If the worries in this post work, then the argument in this one needs improvement.
Suppose there are two groups of people, the As and the Bs, all of whom have headaches. You can relieve the headaches of the As or of the Bs, but not both. You don’t know how many As or Bs there are, or even whether the numbers are finite or finite. But you do know there are more As than Bs.
Obviously:
- You should relieve the As’ headaches rather than the Bs’, because there are more As than Bs.
But what does it mean to say that there are more As than Bs? Our best analysis (simplifying and assuming the Axiom of Choice) is something like this:
- There is no one-to-one function from the As to the Bs.
So, it seems:
- You should relieve the As’ headache rather than the Bs’, because there is no one-to-one function from the As to the Bs.
For you should be able to replace an explanation by its analysis.
But that’s strange. Why should the non-existence of a one-to-one function from one set or plurality to another set or plurality explain the existence of a moral duty to make a particular preferential judgment between them?
If the number of As and Bs is finite, I think we can do better. We can then express the claim that there are more As than Bs by an infinite disjunction of claims of the form:
- There exist n As and there do not exist n Bs,
which claims can be written as simple existentially quantified claims, without any mention of functions, sets or pluralities.
Any such claim as (4) does seem to have some intuitive moral force, and so maybe their disjunction does.
But in the infinite case, we can’t find a disjunction of existentially quantified claims that analysis the claim that there are more As than Bs.
Maybe what we should say is that “there are more As than Bs” is primitive, and the claim about there not being a one-to-one function is just a useful mathematical equivalence to it, rather than an analysis?
The thoughts here are also related to this post.
Alex,
ReplyDeleteHow about the following alternative?
1': You should (all other things equal, assuming no costs, etc.) relieve the As’ headaches rather than the Bs, because on the basis of the information available to you, it is probable that that course of action will result in less suffering than any alternative.
The intuitive probabilistic assessment seems true. That said, it's very difficult to make probabilistic assessments with so little info about the As or the Bs or why you can relieve their headaches, etc.
Side note: In the post on numerosity, why are A and B subsets of S? Your hypotheses imply that both A and B are elements of S, but they might not be subsets. Am I misreading?
But if "more" just means there is no one to one function, then the problem comes back.
DeleteS contains all the sets, and hence all the elements of A and B, since the context is pure sets, I.e., sets whose members are sets, all the way down.
DeleteIn the context of pure or hereditary sets, it is not true that for any sets X, Y, Z, if X is an element of Y and Y is an element of Z, then X is an element of Z. For example, let S1 be
ReplyDeleteS1:={ aleph_n: n is a nonnegative integer}.
The set S1 is denumerable, but it contains non-denumerable elements, and it's a pure set (on some constructions of the cardinals, at least. For example, we can use von Neumann's definition of the ordinals, define the cardinals as a class of ordinals).
With respect to suffering, the fact that there are more As than Bs, is evidence (all other things equal) that the Bs' headaches are a state of affairs with less overall suffering. If you like, we can say that the As' headaches are a state of affairs with more overall suffering, but here, "more" definitely does not mean that there is a 1-to-1 function between the sets of As and Bs. If - for example -, it were stipulated that there are 10000 As, and 10000 Bs, but As' headaches are far more intense, it seems to me there would still be an obligation to relieve the As (all other things equal, no further costs, etc.).
ReplyDeleteGranted, in the scenario you present, the evidence for the assessment that there is more suffering in the situation in which the As suffer headaches is given by the stipulation that there are more As than Bs. But even granting that "more" means there is no 1-1 function, etc., the obligation (in the alternative I suggest) results from their being more overall suffering, not from their being greater numbers. Now, you might ask why greater numbers is evidence (all other things equal) of greater overall suffering. But that seems intuitively obvious to me. Moreover, if it's obvious in the finite case, that seems to suffice for an obligation, since in any scenario like the one you present and without further conditions, one should reckon that very probably, the numbers are finite - else, it would be pretty difficult to have conclusive evidence one can affect them in such a precise way as to relieve the As pain, etc.
If S contains *all* the sets at some world w, and A and B are pure sets at w, then S contains all the members of A and B, since these members are sets, and hence members of S.
ReplyDeleteAs to the suffering, fair enough. But the question comes up again when we ask: What makes it be the case that one situation involves "more" suffering than the other? First, simplify and suppose that suffering is made up of atoms that are minimal and equal units of suffering. Then to say that one situation has more suffering than another seems to mean that there is no one-to-one function from its units of suffering to the other's units of suffering. But, again, why should this fact about the non-existence of a function matter morally?
