“Put differently, just because we count up to n doesn’t mean we pass through n/3”—The first possible objection I’ll deal with is not what I think you are complaining about, but I think it’s worth handling anyway. n/3 mightn’t be an Omnific Number, but in this case we just take the integer part of n/3.
I think the issue you are highlighting is that all finite numbers are less than n/3. And if you define a sequence as consisting of finite numbers then it’ll never include n/3. However, if you define a sequence as all numbers between 1 and x where x is a surreal number then you don’t encounter this issue. Is this valid? I would argue that this question boils down to whether there are an actually infinite number of days on which Trump experiences or only a potential infinity. If it’s an actual infinity, then the surreal solution seems fine and we can say that Trump should stop saying yes on day n/3. If it’s only a potential infinity, then this solution doesn’t work, but I don’t endorse surreals in this case (still reading about this)
If the number of days is, specifically, ω, then the days are numbered 0, 1, 2, …, with precisely the (ordinary, finite) non-negative integers occurring. They are all smaller than ω/3. The number ω isn’t the limit or the least upper bound of those finite integers, merely the simplest thing bigger than them all.
If you are tempted to say “No! What I mean by calling the number of days ω is precisely that the days are numbered by all the omnific integers below ω.” then you lose the ability to represent a situation in which Trump suffers this indignity on a sequence of days with exactly the order-type of the first infinite ordinal ω, and that seems like a pretty serious bullet to bite. In particular, I think you can’t call this a solution to the “Trumped” paradox, because my reading of it—even as you tell it! -- is that it is all about a sequence of days whose order-type is ω.
Rather a lot of these paradoxes are about situations that involve limiting processes of a sort that doesn’t seem like a good fit for surreal numbers (at least so far as I understand the current state of the art when it comes to limiting processes in the surreal numbers, which may not be very far).
I already pointed above to the distinction between absolute and potential infinities. I admit that the surreal solution assumes that we are dealing with an absolute infinity instead of a potential one, so let’s just consider this case. You want to conceive of this problem as “a sequence whose order-type is ω”, but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.
So I don’t think the ordinary notion of sequence makes sense. In particular, it doesn’t account for the fact that two sequences which appear to be the same in every place can actually be different if they have different lengths. Anyway, I’ll try to untangle some of these issues in future posts, in particular I’m leaning towards hyperreals as a better fit for modelling potential infinities, but I’m still uncertain about how things will develop once I manage to look into this more.
So far as anyone knows, no actual processes in the actual world are accurately described by surreal numbers. If not, then I suggest the same goes for the “nearest possible worlds” in which, say, it is possible for Mr Trump to be faced with the sort of situation described under the heading “Trumped”. But you can have, in a universe very much like ours, an endless succession of events of order-type ω. If the surreal numbers are not well suited to describing such situations, so much the worse for the surreal numbers.
And when you say “I don’t think the ordinary notion of sequence makes sense”, what it looks like to me is that you have looked at the ordinary notion of sequence, made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers, and indeed not only that but made the further arbitrary choice that you are only prepared to understand it if there turns out to be a uniquely defined surreal number that is the length of such a sequence, observed that there is not such a surreal number, and then said not “Oh, whoops, looks like I took a wrong turn in trying to model this situation” but “Bah, the thing I’m trying to model doesn’t fit my preconceptions of what the model should look like, therefore the thing is wrong”. You can’t do that! Models exist to serve the things they model, not the other way around.
It’s as if I’d just learned about the ordinals, decided that all infinite things needed to be described in terms of the ordinals, was asked something about a countably infinite set, observed that such a set is the same size as ω but also the same size as 1+ω and ω2, and said “I don’t think the notion of countably infinite set makes sense”. It makes perfectly good sense, I just (hypothetically) picked a bad way to think about it: ordinals are not the right tool for measuring the size of a (not-necessarily-well-ordered) set. And likewise, surreal numbers are not the right tool for measuring the length of a sequence.
Don’t get me wrong; I love the surreal numbers, as an object of mathematical study. The theory is gorgeous. But you can’t claim that the surreal numbers let you resolve all these paradoxes, when what they actually allow you to do is to replace the paradoxical situations with other entirely different situations and then deal with those, while rejecting the original situations merely because your way of trying to model them doesn’t work out neatly.
