Thanks for the suggestion. I took a look at nets, but their purpose seems mainly directed towards generalising limits to topological spaces, rather than adding extra nuance to what it means for a sequence to have infinite length. But perhaps you could clarify why you think that they are relevant?
A net is just a function f:I→X where I is an ordered index set. For limits in general topological spaces, I might be pretty nasty, but in your case, you would want I to be some totally-ordered subset of the surreals. For example, in the trump paradox, you probably want I to:
include n and n/3 for some infinite n
have a least element (the first day)
It sounds like you also want some coherent notion of “tomorrow” at each day, so that you can get through all the days by passing from today to tomorrow infinitely many times. But this is equivalent to having your set I be well-ordered, which is incompatible with the property “closed under division and subtraction by finite integers”. So you should clarify which of these properties you want.
“But this is equivalent to having your set be well-ordered, which is incompatible with the property “closed under division and subtraction by finite integers”″ - Why is this incompatible?
An ordered set is well-ordered iff every subset has a unique least element. If your set is closed under subtraction, you get infinite descending sequences such as 1>0>−1>−2>⋯ . If your sequence is closed under division, you get infinite descending sequences that are furthermore bounded such as 1>12>14>⋯>0. It should be clear that the two linear orders I described are not well-orders.
A small order theory fact that is not totally on-topic but may help you gather intuition:
Every countable ordinal embeds into the reals but no uncountable ordinal does.
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.
A well-order has a least element in all non-empty subsets, and 1 > 1⁄2 > 1⁄4 > … > 0 has a non-empty subset without a least element, so it’s not a well-order.
In general, every suborder of a well-order is well-ordered. In a word, the property of “being a well-order” is hereditary. (compare: every subset of a finite set is finite)
Thanks for the suggestion. I took a look at nets, but their purpose seems mainly directed towards generalising limits to topological spaces, rather than adding extra nuance to what it means for a sequence to have infinite length. But perhaps you could clarify why you think that they are relevant?
A net is just a function f:I→X where I is an ordered index set. For limits in general topological spaces, I might be pretty nasty, but in your case, you would want I to be some totally-ordered subset of the surreals. For example, in the trump paradox, you probably want I to:
It sounds like you also want some coherent notion of “tomorrow” at each day, so that you can get through all the days by passing from today to tomorrow infinitely many times. But this is equivalent to having your set I be well-ordered, which is incompatible with the property “closed under division and subtraction by finite integers”. So you should clarify which of these properties you want.
“But this is equivalent to having your set be well-ordered, which is incompatible with the property “closed under division and subtraction by finite integers”″ - Why is this incompatible?
An ordered set is well-ordered iff every subset has a unique least element. If your set is closed under subtraction, you get infinite descending sequences such as 1>0>−1>−2>⋯ . If your sequence is closed under division, you get infinite descending sequences that are furthermore bounded such as 1>12>14>⋯>0. It should be clear that the two linear orders I described are not well-orders.
A small order theory fact that is not totally on-topic but may help you gather intuition:
Every countable ordinal embeds into the reals but no uncountable ordinal does.
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.
Your second example, 1 > 1⁄2 > 1⁄4 > … > 0, is a well-order. To make it non-well-ordered, leave out the 0.
A well-order has a least element in all non-empty subsets, and 1 > 1⁄2 > 1⁄4 > … > 0 has a non-empty subset without a least element, so it’s not a well-order.
Yes, you’re right.
Adding to Vladimir_Nesov’s comment:
In general, every suborder of a well-order is well-ordered. In a word, the property of “being a well-order” is hereditary. (compare: every subset of a finite set is finite)