ReplyDeleteOf course, it doesn't seem right to suppose that suffering is constituted of such atoms. But we *still* have the question of what it means to say one situation has "more" suffering than another, especially when the situations involve an infinite quantity of suffering.
By the way, I can recover my original case when we consider things other than suffering. For instance, the As and Bs are all innocents in danger, and I can save either the As or the Bs. Which should I do? Well, all other things being equal, the more numerous group. But why should the non-existence of a function affect which group I should save?
In re: sets, I see you're taking that all of the elements of the sets A and B also exist at the world w2 (and also, A and B are pure sets). But in that case, I still see a problem. I'm not sure whether I misunderstood one of your hypotheses, but a contradiction seems to follow:
ReplyDeleteIt follows from 1. and 2. that there is a possible world w3 at which there exists S(A), the set whose elements are all of the sets that exist at the actual world A. Also by hypothesis, all of the ZFC axioms hold at w3. Thus, it follows that the set P(S(A)) (the power set of S(A)) exists at w3. Since there is a bijection between S(A) and the set of natural numbers, there is also a bijection T3 from the power set of the set of natural numbers onto P(S(A)). Let w4 be a possible world at which the set S(w3) exists. There is a bijection T4 from S(w3) onto the set of natural numbers. But P(S(A)) is an element of S(w3), so it is a subset of S(w3). It follows that T4oT3 is an injective function from the power set of the set of natural numbers into the set of natural numbers.
In re: suffering.
ReplyDeleteIt seems to me that we can intuitively make sense of the assertion "Alice experienced more pain than Bob today", or "I experienced more pain yesterday than the day before yesterday", and so on. More pain may involve more intense pain for the same amount of time, or equally intense pain for a longer period, or both. One may wonder whether "more pain" can involve, say, more intense pain for a shorter period. Maybe we can dissolve the question, and simply say it's more intense pain for a shorter period.
That's as far as pain goes. The issue of suffering complicates things, among other reasons because sometimes, more pain may not imply more suffering. Perhaps, intense pain experienced by 1/10 of a second by 1000000000000000 people involves all other things equal less suffering than equally intense pain experienced by 1 individual for 10000000000000 seconds, even if it involves 10 times less pain. Or maybe not, that's debatable. But in the scenario you present, there does not seem to be such a problem.
Granted, one could still ask the question. But one can always keep asking "why", and at some point, I don't see any other answer than its being intuitively clear: if - say - more innocents get killed, that is a worse result, and then, there is (let's say, all other things equal) a moral obligation to prevent the worse result. If for some reason, that is not morally obligatory, then one can say it's morally praiseworthy to save the greater number. As always, one can still ask the question of why, and it does not seem to depend on whether one phrases this in terms of functions (at least, not in the finite case), but I'm not sure why this is problematic.
The phrase "the power set of the set of natural numbers" can denote a different object in different worlds. T4oT3 is a function from w3's power set of the naturals to the naturals, but that's ok as it isn't a function in w3, but in w4. And in w4 the power set of the naturals will be something different.
ReplyDeleteBy "the power set of the set of natural numbers", I mean the set that the ZF axioms of power set and specification yields, which is the set whose elements are all of the subsets of the natural numbers (and has other elements). That the sets are in different worlds does not matter, as long as the set of natural numbers is the same in both cases - i.e., it is the set {1,2,3...}.
ReplyDeleteThe function T4oT3 is a 1-1 function from the set of all subsets of {1,2,3...} into the set {1,2,3...}, but that is a contradiction. Granted, you could stipulate that "set of natural numbers" picks different objects in different worlds. But then it seems to me you would have to provide a definition of "natural numbers".
Further clarification: If you think that the different worlds make a difference, I don't agree, but in any case, one can slightly modify the previous argument and argue as follows:
ReplyDeleteAll pure sets that exist in w3 are elements of P(w3), which exists in w4. They are (by stipulation) also subsets of w4, and so are all of their elements, etc. Since (also stipulated) all elements of a set in w4 exist in w4, in particular, all pure sets that exist in w3, also exist in w4, and so do functions between any two of such sets that exist in w3 (we can always look at functions as sets of pairs, which are also hereditary sets given some standard constructions in set theory).
In particular, the set of subsets of the set {1,2,3...} in w3 exists in w4, and is a subset of P(w3). So, there is a 1-1 function from the set of all subsets of {1,2,3...} in w3 into the set {1,2,3...} in w4. But the set {1,2,3...} is the same set (or an identical counterpart if you like), and so is the sets of subsets (in w3 or w4). A contradiction can be derived by a standard argument.