Maybe I should re-emphasise the caveat at the top of the post: “I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don’t mistake these for formal proofs. I’m also aware that simply noting that a formalisation provides a satisfactory solution doesn’t philosophically justify its use, but this is also not the focus of this post.”
You wrote that I “made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers”. This choice isn’t arbitrary. I’ve given some hints as to why I am taking this approach, but a full justification won’t occur until future posts.
You want to conceive of this problem as “a sequence whose order-type is ω”, but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.
It seems to me that measuring the lengths of sequences with surreals rather than ordinals is introducing fake resolution that shouldn’t be there. If you start with an infinite constant sequence 1,1,1,1,1,1,..., and tell me the sequence has size ω, and then you add another 1 to the beginning to get 1,1,1,1,1,1,1,..., and you tell me the new sequence has size ω+1, I’ll be like “uh, but those are the same sequence, though. How can they have different sizes?”
Because we should be working with labelled sequences rather than just sequences (that is sequences with a length attached). That solves the most obvious issues, though there are some subtleties there
Why? Plain sequences are a perfectly natural object of study. I’ll echo gjm’s criticism that you seem to be trying to “resolve” paradoxes by changing the definitions of the words people use so that they refer to unnatural concepts that have been gerrymandered to fit your solution, while refusing to talk about the natural concepts that people actually care about.
I don’t think think your proposal is a good one for indexed sequences either. It is pretty weird that shifting the indices of your sequence over by 1 could change the size of the sequence.
I assume here you mean something like “a sequence of elements from a set X is a function f:α→X where α is an ordinal”. Do you know about nets? Nets are a notion of sequence preferred by people studying point-set topology.
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)
“Put differently, just because we count up to n doesn’t mean we pass through n/3”—The first possible objection I’ll deal with is not what I think you are complaining about, but I think it’s worth handling anyway. n/3 mightn’t be an Omnific Number, but in this case we just take the integer part of n/3.
I think the issue you are highlighting is that all finite numbers are less than n/3. And if you define a sequence as consisting of finite numbers then it’ll never include n/3. However, if you define a sequence as all numbers between 1 and x where x is a surreal number then you don’t encounter this issue. Is this valid? I would argue that this question boils down to whether there are an actually infinite number of days on which Trump experiences or only a potential infinity. If it’s an actual infinity, then the surreal solution seems fine and we can say that Trump should stop saying yes on day n/3. If it’s only a potential infinity, then this solution doesn’t work, but I don’t endorse surreals in this case (still reading about this)
If the number of days is, specifically, ω, then the days are numbered 0, 1, 2, …, with precisely the (ordinary, finite) non-negative integers occurring. They are all smaller than ω/3. The number ω isn’t the limit or the least upper bound of those finite integers, merely the simplest thing bigger than them all.
If you are tempted to say “No! What I mean by calling the number of days ω is precisely that the days are numbered by all the omnific integers below ω.” then you lose the ability to represent a situation in which Trump suffers this indignity on a sequence of days with exactly the order-type of the first infinite ordinal ω, and that seems like a pretty serious bullet to bite. In particular, I think you can’t call this a solution to the “Trumped” paradox, because my reading of it—even as you tell it! -- is that it is all about a sequence of days whose order-type is ω.
Rather a lot of these paradoxes are about situations that involve limiting processes of a sort that doesn’t seem like a good fit for surreal numbers (at least so far as I understand the current state of the art when it comes to limiting processes in the surreal numbers, which may not be very far).
I already pointed above to the distinction between absolute and potential infinities. I admit that the surreal solution assumes that we are dealing with an absolute infinity instead of a potential one, so let’s just consider this case. You want to conceive of this problem as “a sequence whose order-type is ω”, but from the surreal perspective this lacks resolution. Is the number of elements (surreal) ω, ω+1 or ω+1000? All of these are possible given that in the ordinals 1+ω=ω so we can add arbitrarily many numbers to the start of a sequence without changing its order type.
