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June 15, 2012

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wolfgang

I think what you try to show can be nicely illustrated within (general) relativity. I am thinking about the analytical continuation e.g. of the Reissner metric (electrically charged blackhole).
There are events to the future of us , which the time slicing used by an observer who stays outside of the horizon cannot reach.

Valdi Ingthorsson

There are some aspects of this paper that puzzles me. First, why should we take the earlier than relation to be isomorphic to the first uncountable ordinal + 1? Do you really mean to say that the earlier than relation holding between any two events, or between any given event and every other, is isomorphic to an uncountable infinity of events + 1? Why? Are you perhaps instead describing time as series of events related as earlier/later than, rather than the earlier/later than relation per se. Second, why should we think time as a whole is isomorphic to an uncountable infinity of events + 1? Since the argument turns on the acceptance of the isomorphism and the idea that there is a last event in a series, it would be nice to see an explanation for what it really means and an argument for the suitability of this characterisation. This might also make the paper accessible for a wider audience, i.e. those philosophers that aren’t comfortable with the ideas of uncountable infinities and the like.

Tristan Shawn Johnson

Thank you for your reply Valdi and your time in reading the argument.

The argument should be viewed as one of this type: x is possible, but on account y, x is not possible, therefore y is false or in need of revision.

In actuality I have no reason to take the earlier/later than relation, as a whole, to be isomorphic to the first uncountable ordinal + 1. But I also see no reason it couldn't be that way i.e. I think it's perfectly possible (even if not actual). In fact, for any ordinal x, I think it's possible that the earlier than relation be isomorphic to x. But tensed theorists can't (or may have trouble) accepting these possibilities. If they agree that these are genuine possibilities, then they need to explain how the present can access all times (or why it doesn't need to). If they don't think these are genuine possibilites, then they need to explain why.

Just to be perfectly clear. I don't think that between any two events there are uncountably many events + 1. I'm really thinking about the second possibility you consider viz. that the timeline as a whole is isomorphic to the first uncountable ordinal + 1 so that there's an event later than all others. I think this latter option is possible and that tense theory may have trouble dealing with that possibility.

Thanks again for your response Valdi.


Tristan Shawn Johnson

Thank you as well Wolfgang for the nudge.

I'll look into analytical continuation and the Reissner metric.

Valdi Ingthorsson

Thank you for this clarification. It makes it easier to phrase some other related worries that I have, although the risk is that they just reflect my cursory knowledge of the mathematical jargon you use, but then again perhaps it would help the general reader to sort these worries out. So take this as a response from someone who does not work with the mathematical assumptions you work with on an everyday basis, and who is a little confused about the whole thing.

First, you do not explain in any detail why the last member of the set ω1+1 can never become present? Here are my speculations about why you might think that it is impossible. Since (i) time is sometimes thought to have the structure of the continuum (i.e. be structurally isomorphic to the continuous line), and (ii) if I have understood it correctly, the continuum is sometimes considered to be equivalent to the first uncountable set, then it wouldn’t be far fetched to think of you as elucidating the consequences of the assumption that time has the structure of a continuum but adding to the continuum a last member later than all the rest (see second worry below). The consequence appears to be (as far as I can understand what the argument professes to prove) that this last member cannot possibly enter the present. However, doesn’t this impossibility presuppose a particular rate at which moments enter the present, and that the structure of the sequence in which moments enter the present at that rate must be somehow structurally different from the set of moments that make up time (i.e. that this sequence is not expressible as ω1+1). If the sequence in which moments enter the present is structurally isomorphic to ω1+1 then it would appear that there is a one to one correspondence between every entering into the present and every member in the set of moments of time, right? If, despite the structural isomorphism of the two sets, the last moment could still never enter the present, there must be some other obstacle to this fact, e.g. that one cannot come to the end of an uncountable infinity +1. I’ve wondered whether your brief mentioning of the rate in which moments enter the present, suggests that you assume that if all the moments are ever to enter the present they must do so in a finite time, much like Xeno seemed to think that you cannot go through all the points of a continuous line in a finite amount of time (especially if you go through them one at a time). Bertrand Russell discusses this problem somewhere. Your claim in (3bii) that “either intervals of time become smaller over hyper-time, or the rate at which the present moves increases to a rate of uncountably many units of time per unit of hyper-time” seems to support this reading. However, if we have an infinite time (but with a last time)—something that should be equally possible as having an infinite series of moments (but with a last moment)—then it should be possible that there should be a last entering into the present of the last moment of time.