"The function T4oT3 is a 1-1 function from the set of all subsets of {1,2,3...} into the set {1,2,3...}, but that is a contradiction."
ReplyDeleteIt is a 1-1 function in w4 from the set of all the subsets that {1,2,3,...} has in w3 to the set {1,2,3,...} That is only a contradiction if {1,2,3,...} has the same subsets in w4 as in w3. But in w4, {1,2,3,...} has more subsets than it does in w3.
It is intuitively very strange that a set might have more subsets in some worlds than in others. But if we think of different worlds as coming along with different intended models of ZFC, this is just a twist on standard stuff in set theory. As you vary models of set theory, you get subsets of the naturals coming in and dropping out. By Lowenheim-Skolem, if the "intended model" of ZFC includes an inaccessible cardinal, there will be a countable model M of ZFC. In that countable model M, the naturals will in a sense only have countably many subsets. That sounds like a contradiction, but it's not, because "countably many" here is understood according to the *intended model* of ZFC (that's why I said "in a sense"). Thus, according to the "intended model", M's naturals have countably many subsets; but according to M itself, its naturals have uncountably many subsets.
When we deal with a multiplicity of models of ZFC, we have to keep track of which statements hold according to which model. Thus, phrases like "subset", "countable" and "function" all need to be relativized to a model.
And in the post you're commenting on, I simply assumed that each world comes along with its own privileged model of set theory. So relativizing to a model is the same as relativizing to a world then.
"in particular, all pure sets that exist in w3, also exist in w4, and so do functions between any two of such sets that exist in w3": That is true: all the functions in w3 are also functions in w4. But for the contradiction, you need the converse: that any function f in w4 between two sets A and B in w3 also exists in w3. And that's not true. For just as what subsets there are of a set varies between worlds, so too what functions there are between a pair of sets varies between worlds (on the story told in that post, of course; I take no stance on whether the story is true, only that probably it is coherent). It has to be that way, because there is a 1-1 correspondence between the subsets of a set and the functions from that set to {0,1}.
Does that help?
That might help, thanks, though I'm not sure.
ReplyDeleteMaybe the difficulty is that I'm not sure I get how you see possible worlds. In particular, I don't know how hereditary sets vary between worlds, since I see the sets as existing in some hypothetical scenarios (e.g., in other sets, or the class of all hereditary sets, etc.), whereas possible worlds are scenarios with concreta in them (possibly), and if you like, with some of those sets, etc., but including either all or none.
That aside, based on your latest reply, if I understand it correctly it seems to me you're positing interpretations of the axioms that would no longer make them about sets, or at least not about all hereditary sets.
For example, let N3 be the set {1,2,..3} in w3, and let N4 be the set {1,2,3...} in w4. It follows from the hypotheses that N3 exists in w4. But for every x, if x is in N3, then x is in N4, and vice versa, so it's the same set. So, let A be a subset of N4 in w4. If x is in A, then x is in {1,2,3...}, so x is in N3. Thus, A is in fact a collection of natural numbers. But you're saying that A (for some A) does not exist in N3, even though it exists in N4 and is a collection of natural numbers. Then, the class of all sets in w3 does not seem to match the intuitive concept of hereditary sets that we have: if A is indeed a collection of natural numbers, it's intuitively clear that it's a subset of the set of natural numbers. But in w3, you don't get that. Moreover, w3 is arbitrary, so you get that in any possible world w, at least one coherent/logically possible collection of natural numbers is not a subset of the set of natural numbers (and indeed does not exist) despite the fact that the set of natural numbers exists, and moreover, there is a logically possible injective function f(w) from the subsets of the set of natural number that exist in a world w, into the set of natural numbers (in the actual world, since that one is the same).
Now I guess you might ask at which world does f(w) exist?
I don't think that that is the right view of possible worlds, but at any rate, I would say that the assignment is logically possible, and also if there is an intuitive class of all hereditary sets, it does not exist at any world under those hypotheses.
Am I reading this correctly?
Maybe the following will make what I think is a problem more clear:
ReplyDeleteIt follows from the assumptions that there is a possible world w6 at which there exists a set C of ordered pairs {(A,n)}, where the A's are all of the subsets of the set of natural numbers that exist in the actual world, the n's are natural numbers, and every pair that differs in the first coordinate also differs in the second. Similarly, at some world w7 there is a set D of order pairs {(x,n)} where the x's are all of the real numbers in the actual world, etc.
The main point of that post is that it is coherent to suppose that what pure sets there are varies between worlds.
