animal with no deific attributes at all, which further means that God now lacks the ability to turn Himself back into the God of all creation. At this point, having only a deer's powers of cogitation, God is no longer in a position to say about Himself, "Well, shit... now I'm a deer. What the hell do I do?" Even that thought is beyond Him.

If God's omnipotence includes the ability to become something less-than-God so completely that God loses His God-ness, then the claim that "God can be whatever He wants to be" can be true only once. Otherwise, if God turns into a deer but retains the deific power to revert back to being fully God, then God has not truly become a deer in full: He's retained an ontological escape clause.

I realize you were just being playful with your Bohr/Einstein remark, but I saw an opportunity to engage in some theological whimsy. Apologies.

[TheBigHenry]

Kevin,

Far be it from me to make claims (theological or otherwise) for which I am not certified. I was indeed being playful.

Nevertheless, your "deer" connundrum sounds similar to the old "If God is omnipotent, can He make a rock so heavy that even He can't lift it?" Frankly, I think He probably can, albeit knowing how is way beyond my paygrade.

[spear]

Hi Bill,

I have a brief comment regarding the conditional near the beginning of TheBigHenry's statement. TheBigHenry says, "If God embodies all existence, then God is "existential infinity"..."

That's not quite right, Henry. It assumes that existence is infinite, but it might not be. But, if it's possible that existence might not be infinite, then it would seem that, even if God embodies all existence, God might not be "existential infinity".