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God in Set Theory
Sep. 27th, 2026 09:13 amI have 3 younger brothers, and when one of us has a “round” birthday, we all get together with our families to celebrate. What we do mostly is talk. (There used to be a great deal of drinking involved, but now it’s mostly just talk.) Many years ago we gathered in Philadelphia to celebrate Rudy’s 50th. My brother Gaylord was teaching Linguistics and Russian at the time, so he and I had some interesting discussions about the areas of overlap between linguistics and my fields of Set Theory and Mathematical Logic, e.g. context free grammars, Turing machines, etc.
Gaylord and I had just returned to the church after having turned our backs on God for 30 years or so. While I returned to the faith of my childhood and landed at First Pres, Gaylord somehow ended up with the Anglicans. So it was that on the Sunday of that weekend in 1995 we went to Christ Episcopal Church, spiritual home to many of our country’s founding fathers. The gospel that morning was the parable you know as the Parable of the Prodigal Son, which we first-borns call the Parable of the Unappreciated Elder Brother.
As I sat there listening again to the story of how the younger, irresponsible sibling inveigles his “inheritance” from his father, then blows it all on booze and broads, only to be welcomed home like a prince, while the responsible older sibling gets nothing more than an attaboy and an invitation to the party in honor of his wastrel brother, I could feel my forehead furrowing with puzzlement and displeasure. It just wasn’t fair!
Then a remarkable association occurred. Probably because of my conversations with my brother, I recalled gazing at a room full of such furrowed foreheads, almost 30 years earlier. I was teaching a class on logic and set theory in a summer program for high school teachers at RPI. I had just received my Masters degree and my professor, who was scheduled to teach the class, got a research grant at the last minute and asked me to sub for him.
Now, set theory starts out innocently enough. A set is any well-defined collection of objects, like you, me, and the square root of 3, although mathematicians of course focus on sets of mathematical objects, like the set A containing the integers 1, 3, 5, and 7, written
A = {1,3,5,7}
and the set B containing the integers 2, 3, 4, 5, and 6, written
B = {2,3,4,5,6}
I introduced to the class the concepts of the union of two sets – the set containing all members that are in one or the other of the two:
A U B = {1,2,3,4,5,6,7}
and the intersection of two sets – the set containing just those members that are in both of them:
A ∩ B = {3,5}
The classroom was all smiles. These folks were mostly in Troy for the month with their families, and they were thinking, “beach – I think we’ll go to the beach this afternoon.”
We discussed the concept of the cardinality or size of a set, and the related question of determining if two sets are the same size. For sets like A and B above, this is easy: we count one; then we count the other; then compare counts. In our case, A has four members, B has 5, so B is larger than A. Smiles still reigned.
But the really interesting sets, like the set N of all positive integers, or the set Q of all rational numbers (proper or improper fractions of integers) have more than a finite number of members. If we had to count their members to compare them, we could never finish counting the first one, we could never even begin counting the second one, and we would never get an answer. So mathematicians found another way to compare the sizes of two sets We say that two sets are the same size if there is a correspondence between each member of one with a unique member of the other and vice versa. In the case of A and B, we see that there is an extra member of B whenever we try to do this; they are not the same size; B is larger than A.
But what about these two sets
N = {1,2,3,4,5,. . .} and
C = {2,3,4,5,6,. . .}
Every member of C is a member of N, but N has a member (1) that is not in C. Is N larger than C?
No, because a correspondence between the two can be created:
c = n + 1 and n = c – 1
that ties each member c of C to a unique member n of N and vice versa. They are the same size.
How about these two:
N = {1,2,3,4,5,. . .} and
D = {5281,5282,5283,5284, 5285,. . .}
Since d = n + 5280 and n = d – 5280, they are the same size even though there are 5280 members of N that are not in D.
In fact for any number k, no matter how ridiculously large, the sets
N = {1,2,3,4,5,. . .} and
E = {k+1, k+2, k+3, k+4, k+5,. . .}
are the same size, even though there are k members of E that are not in N. At this point all the smiles in the classroom were gone.
We continued, how about
N = {1,2,3,4,5,. . .} and
F = {2,4,6,8,10,. . .}
where every member of F is a member of N, but there are an infinite number of members of N that are not in F?
It’s easy to see that
n = f / 2 and f = n * 2
assigns a unique member of F to each member of N and vice versa, so here too, both sets are the same size.
