by on May 31, 2004

Following up my three-way classification of paradoxes, I want to argue that paradoxes involving infinity are always of type-3, that is, the result of ill-posed problems or inappropriate ways of taking limits. (Much the same position is defended in the comments thread by Bill Carone). In fact, I’d argue for the following general principle, applicable to all models relevant to human decisionmakers.

Whenever a result, true for all finite n, is strictly reversed for the infinite case, the problem in question has been posed incorrectly

To defend this, I rely on the premise that we are finite creatures in a finite universe. If a mathematical representation of a decision problem involves an infinite set, such as the integers or the real line, it is only because this is more convenient than employing finite, but very large bounds, such as those derived from the number of particles in the universe. Any property that depends inherently on infinite sets and limits, such as the continuity of a function, can never be verified or falsified by empirical data. Since we are finite, any result that is true for all finite n is true for us.

It’s not surprising then that paradoxes frequently introduce beings who can be assumed infinite – God, angels, demons and so on. There is, of course, a long theological tradition of asking what God can and cannot do – for example, make a logically contradictory statement true. It certainly appears from the various paradoxes that, if God is capable of handing out infinite rewards and punishments (which is, I think, generally supposed by believers), it’s not valid to say that, if a given course of action is better than another in every possible case (for some partition of the possible cases), then it is definitely the best choice.

It’s hard to imagine decision theory without this premise, so it’s reasonable to conclude that good theists should not be good decisionmakers, and vice versa – a conclusion supported by consideration of Pascal’s wager. Similarly, to suggest that the Almighty can’t generate a probability distribution giving equal weight to every positive integer seems tantamount to denying Omnipotence altogether, but allowing for this possibility creates all sorts of problems and paradoxes, leading once again to the acceptance of strictly dominated choices.

This leads me on to this paper by Arntzenius, Elga, and Hawthorne, pointed out by Brian. As in the Peter Wakker piece I cited earlier, Arntzenius, Elga, and Hawthorne observe the crucial role of (the absence of) countable additivity in generating a number of paradoxes. But rather than adopting a type-3 solution like Wakker’s (in such circumstances always use a sigma-algebra and not just an algebra), Arntzenius, Elga, and Hawthorne seem to want to derive a type-1 solution, in which this result is supposed to have practical implications, such as that “when infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruin”. Except in relation to dealings with supernatural beings, I think this conclusion is profoundly mistaken.

fn1. Insert obligatory joke about the world being divided into three sorts of people

fn2. To clarify the relevance of “strictly”, consider a sequence xn approaching y from below. It’s true for all finite n, but not in the limit, that xn is less than y. A strict reversal would arise if in the case where for infinite n, we had x strictly greater than y.

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Anarch 05.31.04 at 2:53 pm

You seem to be essentially positing a 0-1 law for all practical applications. This raises the question: given that we know that there are classes of models that do not possess 0-1 laws, are the non 0-1 statements then empirically false or just empirically unobservable?

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Anarch 05.31.04 at 2:59 pm

Also:

To defend this, I rely on the premise that we are finite creatures in a finite universe. If a mathematical representation of a decision problem involves an infinite set, such as the integers or the real line, it is only because this is more convenient than employing finite, but very large bounds, such as those derived from the number of particles in the universe.

As I think Brian’s repeatedly mentioned, you don’t need a completed infinity in order for some of these paradoxes to go through, only a quantity capable of arbitrary finite magnitude. [For example, the decaying nucleus with potentially infinite time of decay.] Provided that time is (physically) unbounded, you still get the same problems.

Which means, AFAICT, that your principle also implies that there’s an absolute bound on all measurable quantities. I’m not sure whether that’s plausible or not…

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Anarch 05.31.04 at 3:00 pm

Gah: my parenthetical should have read “For example, the decaying nucleus with arbitrarily large time of decay.” Sorry about that.

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Dave 05.31.04 at 4:29 pm

For example, the decaying nucleus with potentially infinite time of decay.

An isolated, unstable atomic nucleus will undergo decay eventually, according to a predictable probability distribution over time. The question is only how long it will take. To assert that the time is “potentially infinite” is not only incorrect, but it is not even well-defined.

We can, if you like, make the distinction between probability distributions that become zero at and above some specific value, and those that are finite (but small) at arbitrarily large values.

In terms of time, we can also make a distinction between things that will stop (or cease to be) at a particular time, and those that will persist indefinitely. In the case where both possibilities are present (for example, you have an atom that is either carbon-12 or carbon-13, wan to know if/when it will decay radioactively), there would be a finite probability of each. There would be a probability P that the atom would last indefinitely (ignoring proton decay and other strange quantum tunneling effects), and another probability (1-P) that the atom would decay with probability distribution Q(t).

