Story: Immortal Ike was born, has lived an infinite number of days,

and is alive today.

Question 1: Is this story logically possible?

Question 2: If so, what is the structure of Ike’s days?

I suggest that the answers are: 1) yes, and 2) the structure of an infinite number in a nonstandard model of arithmetic, which looks a bit like:

| | | | | … …| | | | | | … ….| | | | |

Note that everyday, except for the first, needs a yesterday; everyday, except for the last, needs a tomorrow — ruling out such answers as omega + 1.

But interestingly, over a number of informal conversations, many people answer “no” to 1). I have yet to see a knock down argument. Is there such?

Hi Jeremy,

Well, surely the story is not ruled out logically. What theorem would make the story, so far forth, impossible? I wonder what the objectors could be balking at? Is it that no one could live an infinite number of days consecutively? That might come from the assumption that an infinite number of days could fit into a finite amount of time. But suppose time speeds up for Ike, so the normal day passes instantaneously. He of course experiences it as a normal day, but from our frame it’s instantaneous (see Lewis, OPOW, p 72 ff.). No doubt the number of his days could be infinite, I think, though those days pass in a finite amount of (our) time (this would be hidden from Ike). I’m not seeing the pressure for one particular structure on those days, rather than another. Suppose the “number” of his days is uncountably large. In that case, there is no need for the non-standard infinite assumption. Or, I’m not seeing it.

Fascinating.

1. I am concerned about the possibility of causation across an infinite number of intermediate causes. The same Zenonian worry makes me prefer the view that time is infinitely subdivisible to the view that it is infinitely subdivided. If there is no causal series comprising the days of Ike’s life (e.g., because maybe all of Ike’s life is caused “all at once” by an atemporal being), I have no worries about the story.

2. I am inclined to think that the answer to your question is exactly the same as the answer to the much more standard question whether it is possible for a person to have an infinite past. In other words, there is nothing additionally paradoxical about the order type you describe. In regard to the question of the infinite past, my main worry is Grim Reapers.

3. I don’t think you need anything a non-standard model to generate the story you give (nor do you need anything like the Axiom of Choice). Let T be the set of all integers. Then say that x is prior to y iff one of these holds:

(a) x and y are both non-negative and x < y

(b) x and y are both negative and x < y

(c) x is non-negative and y is negative.

There is a first element, namely 0, and a last element, namely -1. Each element but the first has a predecessor (the predecessor of a number x other than 0 is x-1) and each element but the last has a successor (the successor of a number x other than -1 is x+1).

This order results from taking first the sequence of non-negative integers and then following them up with the negative integers:

0,1,2,…; …,-3,-2,-1.

If you want continuous time, no need for non-standard arithmetic. Just paste a copy of the interval (0,1) between each pair of successive elements of T.

This suffices for your story and doesn’t use Choice or non-standard models. I may be missing something.

4. On a related note, I’ve wondered whether there would be any benefit to having eternal life whose days have order type like omega+1 or omega+omega rather than omega.

Apologizes if I run together some replies.

Dennis and Mike: Many peopleâs troubles come from the worry of âescapingâ the initial omega copy (0, 1, 2â¦), which would seem to have to start Ikeâs days. That is, how does Ike âget fromâ his birthday to today? More specifically, how does he leave the initial omega copy? Others, in their Cantorian haze, just seem troubled by anything that is not well-ordered.

Mike: The pressure for structure comes from the assumption that everyday has a yesterday and a tomorrow, as is the case with days (except that, for the purpose of describing Ikeâs days, Ikeâs birthday needs no yesterday and today needs no tomorrow). As mentioned, this rules out omega + 1, and many other structures besides. As Alexander mentioned, the simplest structure satisfying these conditions is 0,1,2,… …,-3,-2,-1.

Alexander: I am looking forward to reading the paper on the Grim Reaper. I actually think that infinite tasks of this structure, the structure of a nonstandard number, are possible, but super-tasks are not. Let me hold off on that argument for now, though I do plan to post on that topic soon (and it is closely related).

With 0,1,2,… …,-3,-2,-1, a potential question that arises is: What is âin the middleâ? That is, imagine that Ike has lived an infinite and even number of days. What is his halfway point? This question motivates placing a copy of the integers (â¦-3, -2, -1, 0, 1, 2, 3â¦) âin the middleâ. We can ask similar questions about Â¼ and Â¾ of Ikeâs days. Etc. This line of reasoning motivates placing copies of the integers (densely linearly ordered without endpoints) âin the middleâ, which moves us to the structure of a nonstandard number.