By now, furrows of puzzlement and displeasure had appeared on many a brow. One clever guy in the back row tried to save the day by suggesting that all sets with an infinite number of members are the same size, causing hopeful expressions to appear on a few faces. But then, using the famous diagonal argument of Georg Cantor, we constructed an infinite set that was larger than N, and furrowed brows were everywhere.
Now what does all of this have to do with Matthew 20? To the workers who were hired first, and to us in our egalitarian culture, paying the same wage to those who worked all day as to those who only worked one hour is wildly unfair. The former worked much harder and their hourly wage is less than one tenth of the latter. Our concept of justice tells us that this is not right. Yet there is no ambiguity in the way Jesus tells the story; He intends to challenge us. Why?
Almost everyone interprets the parable as describing the relationship between God (the landowner) and us (the workers). Some scholars say that the message of the parable is straightforward: all who find their way to God are welcomed regardless of how long it takes them. Although this is a reasonable interpretation, I doubt that it removes the challenge posed by the story to a Christian who has always has led a good life when she considers the “death-bed conversion” of someone who certainly has not.
Other scholars say that the point of the parable is that we receive not what we think we deserve, but that we all receive salvation. Although this is headed in the right direction as we shall see, it is essentially the same as the first explanation, and just as challenging to the faithful Christian, I think. (I wouldn’t know, since I am one of the eleventh-hour workers.)
I think the parable is challenging because we interpret it in terms of justice or fairness, but that is not what it is about. I believe it is about love.
I think that our unease at the unequal wages paid to the workers has the same cause as the unease of my Set Theory students. All the sets they or we have ever encountered in our daily lives have been finite. Our understanding of how sets behave is based entirely on our experience with finite sets, and so our intuition tells us that all sets behave like finite sets. When we find that infinite sets behave very differently, we are confused and troubled. Jesus is telling us in Matthew 20 that our understanding of love, based entirely on our experience with fallible, inconstant, finite human love, is inadequate for comprehending the love of God. He is telling us that our understanding of love and of God is too small. This is still challenging, but in an entirely different way: Instead measuring the landowner’s (God’s) actions against the workers’ (our) expectations, we assess our expectations against the reality of a God who is infinite in time and space and whose love cannot be measured or constrained.
The problem with all parables is that they inevitably shrink God to a humanly-comprehensible scale. But the creator of all that is, is not of human scale, nor can we place our human limits on God’s boundless love.
Thanks be to God!
Gaylord and I had just returned to the church after having turned our backs on God for 30 years or so. While I returned to the faith of my childhood and landed at First Pres, Gaylord somehow ended up with the Anglicans. So it was that on the Sunday of that weekend in 1995 we went to Christ Episcopal Church, spiritual home to many of our country’s founding fathers. The gospel that morning was the parable you know as the Parable of the Prodigal Son, which we first-borns call the Parable of the Unappreciated Elder Brother.
As I sat there listening again to the story of how the younger, irresponsible sibling inveigles his “inheritance” from his father, then blows it all on booze and broads, only to be welcomed home like a prince, while the responsible older sibling gets nothing more than an attaboy and an invitation to the party in honor of his wastrel brother, I could feel my forehead furrowing with puzzlement and displeasure. It just wasn’t fair!
Then a remarkable association occurred. Probably because of my conversations with my brother, I recalled gazing at a room full of such furrowed foreheads, almost 30 years earlier. I was teaching a class on logic and set theory in a summer program for high school teachers at RPI. I had just received my Masters degree and my professor, who was scheduled to teach the class, got a research grant at the last minute and asked me to sub for him.
Now, set theory starts out innocently enough. A set is any well-defined collection of objects, like you, me, and the square root of 3, although mathematicians of course focus on sets of mathematical objects, like the set A containing the integers 1, 3, 5, and 7, written
A = {1,3,5,7}
and the set B containing the integers 2, 3, 4, 5, and 6, written
B = {2,3,4,5,6}
I introduced to the class the concepts of the union of two sets – the set containing all members that are in one or the other of the two:
A U B = {1,2,3,4,5,6,7}
and the intersection of two sets – the set containing just those members that are in both of them:
A ∩ B = {3,5}
The classroom was all smiles. These folks were mostly in Troy for the month with their families, and they were thinking, “beach – I think we’ll go to the beach this afternoon.”