ReplyDeleteYou write: "if A is indeed a collection of natural numbers, it's intuitively clear that it's a subset of the set of natural numbers". Well, there are two further possibilities, which seem to be coherent:
1. A is a set of natural numbers in world w4, all the members of A exist in w3, but A itself does not exist in w3;
2. A is a set of natural numbers in world w4, all the members of A exist in w3, A itself exists in w3, but A is not a *set* in w3.
There might be philosophical reasons to reject options 1 or 2. But I am not aware of any logical or mathematical reasons in the ZFC axioms to reject them (as long as ZFC is consistent with an appropriate large cardinal assumption).
"It follows from the assumptions that there is a possible world w6 at which there exists a set C of ordered pairs {(A,n)}, where the A's are all of the subsets of the set of natural numbers that exist in the actual world, the n's are natural numbers, and every pair that differs in the first coordinate also differs in the second. Similarly, at some world w7 there is a set D of order pairs {(x,n)} where the x's are all of the real numbers in the actual world, etc."
Exactly. That's the point. And as far as we know, this is perfectly mathematically coherent. It would be incoherent with the ZFC axioms to say that a set like C or D exists in the actual world, but it is coherent with the ZFC axioms (or even with the necessary truth of the ZFC axioms) to say that it exists in another world.
Here's a thought that might help you. Given the consistency of ZF with a large cardinal axiom, each of (a) and (b) below on its own (but not together!) is consistent with the ZF axioms:
(a) every set of reals is measurable, and
(b) Vitali nonmeasurable sets exist.
Well, then, if what sets there are varies between possible worlds, it could well be that there is one possible world where ZF+(a) is true and another where ZF+(b) is true. After all, both possibilities are logically consistent (given the large cardinal consistency assumption). So why couldn't one be true in one world and another in another?
Suppose this is so, and suppose that ZF+(b) is actually true while ZF+(a) is true in some world w8. Then there will be a difference in what pure sets exist in the actual world and in w8.
I think there are different senses in which something can be coherent, or consistent. For example, there is a way in which it's coherent to deny any of the axioms of ZF, considering them as first-order formulas: since the first-order formulas are not theorems of a first-order predicate calculus, the denial of any of them will not yield a contradiction using only first-order logic and without further consideration on the meaning of the axioms. However, it seems to me that there is something about our concepts of set (which is primitive) and the class of all hereditary sets that would make some of this denial not coherent (as long as it's meant to hold in the entire class). In general, my impression is that first-order theories may not fully capture some of our intuitive concepts.
ReplyDeleteMaybe we could distinguish between (at least) 3 different sorts of incoherence:
I1: Meaningless expressions, like "loyuporak", or whatever,
I2: Assertions that imply a contradiction because of their logical form, regardless of the meaning of the terms. For example, "John is married and it is not the case that John is married".
I3: Assertions that imply a contradiction but only after considering the (internal, if one makes a distinction) meanings of the words, such as "John is a married bachelor".
First order statements denying any of the axioms of ZF is not incoherent in the sense of I1 or I2, but I'm not sure about I3 (sure, it would be coherent to deny that they hold in some specific set or another, but in the class of all hereditary sets?). For that matter, if F is a first-order formula which, on a standard interpretation, says that 3 is a prime number, then ¬F implies no contradiction, but once we factor in the meanings of "3", "prime", and others involved, it seems to me it does.
"Well, there are two further possibilities, which seem to be coherent:
1. A is a set of natural numbers in world w4, all the members of A exist in w3, but A itself does not exist in w3;
2. A is a set of natural numbers in world w4, all the members of A exist in w3, A itself exists in w3, but A is not a *set* in w3."
Maybe. I admit I have difficulty understanding them, but perhaps your intuitive concept(s) of "possible world" and/or "set" differs from mine in a way that is relevant to the matter at hand. But I'll raise a more concrete issue below.
"There might be philosophical reasons to reject options 1 or 2. But I am not aware of any logical or mathematical reasons in the ZFC axioms to reject them (as long as ZFC is consistent with an appropriate large cardinal assumption)."
If I understand now what you're saying in the post, I'm inclined to agree (at least, I haven't found a reason within the ZFC axioms; but for potential mathematical reasons, I will write more below). Then again, as far as I can tell, it's also consistent in this sense to posit that in some worlds, some of the ZF axioms are false, or even that ¬F holds in some world.
"Exactly. That's the point. And as far as we know, this is perfectly mathematically coherent. It would be incoherent with the ZFC axioms to say that a set like C or D exists in the actual world, but it is coherent with the ZFC axioms (or even with the necessary truth of the ZFC axioms) to say that it exists in another world."
