I examine several arguments by Bruno Whittle against the Cantorian conception of the size of infinite sets. I find that none of them succeed.
Sets
Whittle (2015a, 2015b, 2018) objects to this standard conception. He holds that we do not in fact have good reason for believing either of the following two principles, and that we are thus not in a position to know either of them or to know that there are infinite sets of different sizes:
Size
Size*
Whittle puts forward a series of arguments that aim to establish this. Five of his arguments purport to refute the following widely accepted theses:
Thesis 2. The notion of cardinality defined in set theory is at least “size-like”: it is at least similar to our ordinary notion of size (of sets) and perhaps a natural generalization of the notion of finite size.
Thesis 3. Cantor’s theorem establishes that there are infinite sets of different sizes.3
Here, I examine Whittle’s arguments against theses 1 and 2; I argue that none of them succeed as refutations of either of those theses. The arguments, appearing in their fully developed form in Whittle (2018), are:
(i) an argument against Thesis
1 based on its interpretation as stating that “
(ii) an argument against the same thesis, based on what would be true in a mathematically-impossible situation in which there are no functions from certain sets;
(iii) a Benacerraf-style challenge for accounts of size in terms of functions—accounts that may be offered in support of Thesis 1;
Whittle also argues directly against Thesis 3 in his (2015a) and (2015b). I examine and reject his argument against this thesis in (manuscript).
If I am right in rejecting the arguments against theses 1–3, then Whittle fails to establish his claim that we are not in a position to know that there are different sizes of infinity. For these three theses, if correct, provide routes to such knowledge (this is, indeed, why Whittle attacks them); and this is so whether or not some additional routes are successfully blocked by those of Whittle’s arguments that I will not consider here.4
Although my treatment of Whittle’s arguments can be considered a defense of the Cantorian view of size, I will not offer positive arguments for this view or argue that other views, incompatible with it, are incorrect or unjustified; my point is only that Whittle fails to establish this for Cantor’s view.
Sections 1–4 below are each dedicated to the examination of one of the arguments (i)–(iv). I briefly conclude in section 5.
1 Structured Propositions
Consider the following two sentences:
Size
Function There is a bijection from
According to Whittle, the most straightforward interpretation of Thesis 1 is this:
Whittle thinks of propositions here as structured, Russellian propositions. The propositions that Size and Function seem to express are, according to him,
(
and
(
respectively.
Whittle now suggests that each of the steps in the following argument against (*) is at least very plausible, or prima facie correct (Whittle 2018, 855–856):
(
) and ( ) are about very different sorts of things: ( ) is about a certain sort of property—a size—that and have in common; ( ) is about a function.If (*) is true, then either both Size and Function express (
), or both of them express ( ).Most sentences about functions express propositions that are genuinely about functions.
There is no plausible account on which the following sentence expresses a proposition about properties:
Constant There is a constant function from
to .The sentences Function and Constant seem to be very similar.
From (5): Function and Constant should express similar propositions—not ones about completely different sorts of things.
From (1), (4), and (6); and also from (1) and (3): Function cannot express (
).Size has the same general form as the following:
Color
is the same color as .Height
is the same height as .Sex
is the same sex as .From (8): Size and sentences like Color, Height, and Sex should express similar propositions—propositions of the same general form.
The propositions expressed by sentences such as Color, Height, and Sex are not about functions.
From (7) and (11): It is not the case that both Size and Function express (
), and it is not the case that both of them express ( ).
Whittle’s argument here relies on the following view of structured propositions:
Prop For each declarative sentence such as Size and Function, there is one unique structured proposition that the sentence can be taken to express; the structure of that proposition, as well as what it is about, is given by the most straightforward formalization of the sentence in higher-order predicate logic.
Without Prop, we seem to have no reason to accept the inference from (5) to (6) or the one from (8) to (9). As I will now argue, however, it is illegitimate to rely on Prop in this context. This is so, since Prop is incompatible with Whittle’s reading of “what it is”—a reading on which
- There being a bijection between
and is what it is for and to be the same size
is correctly interpreted as
Prop, together with this reading, yields absurd consequences, as illustrated by the following example.