If we know the atom is carbon-13, the expected lifetime will be E[t*Q(t)]. If we know it is carbon-12, we say it will last forever. If it could be either (but we know P), we could say that on average the atom will last forever, but it makes more sense to say that it will last forver with probability P, or will decay in time E[t*Q(t)] with probability (1-P).

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Dave 05.31.04 at 4:31 pm

Oops – sorry about that Anarch! I left the comment sitting around and came back without seeing you had corrected yourself!

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Matt Weiner 05.31.04 at 6:07 pm

â€œwhen infinitely many decisions are involved, the difference between making the decisions simultaneously and making them sequentially can be the difference between riches and ruinâ€. Except in relation to dealings with supernatural beings, I think this conclusion is profoundly mistaken.

I don’t quite get it. Isn’t your position that, except in relation to supernatural beings, this conclusion has no application whatsoever?

Also:
Similarly, to suggest that the Almighty canâ€™t generate a probability distribution giving equal weight to every positive integer seems tantamount to denying Omnipotence altogether

On most versions of Omnipotence, Omnipotence doesn’t require ability to do the logically impossible; so someone could happily suggest that a probability distribution giving equal weight to every positive integer is logically impossible and retain that version of Ominpotence. (I’m not convinced this doesn’t amount to a denial of Omnipotence myself, but I am completely unable to wrap my head around what it would be to do the logically impossible.)

Anyway, here’s a real-life case in which I think infinity may come up:

Before humanity started doing serious cosmology, how old should we have thought that the universe was? We should have realized that it could be infinitely old, and that it had to be pretty old. But past that I don’t think that in advance there was any principled reason to distinguish between the hypothesis that it was a million years old, or a billion years old, or a trillion…. This looks to me as though there’s a distribution in which a non-zero amount of credence should’ve been spread more or less evenly across an infinite number of possibilities.
[Also, if someone can explain to me why I can never get white space into my comments on this site, I would appreciate it.]

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Dave 05.31.04 at 6:37 pm

Matt – you’re essentially posting in HTML, so if you want to include extra whitespace, you need to use HTML entities. Use &nbsp; for a non-breaking space. You can also (evidently) insert <br /> for a line break and separate your text into paragraphs with <p> and </p>.

As for the age of the universe argument, you are falling into a common trap. You are assuming that if we don’t know a prior distribution, it must be uniform. Assuming a prior is uniform causes the prior to stop being a valid probability distribution, and is the source of a large number of logical fallacies.

First, the math: probability distributions must be normalized to 1. That means that if you have a distribution over a numeric range, the integral of the distribution over the range must be 1. Taking the integral of any constant (uniform probability) over a range of time of length T requires the constant to evaluate to 1/T. If we let T tend to infinity, the probability tends to zero – but since P(t) = 0 is not a valid probability distribution, we cannot have a uniform probability over an infinite range of times.

Now, what problems does assuming a uniform prior (or for that matter, any arbitrary prior) cause? Pascal’s Wager is the easiest example of the mis-application of a prior. By assuming that the probability of Pascal’s notion of God existing is non-zero and that the probability of any other God(s) existing is zero, Pascal forces the conclusion he wants. However, since even Pascal posits that the probability of the existence of his God is unknown, it is not reasonable to apply an arbitrary prior.

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Matt Weiner 05.31.04 at 7:11 pm

Well, I’m not trying to say that there has to be a valid distribution that’s uniform over the possible times. I’m thinking more along these lines: This case shows that there will be cases where the credence it is rational to have can’t be captured by a single valid probability distribution. If we can’t have a uniform probability over an infinite span of time, perhaps our pre-scientific estimate of the age of the universe can’t be captured using any well-defined distribution as a prior. I certainly don’t find any reason in probability theory to think that, before we started out, we must have been able to assign higher probability to one age than another age.

I don’t think Pascal’s wager is particularly relevant here. Pascal’s wager shows the danger of assigning wacked-out priors, but that’s got nothing to do with infinity. (Actually, Pascal has arguments that no other God than his can exist; the problem is that these arguments stink.) The problem with Pascal is that his priors are biased toward his religion; the question I’m asking is whether, in the age-of-the-universe case, it’s possible to assign a prior without introducing bias to some time.

For instance, it would be possible to assign a prior where older ages get progressively more unlikely (so the curve tails off as it gets to infinity), but would that be a good representation? Say I do one experiment that yields a result that has probability 1 if the universe is a billion years old and probability 1 if the universe is a quintillion years old and probability 0 otherwise. If we apply Bayes’ theorem to a tailing-off distribution, the result is that after this experiment it’s much more likely that the universe is a billion years old (I think). I just don’t see why that should be so.
(and thanks for the HTML tip.)