Second, I find the idea of a time that has the structure of an uncountable infinity and yet has a last member later than all the others a little unintuitive, and I even wonder if it makes good mathematical sense. Isn’t the lack of a last member the defining difference between a finite and infinite set? If ω1 is an uncountable infinity with no last member, then the additional moment that is later than all the rest, would have no determinate latest preceding moment. Wouldn’t that be a problem for a B-theorist that assumes time is a well ordered sequence with one and only one moment later than all the rest.

Third, you make it clear that you work under the assumption that it is the role of a metaphysical theory to accommodate for every conceivable possibility, not just those we have reason to believe is actually possible. You should make that assumption clear at the outset, because this is not universally accepted. On the whole, however, I don’t think you could really be seen to establish that the tensed view generally has a problem, but merely those who think that the tensed view must allow for the possibility that time has the structure you describe.

Fourth, ‘Fairness’ is stated as “Every time, related by the earlier than relation, eventually becomes ontologically privileged”, but the moment that you say can’t possible become present does not stand in the earlier than relation to any event; it is later than all the others.

All the best,

Tristan Shawn Johnson

Thanks again for the thoughtful comments. Far from confused you seem to be at the heart of the argument. I'll respond in order.

FIRST

I do need to explain in detail why the last time can't become present. This is a major weakness of the argument and it may be decisive. The argument is something like this:

Suppose, the present is moving at 1 second per 1 hyper-second. Then at this rate the present will never (this `never' is hyper-time never) complete the times in ω1 and therefore never (again, hyper-time never) get to the last time.

This does not assume that time or hyper-time are finite or that they disagree in structure. I'm actually thinking of them both as having the same structure ω1+1.

What needs explaining is this phrase ``never complete the times in ω1 + 1''. A similar issue arises in the Tristram Shandy paradox (though there it's just ω, which has the order of the natural numbers). Tristram Shandy is writing his autobiography at a rate of one day (of his life) per day of his life (it's actually one day per year but I want to bring it in alignment with our scenario). The claim is that if he lives forever he can complete his autobiography. But there's an ambiguity in the word `complete'. The first sense is that he leaves nothing out. The second is that the book is finished.

1) If `completes' means `For every day d, of Tristram's life, there exists a day for which Tristram writes about d' then we must say Tristram completes his autobiography. (He leaves nothing out.)

2) But if `completes' means `There exists a book, such that, for every day of Tristram's life, there is a section of that book for which that day is discussed' then we must say that he has not completed the book. (He hasn't finished it.)

So in what sense can I say the present completes the task of making every time in ω1+1 privileged? By analogy the following two thoughts suggest themselves.

3) For every time t, there's a corresponding hyper-time T, such that t is present (ontologically privileged) at T. (The present leaves no times out.)

4) There exists a hyper-time such that no time is present and all times are past. (The present finishes making all times ontologically privileged.)

But neither of these seem particularly bad to me (though hyper-time will need to have structure ω1+2 to fulfill reading 4). So maybe the argument is not so good. Thanks for pushing me on this.

SECOND

a) Infinity is usually cashed out in terms of cardinality not ordinality. So a set is finite if and only if it has less members than ω. A set is infinite just in case it's not finite. In short, it does make good mathematical sense.

b) You're correct that the last member of ω1+1 would have no determinate latest preceding moment.

c) I don't think the B-theorist needs to take a stand on what the ordering actually is. The eternalist block gives that order but I'm not sure what evidence we can produce for determining what it actually is. It may be discrete, dense, continuous, it may have no beginning or no end, it may have gaps like I've considered, or it may be circular. Who knows?

THIRD

I think it's a virtue of a metaphysical theory that it doesn't rule out possibilities that we, pre-theoretically, think are possible (e.g. the possible orderings just mentioned). You're right, I should state this early and clearly.

FOURTH

I'm not quite sure what the objection is here but here's what came to mind reading it.

Why can't the tense theorist just say `Look, the ω1+1 structure for time is not possible, but on the tenseless picture it is possible. So the tenseless theory is false'? This is similar to the "possibility'' of time without change discussed between substantivalists and relationalists. On one theory it looks possible, on the other it doesn't. Maybe the thought that time could be ordered like ω1+1 is not pre-theoretically possible. Maybe my bias toward B-theory makes me think it is. I'm not sure what to say.

Thanks again Valdi. This has been very helpful. I'm starting to think there's less here than I had initially thought. I hope I haven't wasted anyone's time.

Valdi Ingthorsson

Hi again Tristan, and thanks for a very clear response.