Perhaps it is, though I would worry about the different meanings of "coherent", or how to combine possible worlds with ZFC (I'm going to raise a couple of perhaps more concrete objections, but I have to split the post).
"Here's a thought that might help you. Given the consistency of ZF with a large cardinal axiom, each of (a) and (b) below on its own (but not together!) is consistent with the ZF axioms:
ReplyDelete(a) every set of reals is measurable, and
(b) Vitali nonmeasurable sets exist.
Well, then, if what sets there are varies between possible worlds, it could well be that there is one possible world where ZF+(a) is true and another where ZF+(b) is true. After all, both possibilities are logically consistent (given the large cardinal consistency assumption). So why couldn't one be true in one world and another in another?"
If we're going in that direction, we can go much further. We may similarly ask: why wouldn't the axiom of infinity, or the power set axiom, or the axiom of union, etc., be false in some world?
My worry is that at some point, we may lose sight that the first-order formalism is a model of our intuitive concepts, but can be interpreted in different ways, and denying some things raise (in my view) worries about the coherence with the concept of set (and other relevant concepts involved).
Anyway, here's something a bit more concrete:
O1: Let w9 be a possible world at which there is a function G from the set of natural numbers in w9 onto the set of real numbers in the interval (0,1) that exist at the actual world.
Let x be the following real number (which is logically possible and exists in some possible worlds).
x=a0*10^0+a1*10^1+....,
where a0=0, a1=2 if the second term in the decimal expansion of G(1) is 5, and a1=5 otherwise, a2=2 if the second term in the decimal expansion of G(2) is 5, and a2=5 otherwise, and so on (the set of natural numbers in w9 is stil {1,2,3...}). It follows right away that from any n, G(n)=\=x. Thus, x is a real number between 0 and 1 which does not exist at the actual world!
Of course, one could argue that when we do mathematics, we are not talking about existence in the actual world or in any possible world, but rather, existence in some intuitive mathematical realm which varies from context to context. I actually think this is much more probable (well, I'm probably saying that! Maybe different people mean different things and we don't realize that we're talking past each other when we do math!), but that's off-limits in the context of this argument (at least, if I read it correctly), in which we're accepting talk about existence of sets, functions, etc., at possible worlds.
So, it seems to me that if the theory in question (about possible worlds with different sets, etc.) is correct, then there is much of mathematics we cannot properly do - in the usual sense in which most people including mathematicians do math, which is not by engaging in pure first-order reasoning with formulas representing natural numbers, etc. (a superintelligent AI might be able to do that, but I think it's beyond human capability, at least for much of mathematics).
In fact, I reckon probably many proofs in math papers (not on mathematical logic) published in peer-reviewed, well-regarded journals (not sure how many, but I think really a lot) fail under the assumptions in question, either directly or because they rely on some more basic proofs that fail. Moreover, if one should assign a middling probability (or even a not-very-low probability) to the hypotheses about possible worlds, then one should significantly reduce one's confidence in much of mathematics, it seems to me.
To be clear, I'm not trying to make this an argument from consequences. Rather, I'm saying that it's extremely probable that the vast majority of math papers (so published, etc.) are generally on firm ground.
Granted, you might still say it's coherent to say otherwise. I would say that maybe so, but I'm not sure; I think it might depend on the meaning of "coherent". However, even if it is coherent, I'd say it's extremely improbable.
I thought of two potential rejoinders to my objection above:
ReplyDeleteRejoinder 1. But the function G does not exist in the actual world, so how can you use it to make an argument?
Reply:
1.a. If we assume that worlds exist at worlds, then w9 exists at the actual world, and G exists at w9. If not, then at any rate, this does not seem different from ordinary discussions about possible worlds, using what happens at different worlds, etc.
1.b. Surely, whether x exists at the actual world does not depend on who makes the argument. Let's say that Bob, who exists at w9, makes the argument. Bob would not be calling our world "actual", but that's not the point. He concludes that if the hypotheses about worlds, sets, etc., is true, then some x does not exist in our world (whatever he calls it), and moreover, that goes for more than n reals (which exist in some world but not in other), for any natural number n. He concludes that if mathematicians in our world properly assign not very low probability to the theory about worlds, etc., they should reckon that their statements about existence of reals are suspect (assuming always that those statements are regularly about existence in the world in which they are).
Rejoinder 2: But if you can do that, why can't mathematicians generally make a similar move to make their arguments?
Reply: Perhaps they can, but it seems to me the matter would have to be considered on a case-by-case basis, depending on the proof we're considering. But as long as ordinarily, mathematicians who assert existence are talking about existence in the actual world (or something equivalent to that), it seems that many mathematical proofs become suspect.
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