Okay, I now understand why closure under those operations is incompatible with being well-ordered. And I’m guessing you believe that well-ordering is necessary for a coherent notion of passing through tomorrow infinitely many times because it’s a requirement for transfinite induction?
I’m not so sure that this is important. After all, we can imagine getting from 1 to 2 via passing through an infinite number of infinitesimally small steps even though [1,2] isn’t well-ordered on <. Indeed, this is the central point of Zeno’s paradox.
Yes, there are good ways to index sets other than well orders. A net where the index set is the real line and the function f:I→X is continuous is usually called a path, and these are ubiquitous e.g. in the foundations of algebraic topology.
I guess you could say that I think well-orders are important to the picture at hand “because of transfinite induction” but a simpler way to state the same objection is that “tomorrow” = “the unique least element of the set of days not yet visited”. If tomorrow always exists / is uniquely defined, then we’ve got a well-order. So something about the story has to change if we’re not fitting into the ordinal box.
Okay, I now understand why closure under those operations is incompatible with being well-ordered. And I’m guessing you believe that well-ordering is necessary for a coherent notion of passing through tomorrow infinitely many times because it’s a requirement for transfinite induction?
I’m not so sure that this is important. After all, we can imagine getting from 1 to 2 via passing through an infinite number of infinitesimally small steps even though [1,2] isn’t well-ordered on <. Indeed, this is the central point of Zeno’s paradox.
Yes, there are good ways to index sets other than well orders. A net where the index set is the real line and the function f:I→X is continuous is usually called a path, and these are ubiquitous e.g. in the foundations of algebraic topology.
I guess you could say that I think well-orders are important to the picture at hand “because of transfinite induction” but a simpler way to state the same objection is that “tomorrow” = “the unique least element of the set of days not yet visited”. If tomorrow always exists / is uniquely defined, then we’ve got a well-order. So something about the story has to change if we’re not fitting into the ordinal box.