We discussed the concept of the cardinality or size of a set, and the related question of determining if two sets are the same size. For sets like A and B above, this is easy: we count one; then we count the other; then compare counts. In our case, A has four members, B has 5, so B is larger than A. Smiles still reigned.
But the really interesting sets, like the set N of all positive integers, or the set Q of all rational numbers (proper or improper fractions of integers) have more than a finite number of members. If we had to count their members to compare them, we could never finish counting the first one, we could never even begin counting the second one, and we would never get an answer. So mathematicians found another way to compare the sizes of two sets We say that two sets are the same size if there is a correspondence between each member of one with a unique member of the other and vice versa. In the case of A and B, we see that there is an extra member of B whenever we try to do this; they are not the same size; B is larger than A.
But what about these two sets
N = {1,2,3,4,5,. . .} and
C = {2,3,4,5,6,. . .}
Every member of C is a member of N, but N has a member (1) that is not in C. Is N larger than C?
No, because a correspondence between the two can be created:
c = n + 1 and n = c – 1
that ties each member c of C to a unique member n of N and vice versa. They are the same size.
How about these two:
N = {1,2,3,4,5,. . .} and
D = {5281,5282,5283,5284, 5285,. . .}
Since d = n + 5280 and n = d – 5280, they are the same size even though there are 5280 members of N that are not in D.
In fact for any number k, no matter how ridiculously large, the sets
N = {1,2,3,4,5,. . .} and
E = {k+1, k+2, k+3, k+4, k+5,. . .}
are the same size, even though there are k members of E that are not in N. At this point all the smiles in the classroom were gone.
We continued, how about
N = {1,2,3,4,5,. . .} and
F = {2,4,6,8,10,. . .}
where every member of F is a member of N, but there are an infinite number of members of N that are not in F?
It’s easy to see that
n = f / 2 and f = n * 2
assigns a unique member of F to each member of N and vice versa, so here too, both sets are the same size.
By now, furrows of puzzlement and displeasure had appeared on many a brow. One clever guy in the back row tried to save the day by suggesting that all sets with an infinite number of members are the same size, causing hopeful expressions to appear on a few faces. But then, using the famous diagonal argument of Georg Cantor, we constructed an infinite set that was larger than N, and furrowed brows were everywhere.
Now what does all of this have to do with Matthew 20? To the workers who were hired first, and to us in our egalitarian culture, paying the same wage to those who worked all day as to those who only worked one hour is wildly unfair. The former worked much harder and their hourly wage is less than one tenth of the latter. Our concept of justice tells us that this is not right. Yet there is no ambiguity in the way Jesus tells the story; He intends to challenge us. Why?
Almost everyone interprets the parable as describing the relationship between God (the landowner) and us (the workers). Some scholars say that the message of the parable is straightforward: all who find their way to God are welcomed regardless of how long it takes them. Although this is a reasonable interpretation, I doubt that it removes the challenge posed by the story to a Christian who has always has led a good life when she considers the “death-bed conversion” of someone who certainly has not.
Other scholars say that the point of the parable is that we receive not what we think we deserve, but that we all receive salvation. Although this is headed in the right direction as we shall see, it is essentially the same as the first explanation, and just as challenging to the faithful Christian, I think. (I wouldn’t know, since I am one of the eleventh-hour workers.)
I think the parable is challenging because we interpret it in terms of justice or fairness, but that is not what it is about. I believe it is about love.
I think that our unease at the unequal wages paid to the workers has the same cause as the unease of my Set Theory students. All the sets they or we have ever encountered in our daily lives have been finite. Our understanding of how sets behave is based entirely on our experience with finite sets, and so our intuition tells us that all sets behave like finite sets. When we find that infinite sets behave very differently, we are confused and troubled. Jesus is telling us in Matthew 20 that our understanding of love, based entirely on our experience with fallible, inconstant, finite human love, is inadequate for comprehending the love of God. He is telling us that our understanding of love and of God is too small. This is still challenging, but in an entirely different way: Instead measuring the landowner’s (God’s) actions against the workers’ (our) expectations, we assess our expectations against the reality of a God who is infinite in time and space and whose love cannot be measured or constrained.
The problem with all parables is that they inevitably shrink God to a humanly-comprehensible scale. But the creator of all that is, is not of human scale, nor can we place our human limits on God’s boundless love.
Thanks be to God!