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armando 05.31.04 at 8:45 pm

For any function to the reals, there is a minimum a maximum and some number in between not in the range of that function.

Always true for finite sets.

Now, when I throw a ball into the air, it has a maximum height and minimum height. Is there some height in between that it doesn’t reach?

Perhaps this problem is beyond the realm we are considering, or maybe we get round it by denying the existence of the reals. But then we have to be careful to not make the statement true solely by definition or, alternatively, we have to accept some rather awkward consequences of finitism.

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armando 05.31.04 at 8:46 pm

For any function to the reals, there is a minimum a maximum and some number in between not in the range of that function.

Always true for finite sets.

Now, when I throw a ball into the air, it has a maximum height and minimum height. Is there some height in between that it doesn’t reach?

Perhaps this problem is beyond the realm we are considering, or maybe we get round it by denying the existence of the reals. But then we have to be careful to not make the statement true solely by definition or, alternatively, we have to accept some rather awkward consequences of finitism.

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Lance Boyle 05.31.04 at 9:26 pm

I’ve no math for this, but it seems clear that we’re only finite in an exterior sense. We have exterior boundaries. Skin etc.
The interior limits are simply the reach and precision of our investigative tools. Those priestly accolytes coming back from the edge with their “quark”s and their “Eureka!”s; the stopped progression down and in; that always seemed kind of strange to me. Zeno’s traffic cops. Or hall monitors.
It doesn’t stop in there, any more than it stops outside our skin. There may well be a jumping-off point, inside as well as out, but an end? Oh sure. And the moon is made of green cheese – really hard, stale green cheese. Way down inside.
The reduction of the infinite interior to inconsequence is due to its unavailability as resource alone and not some physical property of the universe.
But, again, I have no formulae to back that up.

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Dave 05.31.04 at 10:03 pm

If we canâ€™t have a uniform probability over an infinite span of time, perhaps our pre-scientific estimate of the age of the universe canâ€™t be captured using any well-defined distribution as a prior.

I think the problem is that “our pre-scientific estimate of the age of the universe” is not being properly defined. Since there was no consensus, the “our … estimate” part can’t exist (since there was not one estimate). If you’re asking about the perception of the age of the universe among all pre-scientific people, then that does define a valid distribution, though offhand I would have no idea was it was. You would almost certainly get a percentage of people who would have claimed that the universe had been around forever, so the “average” age would be infinite. However, as I pointed out in a previous post, it’s the distribution itself that is interesting, and not some sort of “average”. Perhaps you want the median instead?

As for the ball traversing infinitely many points in space, I think you have a couple of problems. First, it’s possible (in fact quite likely) that spacetime itself is quantized, so that there are indeed only a finite number of possible locations for an object (though that number is quite large). Second, even if it is not, at arbitrarily small scales it becomes arbitrarily hard to figure out where exactly objects – or rather, the particles that compose them – are.

Third, even if you could pin down exactly the location of an object traversing a known course at a particular time (in violation of all the laws of nature), you’re talking about a completely different kind of infinity than the previous conversation. There is no infinite value in [0, 1], only an infinite number of values – and we can sum infinite numbers of values just fine using integration. Since any probability distribution normalizes to 1, we can show that E[X] where X is any random variable over [0, 1] satisfies 0 ≤ E[X] ≤ 1.

If you’re dealing with values in the range [y, ∞), you’re still okay as long as your distribution P(x) goes to zero sufficiently rapidly as x -> ∞. It’s only when you try to have a distribution over [y, ∞] (whatever that menas) that you have a problem. I think you have to take as a separate case whenever the value is actually infinity.

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WillieStyle 05.31.04 at 10:38 pm

Now, when I throw a ball into the air, it has a maximum height and minimum height. Is there some height in between that it doesnâ€™t reach?

Quantum Mechanics would suggest the answer is yes.

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armando 05.31.04 at 11:06 pm

To a large extent I think that the acceptance of an infinite set and the rejection of an “infinity” is a touch incoherent. To the extent that one is opposing what no one would support – that “infinity” is not a real number – there is no actual argument.

(Having said that, infinitesimals and the corresponding infinities can be put on a sound footing.)

As for quantum mechanics…I think that is perfectly defensible, and comes under the heading of rejecting the reals. Though to form the basis of mathematics, you would essentially have to throw out a lot of calculus. Intuitively, this seems like a problem to me.

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Jason 05.31.04 at 11:46 pm

You probably wouldn’t have to throw out much calculus, but the notation and work involved would become a lot more challenging (you’d have to keep the error terms all the time).

