Brief comment as costs are high:
"As it turns out, the definition of the number one is the set of all singletons, where a singleton is defined to be a set consisting of a single element. The number two is likewise defined to be the set of all pairs. Three is the set of all triples, and so on."
May I submit that this overlooks something.
"1" is a shape on a page, a sound in the air; a label, a name. It is a match of the label to the object labelled. The object could be a car or a bus.
"2" is another shape on a page, another sound in the air; another label, another name. It represents a compulsory match between objects which are grouped and called "2". They could be very different, one a car, one a bus.
The car and bus here are forcibly labelled as "1" + "1".
The problem here is that "number" seems to involve coercion; compulsory association.
Think of the parable of the vineyard and the workers: the worker arriving early agreed to one denari or something for a day's work; the worker arriving in the middle of the day agreed to one denari for his work; the worker arriving in the afternoon agreed to one denari for his work.
But at the end of the day when they were paid, there were complaints. That is not surprising one might say; but what is Jesus Christ telling us when he tells us parables about the "Kingdom of Heaven" and tells us that "My ways are not your ways"?
I figured this much out: If I want to ski a particular run and there I am on the slope ready to go: do I need a bilion dollars? No, I need only what I have. If each of us in living internally consistent and creating the life he truly wants to live, he will not need to worry about differences with others as he will be O.K. with what he has.
When you get to the number "3"; another thing happens:
The order in which the labelled objects appear is not specified.
So may I submit the following omissions from your description of number:
The notion of compulsory association (it looks like it doesn't it? Have I missed something?)
The fudging of the order: order is not specified
for the "ones" inside a large number (or the "twos" and so on.
The fudging of the question of the spaces between the number-labels: you may place several dividing walls along a long oblong box; but that does not say where-abouts along that oblong box those walls are placed.
The other thing is: I learned from a primary school math book that a number does not specify how you got to it (56 can be 478 - 422, or 50 + 6).
I have discovered that Dr. Stafford is correct that physics can be described (to some extent anyway) as a cataloguing system. I maintain that by dealing with "sets of numbers" he has (perhaps not realising very much?) to some extent broken down the assumptions built into "number" that I listed above.
There are may examples in the Gospels of issues regarding "number" that teach us much, I think.