Infinite Execution: He didnât. Such a structure has no first infinite number. Such considerations do worry people, and tie into the concern about âescapingâ the 1st copy of omega (0, 1, 2â¦).

“…how does he leave the initial omega copy?”

…And which is the answer? How does he leave it?

(This would be an equivalent question: The second omega copy can’t begin until the first one has ended, but the first one never ends, because is infinite. So, how can the second copy begin?).

Jeremy:

“What is âin the middleâ?”

Sure, one can add to the story that there is in some sense a middle. But I don’t think it makes the story any more (or less) interesting to do that.

“I actually think that infinite tasks of this structure, the structure of a nonstandard number, are possible, but super-tasks are not. Let me hold off on that argument for now, though I do plan to post on that topic soon (and it is closely related).”

I won’t hold off. 🙂

I think that if infinite tasks of this structure are possible, so are more ordinary super-tasks, with an omega+1 structure such as Thompson’s lamp (where we toggle a lamp switch at every point, and ask what the position of the switch at omega+1 is–except that the times get bunched up).

Here’s why. Take the non-standard embodiment of your story. Suppose Ike presses a lamp toggle switch every day. That’s an infinite task, but Ike’s lamp, unlike Thompson’s lamp, does not as it stands generate a paradox–there is a well-determined final state.

But it can be made to. Since every day has a (unique, I assume) successor, there must be a copy of omega at the beginning of your non-standard sequence of days. (Just prove by induction that for each positive integer n there is a unique day with exactly n-1 predecessors. For some reason I feel diffident on this point, but it seems right.) Let A be such a copy of omega. Intuitively, if the Ike’s lamp story is coherent, so is the story with the modification that Ike only presses the toggle switch on days in A. (Argument: Surely we can imagine Ike freely choosing each day whether to press or not. And any sequence of presses and non-presses should be possible, including the sequence where he presses in all and only the days in A. I suppose A will be an external set, but it would be weird if there was a cross-time free-will restriction that forbade press-patterns that involve external sets.) But the modified Ike’s lamp story involves a isomorph of Thompson’s lamp.

So if Ike’s lamp is possible, supertasks with a classical structure should be possible as well.

On my view, you can have Ike, but only if there aren’t causal connections between infinitely many of the days, and by the same token you can’t have Ike’s lamp as it requires causal connections (the state of the switch each day is caused by what happened a day earlier).

Jeremy,

The problem does not make any explicit metaphysical assumptions on the nature of individuals, so let individuals be 4D beings whose temporal stages have no temporal dimension. Take an individual, Ike, existing over the interval [0, 1] which is exactly one second long. Ike has existed over an infinite number of instants, since each stage of Ike exists at an instant in [0,1]. The structure is one in which 0 has no instant before, and (assuming 1 is the last instant of Ike’s short life) 1 has no instant after. This seems to me possible, but it does not have the structure you note. Is it important to your problem that ‘days’ are 24 hrs or that ‘days’ have duration? Is it important to the problem that 4D is false?

The answers to a number of the comments tie together, so Iâll post one reply.

I am using days, where the assumption is that everyday has a yesterday and a tomorrow, to stay in the discrete case. The question about Ike is the same as asking if the universe could be infinite in age and have had a beginning. To my knowledge this question has not been asked, and I think it an important one. [0, 1] does not address this question. I donât think that the metaphysics of time is important here, though I havenât considered that.

Then because in the finite-day case we can ask about âin the middleâ, and since the finite is our only guide to the infinite, so too are we able to ask about âin the middleâ in the infinite case. To leave the discrete case, we can ask: Is it possible to have two points infinitely far apart (spatially or temporally)? If so, what is the structure? I suggest that the answers are: yes, and the structure of a nonstandard number, though now âfilled inâ. I actually think that this structure is the correct answer to the question: What is the correct conception of infinite distance? Just as I think that the structure of a nonstandard number is the correct answer to the question: What is the correct conception of infinite number? With two points infinitely far apart, it again makes sense that there is space âin the middleâ, and space that cannot be reached by going any finite distance from either endpoint. (Actually, this structure for infinite distance, I believe, is the same structure as [0, 1] on the nonstandard number line.) For further discussion of some of these points, see:

http://www.univie.ac.at/constructivism/journal/articles/7/2/126.gwiazda.pdf

That paper also presents one way that it is possible to âleave the initial omega copyâ.