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Jay 05.31.04 at 11:50 pm

As for quantum mechanicsâ€¦I think that is perfectly defensible, and comes under the heading of rejecting the reals.

*Sigh* Statements like this are why mathematicians often lose patience with philosophers, or at least the subset of philosophers who make muzzy-headed statements like this.

“Rejecting the reals?” What does that even mean? The set of real numbers, like all mathematical objects, is merely an abstract object that is consistent with a set of agreed upon axioms. They just happen to be used in science because they provide a more-or-less accurate model for the universe that we observe. The fact that the quantization of matter and energy does not agree with a non-terminating representation of a real number just happens to show the limitations of the model.

To say that the fact that our construction of the real numbers doesn’t perfectly correspond to our physical observations is a cause for “rejecting” them is essentially a statement of a lack of understanding of the relationship between mathematics and science.

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Jay 05.31.04 at 11:52 pm

As for quantum mechanicsâ€¦I think that is perfectly defensible, and comes under the heading of rejecting the reals.

*Sigh* Statements like this are why mathematicians often lose patience with philosophers, or at least the subset of philosophers who make muzzy-headed statements like this.

“Rejecting the reals?” What does that even mean? The set of real numbers, like all mathematical objects, is merely an abstract object that is consistent with a set of agreed upon axioms. They just happen to be used in science because they provide a more-or-less accurate model for the universe that we observe. The fact that the quantization of matter and energy does not agree with a non-terminating representation of a real number just happens to show the limitations of the model.

To say that the fact that our construction of the real numbers doesn’t perfectly correspond to our physical observations is a cause for “rejecting” them is essentially a statement of a lack of understanding of the relationship between mathematics and science.

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JSN 05.31.04 at 11:52 pm

Is what is possible finite?

The universe is effectively infinite.

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armando 06.01.04 at 12:09 am

Statements like this are why mathematicians often lose patience with philosophers, or at least the subset of philosophers who make muzzy-headed statements like this.

In the context of the statement, it seemed a reasonable summary of the response I got. I am perfectly aware of the status of the reals. By “rejecting the reals” I meant rejecting their applicability. Of course, there are philosophies of maths that *do* reject the reals, and it is a position that one should at least consider in a discussion like this. I don’t think that is particularly “muzzy headed”, even though I don’t buy into finitism myself.

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Bill Carone 06.01.04 at 1:41 am

Anarch,

“you don’t need a completed infinity in order for some of these paradoxes to go through, only a quantity capable of arbitrary finite magnitude.”

You do need to use an infinite set to describe all possibile arbitrary finite magnitudes. The problems start when you e.g. sum over this infinite set either

1) Without taking a limit, or

2) Taking limits differently in different parts of your argument (e.g. sum of limits and then limit of sums), or

3) Taking limits too early (you should finish your argument all the way to its conclusion, then take the limit, when possible).

I claim that if you avoid these things by always modelling infinities as limits, specifying the exact limiting procedure used to generate the infinite set, and taking the limits as late as possible, you will never produce paradoxes, and you will produce intuitively satisfying results, when such exist.

You will often be put into the situation of the mathematics refusing to give an answer, when different limits give different results, or when limits do not converge.

Counterexamples would be welcome: a useful infinite calculation that can’t or shouldn’t be modelled as a limit.

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Bill Carone 06.01.04 at 2:34 am

Matt,

“If we can’t have a uniform probability over an infinite span of time, perhaps our pre-scientific estimate of the age of the universe can’t be captured using any well-defined distribution as a prior.”

Let me start out by agreeing, then disagreeing :-)

Say I model my information about the age of the universe as equally likely between 0 and T (not how I would do it, but for now lets go with it). Then, I do whatever calculation I need to, then take T to infinity.

Problem is, this only works in some cases. Let’s say God told me that I had to guess the age of the universe. He would give me a huge reward if I was right, but if I was wrong He would reduce that reward by the size (absolute value) of my error. I work through the math and, alas, my best guess (the median of my prior) diverges when T increases without bound.

My interpretation of this is that mathematics cannot help me make my guess if I really have this much ignorance about the age of the universe (can’t put a finite upper bound on T, for instance). Human mathematics just isn’t always successful when dealing with infinity; things sometimes fail to converge (maybe angelistic mathematics is better, but I don’t know anything about that :-). By refusing to give an answer, the model is telling me that there is an infinity in the problem that mathematics simply cannot handle.

However, this “prior” will be useful in many other cases. Let’s say I take a measurement that is relevant to the age of the universe. I can use my [0,T] prior, and often will end up with the same answer no matter how large T gets, and so I would be able to find an answer. Similar things can happen when rewards and losses are bounded; the median of my prior might diverge, but my best guess might not.