As for your response to my first worry, then I agree that it is one of the more serious problems with the argument. One thought arose from your discussion about the ambiguity of ‘complete’ is that the concept of ‘never’ is also ambiguous, and neither the concept of ‘complete’ or ‘never’ may really make sense once we have assumed that the time-series is infinite. If it is infinite there is never an end to either time or hyper-time and we should not expect a completion of it. I think you put my point very nicely in 3) i.e. ”For every time t, there's a corresponding hyper-time T, such that t is present (ontologically privileged) at T. (The present leaves no times out.)”. But note that this will only work if the structure of the time-series is ω1, but not ω1+1, which leads again to the question of whether time really can be isomorphic to ω1+1. I think this is a really important issue.

For the argument to go through, you need to characterise the time series as ω1+1, instead of just ω1 because you need something whose entrance into the present can count as the completion of the time series, and which can never come about. I don’t have an argument against the possibility of that structure, except to say again that the conception of a time series that is infinite and yet has a last member strikes me as strange; it is as if you are saying it is infinite and yet finite. One thought is that, as you say, we are dealing here with cardinality, which, as I understand it, is about the relational sizes of infinities. Hence I wonder if it really can make sense to speak of a set that is greater than one infinity (ω1) by a finite number of members (i.e. one more than ω1)?

About my forth worry, then I really didn’t explain as I should have the underlying assumptions. It has to do with whether ‘earlier than’ is a relation distinct from ‘later than’, or if we instead just have one asymmetric relation that makes one relata earlier than the other and the latter later than the first. I personally think there is just one relation, and that we can choose to call it the ‘earlier than’ or the ‘later than’ by arbitrary choice, or, perhaps better, the ‘earlier/later than’ relation. But then one should declare that this is the choice you make. It really just sounded awkward to state that every event that is earlier than some other event must eventually become present, and then to illustrate the impossibility of that with an event that is not earlier than any event. If you make clear that you mean any event that is either earlier or later than there is no problem. However, it does seem to me that the B-theory is bothered by the worry I addressed above, notably whether the postulation of a structure that is infinite and yet has a member later than all the rest is to postulate and infinity that yet is finite.

All the best

Tristan Shawn Johnson

Hey Valdi, thanks again for the comments.

FIRST

Supposing ω1+1 is the structure of both time and hyper-time it looks to me that 3) will still hold. Let t* be the last time and let T* be the last hyper-time. Then T* will be the hyper-time for which t* is present (ontologically privileged). At least I think the tensed theorist could argue this.

SECOND

Remember I'm merely using ω1+1 as an example. I was also thinking of even odder ordinals and it may be helpful to consider two more.

1) ω1+2 which I briefly mentioned last time. This ordering is not dense even though ω1 and ω1+1 are, because there are two members (the last two) that don't have a member between them.

and,

2) ω1 + ω1 which is infinite, dense, but doesn't have a last member like ω1+1. It's like sticking two real number lines side by side.

My concern wasn't the fact that there's a latest member. It was that there's some sort of "gap" in the ordering the present can't seem to "jump" because it would involve "completing" an infinite series. But now, having been pushed on it, I'm not sure I can make this intuitive problem clear.

Let me also be clear that I agree with you that the time series we are considering are counter-intuitive and strange. I do (or at least did) think they are possible.

It does not make good sense to say that adding a finite set to an uncountable set changes the cardinality. But it does make good sense to say that adding a finite set to an uncountable set changes the ordinality (see below for more).

THIRD

You're right I should have been more clear about the relation. I'm just thinking there's one relation and my wording is awkward.

I would assuage this last worry by noting that the cardinality (how many members of the set there are) of ω1+1 is just ω1 (that is, there are just as many members of ω1+1 as there are in ω1, adding the last one doesn't change how many there are). But the ordinality (the order of members) is substantially different. Since infinity is defined in terms of cardinality the B-theorist can respond that by postulating a structure that is infinite and yet has a last member they haven't postulated an infinity that is finite.

Best,

Anthony P. Stone

Thinking of ω + 1 is interesting. I work with a similar situation, where the equivalent of ‘less than’ in a tensed hyper-time isomorphic to an arbitrary infinite ordinal. The difference is is that this is for an unmeasured time in God. There is no problem of small intervals, To understand accessing ω in a series like 1, 2, …, ω, I suggest an analogy with a human being taking one step and crossing an infinite sequence of decreasing length intervals of space. That is, I assume that God knows the elements of the sequence in an appropriate way to access ω, and similarly for other inaccessible ordinals. Needless to say, I take the associated physical time to be nothing unusual.
See http://homepage.ntlworld.com/stone-catend/TheisticModel-Conf.pdf.

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