Here is the question:
What's the difference between infinite divisibility and infinite extension?
Since we are talking about "divisibility" we need an algebraic structure that has order and a division operation defined. One such structures is a field. Here is how Wikipedia defines a field:
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as the corresponding operations on rational and real numbers do.
The rational and real fields are also totally ordered. Here is Wikipedia's definition:
In mathematics, a linear order, total order, simple order, or (non-strict) ordering is a binary relation on some set X, which is antisymmetric, transitive, and a connex relation.
That is, given any two elements from one of these domains there are only two possibilities:
- They are the same.
- One is bigger than the other.
So take two different elements from the domain of rational or real numbers and subtract the smaller from the larger. Call this a line segment. The length of the line segment is another rational or real number. We will use this length to show infinite divisibility.
Since the domain is a field, we can divide that line segment by 2, a number in both the rationals and reals. Add that result to the smaller number to get another rational or real number which is different from the smaller one. Those two numbers are the endpoints of a smaller line segment.
We can repeat this process indefinitely. How do we know this? Assume we cannot. Then we have reached, after some finite process, a line segment with two end points that are different. The assumption is we cannot take those two different numbers, subtract them and divide the result by 2, but that would violate the fact that they are members of an algebraic field in which we can always subtract them and given any member of the domain divide it by 2. Given that the assumption that we cannot indefinitely divide the line segment leads to a contradiction, we assert that we actually can indefinitely divide the line segment.
Define "indefinitely" as "infinitely" and one has infinite divisibility.
Infinite extension doesn't require an algebraic field. We don't need divisibility although we do need addition. Given a member of a domain we should always be able to add 1 to that member indefinitely. The algebraic structures with such properties are called ordered rings. Here is Wikipedia's definition of it:
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.
Take a number from the ordered ring and add 1 to it. This gives us a new number. One can indefinitely repeat that process as well.
That covers the description of infinite divisibility and infinite extension for algebraic structures. The OP has a more basic question about time:
I understand that most people would say that time, such as the time it takes to write this question, is infinitely divisible. But I'm guessing it's unusual to suppose that makes time infinitely extendable, at least as it is happening.
There are different ways to look at time. See Bradley Dowden's survey of the subject. The basic question is whether time can be modeled using fields and rings such as the rational or real numbers. If it cannot then the definitions of infinite divisibility and infinite extension given by those algebraic structures would not be relevant to time. If it can, then the fact that an algebraic field is also a ring would imply that if time is infinitely divisible, then it is also infinitely extendable.
Bradley Dowden, "Time" Internet Encyclopedia of Philosophy https://www.iep.utm.edu/time/#H7
"Field (mathematics)" Wikipedia https://en.wikipedia.org/wiki/Field_(mathematics)
"Ring (mathematics)" Wikipedia https://en.wikipedia.org/wiki/Ring_(mathematics)
"Total order" Wikipedia