So I don’t think the ordinary notion of sequence makes sense. In particular, it doesn’t account for the fact that two sequences which appear to be the same in every place can actually be different if they have different lengths. Anyway, I’ll try to untangle some of these issues in future posts, in particular I’m leaning towards hyperreals as a better fit for modelling potential infinities, but I’m still uncertain about how things will develop once I manage to look into this more.
So far as anyone knows, no actual processes in the actual world are accurately described by surreal numbers. If not, then I suggest the same goes for the “nearest possible worlds” in which, say, it is possible for Mr Trump to be faced with the sort of situation described under the heading “Trumped”. But you can have, in a universe very much like ours, an endless succession of events of order-type ω. If the surreal numbers are not well suited to describing such situations, so much the worse for the surreal numbers.
And when you say “I don’t think the ordinary notion of sequence makes sense”, what it looks like to me is that you have looked at the ordinary notion of sequence, made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers, and indeed not only that but made the further arbitrary choice that you are only prepared to understand it if there turns out to be a uniquely defined surreal number that is the length of such a sequence, observed that there is not such a surreal number, and then said not “Oh, whoops, looks like I took a wrong turn in trying to model this situation” but “Bah, the thing I’m trying to model doesn’t fit my preconceptions of what the model should look like, therefore the thing is wrong”. You can’t do that! Models exist to serve the things they model, not the other way around.
It’s as if I’d just learned about the ordinals, decided that all infinite things needed to be described in terms of the ordinals, was asked something about a countably infinite set, observed that such a set is the same size as ω but also the same size as 1+ω and ω2, and said “I don’t think the notion of countably infinite set makes sense”. It makes perfectly good sense, I just (hypothetically) picked a bad way to think about it: ordinals are not the right tool for measuring the size of a (not-necessarily-well-ordered) set. And likewise, surreal numbers are not the right tool for measuring the length of a sequence.
Don’t get me wrong; I love the surreal numbers, as an object of mathematical study. The theory is gorgeous. But you can’t claim that the surreal numbers let you resolve all these paradoxes, when what they actually allow you to do is to replace the paradoxical situations with other entirely different situations and then deal with those, while rejecting the original situations merely because your way of trying to model them doesn’t work out neatly.
Maybe I should re-emphasise the caveat at the top of the post: “I will provide informal hints on how surreal numbers could help us solve some of these paradoxes, although the focus on this post is primarily categorisation, so please don’t mistake these for formal proofs. I’m also aware that simply noting that a formalisation provides a satisfactory solution doesn’t philosophically justify its use, but this is also not the focus of this post.”
You wrote that I “made the entirely arbitrary choice that you are only prepared to understand it in terms of surreal numbers”. This choice isn’t arbitrary. I’ve given some hints as to why I am taking this approach, but a full justification won’t occur until future posts.
OK! I’ll look forward to those future posts.
(I’m a big surreal number fan, despite the skeptical tone of my comments here, and I will be extremely interested to see what you’re proposing.)
It seems to me that measuring the lengths of sequences with surreals rather than ordinals is introducing fake resolution that shouldn’t be there. If you start with an infinite constant sequence 1,1,1,1,1,1,..., and tell me the sequence has size ω, and then you add another 1 to the beginning to get 1,1,1,1,1,1,1,..., and you tell me the new sequence has size ω+1, I’ll be like “uh, but those are the same sequence, though. How can they have different sizes?”
Because we should be working with labelled sequences rather than just sequences (that is sequences with a length attached). That solves the most obvious issues, though there are some subtleties there
Why? Plain sequences are a perfectly natural object of study. I’ll echo gjm’s criticism that you seem to be trying to “resolve” paradoxes by changing the definitions of the words people use so that they refer to unnatural concepts that have been gerrymandered to fit your solution, while refusing to talk about the natural concepts that people actually care about.
I don’t think think your proposal is a good one for indexed sequences either. It is pretty weird that shifting the indices of your sequence over by 1 could change the size of the sequence.
I assume here you mean something like “a sequence of elements from a set X is a function f:α→X where α is an ordinal”. Do you know about nets? Nets are a notion of sequence preferred by people studying point-set topology.
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)
I think gjm’s response is approximately the clarification I would have made about my question if I had spent 30 minutes thinking about it.