In some sense, then, while we cannot encode this information as a probability distribution, we can as a limit of distributions (taking the limit at the very end of the problem), and this will give good answers to some problems, while mathematics will refuse to give answers to others\, when it can’t deal with the infinities.

Now, I am making some strong statements here, based on what I know about Gauss’s position on these matters. So, one counterexample would be a problem that “direct infinite calculation” could answer that limits cannot. Then, all my “mathematics fails” arguments above would fail.

Also, I may not know what I am talking about when I say “mathematics.” I probably just mean elementary analysis, but maybe not. For example, maybe there is no problem in defining infinite sets, only in calculating with them.

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Mike Ruchlis 06.01.04 at 2:51 am

I find Matt Weiner’s example fascinating. I agree with him that a priori I see no basis on which to say that the probability of the universe being 1 trillion years old is smaller than or greater than it being 1 billion years old. If we substitute STUs (Standard Time Units) for “years” this becomes clear. If there is no knowledge of the time scales involved then there is no way to distinguish between 1 trillion years and 1 trillion STUs which can of course be an arbitrary number of years.

Now mathematically it is clear that we cannot have a probability distribution that is equal for all values. So it would seem that there is no way to have a probability distribution at all for this problem. But perhaps this isn’t a paradox. Perhaps it just means that probability distributions make no sense unless we have some information that can be used to assign probabilities.

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Bill Carone 06.01.04 at 2:57 am

Armando,

“For any function to the reals, there is a minimum a maximum and some number in between not in the range of that function. Always true for finite sets.”

I think John’s “strictly” clause solves this problem. Real numbers can be represented as limits of numbers with finite decimal expansions.

“Now, when I throw a ball into the air, it has a maximum height and minimum height. Is there some height in between that it doesn’t reach?”

Start with a finite set of N possible positions the ball could be. Pick a (real-valued) point between the min and the max. As N increases, the ball will get as close as you want to that point. So, in the limit, the ball will get as close as you want to any point, right?

I don’t think there is any problem with this “kind” of infinity, one that is the result of a limiting procedure. Such a limiting procedure cannot produce one of John’s strict reversals, right? I therefore don’t think this is a counterexample to John’s point.

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Kenny Easwaran 06.01.04 at 7:54 am

This sounds a lot like a version of the Lowenheim-Skolem-Tarski theorem, which says that if a first-order theory has arbitrarily large finite models, then it has infinite models. Thus, anything that’s true in all but finitely many of the finite models has to be true for (at least some of, and probably all of, if we can get it phrased right) the infinite models, provided that it can be formulated in first-order logic. Of course, there are some theories that only have infinite models, which may pose some problems for this view. But at least any theory that has an infinite model has infinite models of every cardinality. (At least, every cardinality at least as great as that of the language the theory is phrased in, which we generally assume is countable.)

One other point to a lot of the commenters here – if we don’t assume infinite additivity of probabilities, they we can in fact have a probability distribution that assigns equal probability to an infinite collection of events: just set them all to have probability zero. By finite additivity, we see that every finite set has probability zero, and that any cofinite set has probability one. But it doesn’t tell us anything about the probabilities of events that are infinite and coinfinite, like the set of evens or odds, for instance. This seems to fit with the intuitions about the age of the universe, which we think is extremely unlikely to be X, whenever X is a number that is given to us, or even a range that doesn’t go off to infinity.

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dsquared 06.01.04 at 8:01 am

I tend to regard most of these paradoxes as being of John’s type 2, so long as you consider the human ability to ignore genuine problems and arbitrarily chop down a set without a well-formed probabiltiy maesure to one with such a measure, as a “triumph of the human spirit”.

I would also caution John and Bill that I suspect the dastardly Weatherson’s next move is to recast all of his paradoxes in a Goedelian form …

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plover 06.01.04 at 8:57 am

Armando said:
Now, when I throw a ball into the air, it has a maximum height and minimum height. Is there some height in between that it doesnâ€™t reach?

Bill Carone responded:
Start with a finite set of N possible positions the ball could be. Pick a (real-valued) point between the min and the max. As N increases, the ball will get as close as you want to that point. So, in the limit, the ball will get as close as you want to any point, right?

This is all fine mathematically, but as a physical example becomes problematic for the case of a quantized (granular) spacetime (which is what the universe increasingly appears to be).

Taking the naive case of a granular spacetime as a medium that a) consists of pixel-like packed 4-dimensional hypercubes, and b) whose structure is unaffected by its contents, there is just a (large) discrete set of positions that an object may occupy, so any talk of a length measurement more precise than the length of one of the space-like dimensions of a single granule must be reduced to the discrete case in some fashion.