I do think that Ikeâs task is possible, and that super-tasks are impossible. Admittedly it is odd to say that a proper part (a super-task) of a possible task (Ikeâs task) is impossible, but that is exactly what I claim. So I do deny Alexâs: âIntuitively, if the Ike’s lamp story is coherent, so is the story with the modification that Ike only presses the toggle switch on days in A [the initial omega copy].â How can this be so? Because omega is only potentially infinite, not actually infinite! Omega does not exist in any static, determined sense, and it is impossible to complete exactly omega many tasks. Cantorâs notion that omega is an actually infinity, and that any potential infinity presupposes an actual infinity, will someday be seen as one of the oddest delusions into which any academic discipline ever fell. (Master of understatement that I am, there is always the worry that I just donât âget itâ.)

By contrast, nonstandard numbers are actually infinite. Ikeâs lamp has a well determined final state, and Iâd go further and argue that once we arrive at the correct conception of infinite number (e.g., when you say âThere are infinitely many starsâ you mean a nonstandard number), no paradoxes of the infinite remain. Hereâs one, the St Petersburg paradox. Itâs accepted that if there is a finite limit to the money in the universe, there is no paradox. But if there is an infinite amount of money in the universe, and this infinite amount is N, an infinite nonstandard number, then again there is no paradox. The finite and infinite cases are resolved the same way. Getting back to the impossibility of super-tasks, I argue for that here:

http://onlinelibrary.wiley.com/doi/10.1111/j.1468-0114.2011.01412.x/pdf

I do think that *if* omega were actually infinite, that is, if it were complete, static, determined, present-all-a-once, etc, then it would be possible to complete a super-task. Just line-up balls numbered 1, 2, 3â¦ at Â½, Â¾, 7/8â¦ feet, and have a turtle with a torch on his back walk the foot under the balls. The turtle has burned all omega-many balls, and so has completed infinitely many tasks. But in the paper above, I argue that it is not possible to complete a super-task. I think that this argument, among other considerations, shows that omega is not actually infinite. For a âtestâ to see if an infinite structure is potentially or actually infinite, see here:

http://www.springerlink.com/content/e2746w1kn6913580/fulltext.pdf

Essentially, ask if random selections from the structure âgrowâ through time. For example, if we donât care about the failure of countable additivity, we can consider random selections from the positive integers. Make such a selection. Then make another. The probability is 1 (or 1 minus an infinitesimal) that the second selection is larger than the first. Such growth of subsequent random selections indicates that omega is potentially infinite. Growth of random selections from a structure demonstrates that that structure is a potential infinity. Note that random selections from a nonstandard number do not grow, indicating that it is an actual infinity. Or so I claim anyway.

I apologize for having gone on a bit longer than intended. I do think that the fundamental question is an important one: Is it possible that Ike was born, has lived an infinite number of days, and is alive today? The question can be considered in both the discrete and the continuous cases.

“Omega does not exist in any static, determined sense, and it is impossible to complete exactly omega many tasks”

It’s weird, though, that a larger collection would be actual while a subcollection would be merely potential.

Maybe not *too* weird, but still weird.

I agree Alex, it is weird. In the 3rd paper of my last post, I write:

‘At this point a difficult question arises. I have suggested that omega is a potential infinity. I have also suggested that an infinite integer is an actual

infinity. And yet, an infinite integer is comprised of nothing but omegas and omega*s. How

can it be that many potential infinities are able to comprise an actual infinity? The

answer to this question lies beyond the scope of this paper…’

Then I refer to the 1st paper of that last post, where I give a possible answer (infinite numbers are large finite numbers — I do recognize how that sounds). But definitely weird.

I feel like the set theory is ‘telling’ us something — insofar as, as you noted, the initial omega copy, A, is an external set.

First, I’m with you here, it seems that it is logically possible.

Could you say anything about the things people have said to cast doubt on this (even though they weren’t knock-down arguments)?

My only thought is that they might be disputing that it is possible in some other relevant sense (nomological, for instance).