For the example of the ball: using the same naive granular spacetime, putting aside quantum considerations, and taking the length a granule to be P, it becomes necessary to define what is meant by “reaching” a given height n*P if n is not an integer.

Of course, a model this simplistic immediately raises a problem akin to Zeno’s paradox – that of how the ball can move from granule n to granule n+1.

On the other hand, if the effective size of the granules can be affected by the local distribution of matter, there’s all sorts of fun to be had with hyper-precise measuring (providing that quantum effects are still neglected…).

If the universe is not granular, however, quantum effects also guarantee that measurements beyond a certain precision become impossible. In order to localize a measurement more and more precisely, one needs a measuring probe of higher and higher energies. Beyond a certain point the concentration of energy required to create a suitable probe will either produce a black hole (destroying any way of recovering the measurement), or break something (and by “something” I mean the actual physics of spacetime).

So now the problem becomes one of defining “reaching” a given height with respect to measurements smeared out by quantum effects.

I’m not sure how any of this might or might not affect Bill’s thesis about infinities. I’m more just pointing out the degree to which the physics may need to be specified to make even an idealized version of one of these arguments.

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armando 06.01.04 at 9:58 am

I donâ€™t think there is any problem with this â€œkindâ€ of infinity, one that is the result of a limiting procedure. Such a limiting procedure cannot produce one of Johnâ€™s strict reversals, right? I therefore donâ€™t think this is a counterexample to Johnâ€™s point.

I think you are making the same point as before, Bill, that we should be careful not to treat the infinite sloppily. But, if that is John’s point, it is rather a weak one since I don’t think there is anyone who disagrees.

More precisely, if you allow the infinite obtained by a suitable limit, I am not sure that you exclude anything except the contradictory. But what about the paradoxes that are mathematically careful, like Banach-Tarski?

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Anarch 06.01.04 at 9:59 am

[Warning: long and technical.]

Bill Carone: You do need to use an infinite set to describe all possibile arbitrary finite magnitudes…

If you mean you need an infinite object in the metatheory then yes, I completely agree; but then again, I don’t know any system of logic that disallows infinite objects in the metatheory so I don’t think that’s the contentious issue.

If you mean you need infinite objects in the theory then I think you’re wrong. Following along Matt Weiner’s argument above, you can work as follows:

• Define an effective model of the field of the rationals with less than and (perhaps unnecessarily) a predicate for the integers, i.e. a computable model of (Q; 0, 1, +, x, -, <, N)
• We can then define a computable function on that model that I’ll call d(t), for decay, as follows. Define d(n) = 1/2^n then extend to the rationals via the enumeration of Q and (dyadic) interpolation. For example, if d(2/3) = 2/3, d(3/4) = 1/5 and the next rational in the enumeration of Q is 7/10, then set d(7/10) = (1/2) [2/3 + 1/5] = 13/30. Note that d(t) is a strictly decreasing computable function on Q that is O(2^-n).
• In a similar way, we can define payoff functions f and g, with the appropriate limiting behaviors; for example, we could set f = 1/d and g = f – 1. These are, again, computable functions.
• The next step is a little tricky, but still manageable. We’d like to define the integral of these functions over Q in a finitistic fashion. There are two ways of approaching this:

Method 1: Since f, g and d are all strictly monotonic, they have unique continuous extensions f*, g*, d* to the reals (either by Dedekind cuts or a Urysohn-style argument). Since these extensions are continuous, they are locally integrable — and note, in particular, that d* is integrable over the entire real line (being O(2^-n)) and that f*d* and g*d* have infinite integrals over R.

Method 2: We define the “integral” of a function h over the interval [a, b], a & b rational, as follows. Let {x_1, …, x_n} be a finite — and thus computable — sequence of rationals between a and b. Let L(h, {x_1, … x_n}) be the usual lower Riemann sum associated with that partition; by the usual theorems on lower Riemann sums there is a well-defined limit alpha over R. Since we’re working over Q, however, alpha won’t necessarily exist in our model. We can, however, define a predicate “e-integral” E(z, e) as follows:

E(z, e) iff every sequence {x_1,…,x_n} has a (definable) refinement {y_1,…,y_n} with | L(h, {y_1,…,y_n}) – z | < e.

This is a definable predicate over PA, although no longer a computable one.

• Via a similar limiting trick (i.e. up to a fudge factor of e) we can define integrals over all of Q. That is, a function h is integrable over all of Q if there is a N such that given any sequence xbar, L(h, xbar) <= N; and, in that case, we can define the e-integral over all of Q as above.
• Finally, with all that under our belt, we can set up the appropriate paradoxes as before. Since fd and gd have infinite integrals over Q, we can (with finitely many parameters, if need be) prove the claim that, given any actual value of f relative to the distribution d, the expectation of g relative to the distribution d strictly exceeds that value — and is, indeed, infinite.

In short: we can set up the whole paradox definably in PA, without any need for recourse to infinite objects in the theory.

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Anarch 06.01.04 at 10:11 am

kenny easwaran: This sounds a lot like a version of the Lowenheim-Skolem-Tarski theorem, which says that if a first-order theory has arbitrarily large finite models, then it has infinite models.

FWIW, I’d consider that a direct application of Compactness rather than LST, but YMMV.

Bill Carone: Counterexamples would be welcome: a useful infinite calculation that canâ€™t or shouldnâ€™t be modelled as a limit.

Well, the above paradox works for one — unless you alter the problem to specify joint limiting behavior. As I mentioned on a previous thread I have a bunch of counterexamples, for appropriate definitions of “useful” of course :)

[One trivial example: compute lim_{m, n \to \infty} m/n. It’s not that mathematics “doesn’t give an answer”, it’s that the limit simply doesn’t exist.]

Also, one other crucially important point: if you’re truly trying to be careful about dealing with infinitary objects, saying that “Real numbers can be represented as limits of numbers with finite decimal expansions” isn’t actually correct, since the sequence of approximants might not be definable in a finitistic fashion. In fact, you can show that the collection of reals “effectively specifiable” in this manner is countable; so in some very strong sense almost no real is given in this matter.

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Anarch 06.01.04 at 3:25 pm

Blecch. Mistakenly hit post instead of preview. Let me try that “crucially important point” again…

If you’re being truly careful about dealing with infinitary objects, saying that “Real numbers can be represented as limits of numbers with finite decimal expansions” doesn’t actually work, since the (infinite) sequence of approximants might not be definable in a finitistic way. In fact, the collection of “computable reals” is countable, so in some very strong sense almost no real can be specified in this fashion.

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Bill Carone 06.01.04 at 3:43 pm

Anarch,

“[One trivial example: compute lim_{m, n \to \infty} m/n. It’s not that mathematics “doesn’t give an answer”, it’s that the limit simply doesn’t exist.]”

My “mathematics doesn’t give an answer” is a glib way of saying that the limit doesn’t exist. It is as if our model is telling us “No way, pal. I ain’t touching that.”

“Real numbers can be represented as limits of numbers with finite decimal expansions”

All I mean is that, e.g. pi/4 = 1 – 1/3 + 1/5 – 1/7 + … Pi is also equal to the limit of this series: 3, 3.1, 3.14, 3.141, 3.1415 … I’m not sure what you mean by “not definable in a finitistic fashion.”

Let me use another example: say we have an interval between 0 and 1. We fill it with N equally spaced points. Pick a real number. As N goes to infinity, one of our equally spaced points will get arbitrarily close to that real number, right? Or am I confused?

PS I would like to see more counterexamples, no matter how useful :-) Thanks!

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q 06.01.04 at 3:53 pm

_Any property that depends inherently on infinite sets and limits, such as the continuity of a function, can never be verified or falsified by empirical data. Since we are finite, any result that is true for all finite n is true for us._

JQ-

Can you clarify this statement? What do you mean by “verified by empirical data”? A finite dataset has an infinite number of solutions. Don’t we use Occams Razor to choose the best one? Sometimes the best one involves a concept of infinity.

Intrumentalist Q.

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Bill Carone 06.01.04 at 4:00 pm

Armando,

“I think you are making the same point as before, Bill, that we should be careful not to treat the infinite sloppily. But, if that is John’s point, it is rather a weak one since I don’t think there is anyone who disagrees.”

Did you look at the Elga et al. paper John cited? They seem to think they are on to something, so they would disagree that they are being sloppy. However, if you look at all their problems using limits, you get non-paradoxical results. This is why I am making my point ad nauseum (that and to collect possible counterexamples to think about).

“More precisely, if you allow the infinite obtained by a suitable limit, I am not sure that you exclude anything except the contradictory.”

That’s all I ask for :-) All I’m trying to do is to argue against people using “sloppy” infinity-type paradoxes to support statements like “The indifference principle/the sure-thing principle/dominance arguments/etc. lead to paradox, so we should scrap them.”

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armando 06.01.04 at 4:10 pm

Bill,

That infinity is a tricky fellow. For instance, this

As N goes to infinity, one of our equally spaced points will get arbitrarily close to that real number, right?

rather begs the question of what a real number is. Seriously, it is not a priori obvious that allowing all possible sequences – even ones which are non-computable – is valid. I mean, describing a real which is given by a non-computable sequence is…tricky. And yet the standard construction of the reals says that “most” reals are exactly of this form.

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Anarch 06.01.04 at 4:23 pm

Let me use another example: say we have an interval between 0 and 1. We fill it with N equally spaced points. Pick a real number. As N goes to infinity, one of our equally spaced points will get arbitrarily close to that real number, right? Or am I confused?

You’re correct. The problem is that you may not be able to determine in an effective fashion which of those N points will be the closest.

I have to run, but here’s a specific example of a real where things get weird: let K be the canonical halting problem set (i.e. the set of all n such that the nth machine halts on input n), and let x be the real whose binary expansion is the characteristic function of K. [For example, if K = {1, 2, 4, 7, …} then x = 0.1101001… ] Then since K is not computable, there is no computable way of specifying the digits of x — i.e. there is no computable function f such that the nth digit of x is f(n).

Now that particular x did have the property that it had computable approximations. That is, I can find a computable sequence x_s that converges to x in the usual analytic way (by taking the canonical K_s computable approximants to K). However, only particularly “non-complex” sets — I think Delta_2 is as far as it goes — can be approximated in this way.

So take a set that’s “sufficiently complicated” (the theory of arithmetic ought to work) and let y be the real whose binary expansion is its characteristic function. Then y has the following two properties:

1) Every individual rational approximation to y is computable, because any rational number is computable.
2) There is no computable sequence of rationals y_s converging to y.

As I mentioned above, it turns out that there are only countable many reals which can be computably approximated, i.e. which are the limit of a computable sequence of rationals. Most of the usual real numbers are c.e. — in particular, any algebraic number is c.e., as well as pi, e, and so forth — so I’m not sure how much you’d lose for practical applications… but in theoretical terms, almost no reals are approximable in this way.

[My former officemate is writing his PhD on c.e. reals and random reals, so I can probably shag some more stuff about them if you’re interested. Although not for about six weeks :)]

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Dee Lacey 06.01.04 at 4:24 pm

Musing about the a priori probability of the age of the universe, and reasonable distribution.

I think that though it does seem intuitive that the likelihood of the age of the universe being either a billion or a trillion years is about equal, it seems counter-intuitive that the likelihood of the age of the universe being between a trillion and a quadrillion years is 1000 times more likely than being between a billion and trillian. So the real intuitive concept is that each “even number” is a stand-in for its order of magnitude, and that it is each order of magnitude that is equally likely, and so the probabilities actually shrink as the age grows, so it does eventually sum to 1 at the limit.

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Dee Lacey 06.01.04 at 4:31 pm

Here’s another musing:

If the probability were evenly distributed over all positive integers, then if you take any positive integer, no matter how large: there are a finite number of integers below it, and an infinite number above it. So the odds are .99999….. that the actual answer is higher. This makes the answer effectively infinity….

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DR 06.01.04 at 6:45 pm

It certainly appears from the various paradoxes that, if God is capable of handing out infinite rewards and punishments (which is, I think, generally supposed by believers), itâ€™s not valid to say that, if a given course of action is better than another in every possible case (for some partition of the possible cases), then it is definitely the best choice.

I’ve been having an email discussion with someone on what exactly the above means. I’m on the side of “nothing”.

If I have an ordered list of courses of action (which is given by your statement that “a given course of action is better than another in every possible case”), what is it about the fact that some of those courses result in infinite rewards/punishment that keeps me from simply performing a max() on the list?

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John Quiggin 06.01.04 at 9:40 pm

dr, This post by Brian gives an example of the kind of problem I’m talking about, as does the paper by Elga and others cited in the post.

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Matt McIrvin 06.02.04 at 4:51 am

Here’s a counterexample. Consider the truth value of the following sentence:

If Georg Cantor proved the uncountability of the real numbers, then this sentence is false.

Now, it is an empirical fact that Georg Cantor did prove the uncountability of the real numbers, so the sentence reduces to “This sentence is false” and therefore has no definite truth value. The paradox applies only to the infinite case, since Georg Cantor did not prove the uncountability of any finite set, just of the infinity of the reals. However, the paradox manifestly has nothing to do with ill-posed statements about infinity; it’s just the old Epimenides paradox with another clause tacked on.

So here is a paradox depending on an empirical fact about infinity that is not resolved by anything having to do with infinity or limits.

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John Quiggin 06.02.04 at 7:03 am

Neat! We could also do the self-reference paradox, by observing that “infinite” is a finite word and therefore a paradigm instance of the class of non-self-descriptive adjectives, while “finite” is self-descriptive. Hence, we might as well use the words “infinitistic” and “finitistic” to describe these classes, and ask the question
“Is “infinitistic” infinitistic?”.

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