Being a man who has never been married is just what it is to be a bachelor. But Prop, together with the aforementioned reading of “what it is,” entails that this is not so. To see this, note first that the most straightforward formalizations in predicate logic of the following differ in structure:
(i)
(ii)
(i) and (ii) are most straightforwardly formalized as, respectively:
(iii)
(iv)
This, by Prop, means that (i) and (ii) express different propositions. Hence, on Whittle’s interpretation of “what it is,” being a never-married man is not what it is to be a bachelor.
One can, of course, hold that, in formalizing (i), we should take into account the analysis of “bachelor” as “never-married man” and correspondingly allow the formalization of both (i) and (ii) as (iv). This is, I think, a reasonable approach, but note that it deviates from Prop. Moreover, if a similar deviation is allowed in the case of Size and Function, then Whittle’s argument does not go through, as there is no longer reason to accept, e.g., the inference from (8) to (9).
Whittle’s approach entails, then, that being a never-married-man is not what it is to be a bachelor. Similarly, it entails that:
There being an isometry that transforms a figure
in the Euclidean plane into a figure is not what it is for and to be congruent.There being a formal proof of a first-order sentence
from a first-order theory is not what it is for to be a theorem of .
Additional, similar examples are not hard to produce.
I conclude that Whittle’s argument from structured propositions fails by relying on two incompatible principles: Prop and a reading of “what it is,” on which Thesis 1 is correctly interpreted as (*). If Prop is given up, then, as noted above, Whittle’s argument seems not to go through; if the aforementioned reading of “what it is” is given up, then the argument does go through, but its conclusion, (13), no longer means that Thesis 1 is false.
2 Sets in an Impossible Situation
Thesis 1, recall, was this: There being a
bijection between
Whittle specifically considers a functional account on which complex properties are “Russellian propositions with gaps” and sizes are properties of the form
where
Whittle’s argument involves the sets
(1) In
(2) In
(3) From (1): In
(4) From (3): If the functional account under consideration
is correct, then, in
(5) From (4): If the functional account is correct, then,
in
There is a good reason to suspect this argument, if not to reject it outright: Arguments very similar to it, and ones that are not any less appealing, lead to absurd conclusions. Consider, for example, the following account of what a circle is:
Circle To be a circle is to be a set of all the
points in the Euclidean plane that are at the same given distance
I submit that this is a correct account of what a circle is.
But an argument similar to Whittle’s leads to the conclusion that this
is not so: Let
(1) In
(2) In
(3) From (1): If Circle is correct, then, in
(Note that I am not claiming that this is a very convincing argument—just that it is similar to, and not less appealing than, Whittle’s argument.)
This “bad company” indicates that something is wrong with Whittle’s
argument. But what? There are, I believe, two major problems with the
argument. First, it is not at all clear that we can make sufficient
sense of mathematically impossible situations like
Second, assuming (for the sake of argument) that we can make
sufficient sense of mathematically impossible situations like
Perhaps Whittle holds that, e.g.,
Size2
The size of
But this, it seems, cannot serve as a ground for Whittle’s premise
(2), since an argument similar to his leads to
the conclusion that Size2 is false: Consider a
situation
(1) It is not the case that, in
(2) If Size2 is true, then, in
(3) From (2): If Size2 is true, then, in
I am not claiming that (1)–(4) is a convincing argument or that
reasoning in this way about
I conclude that Whittle’s argument from
3 Benacerraf’s Problem
Whittle (2018, 862–863) argues that the specific functional account that featured in his argument from an impossible situation faces a version of “Benacerraf’s problem.”8 That functional account, recall, identified sizes with properties of the form
where
This problem does not seem to afflict all functional accounts of size. Consider, for instance, the account I mentioned in the previous section, on which a size property is the property of belonging to a given bijection type. This account does not commit to a particular metaphysical theory of what properties are, and the Benacerraf problem, it seems, does not arise for it. Whittle would seem to agree, but he does not consider this to be a problem for his attack on Thesis 1, since he takes the functional accounts immune to Benacerraf’s problem to be refuted by the argument discussed in the previous section (see his 2018, fn.15). If my arguments in the previous section are correct, however, then there is a genuine problem here for Whittle; for the argument he is relying on fails.
4 Are Cardinalities Size-Like?
Thesis 2 was this:
The notion of cardinality defined in set theory is at least “size-like”: it is at least similar to our ordinary notion of size (of sets) and perhaps a natural generalization of the notion of finite size.
Whittle takes cardinalities to be properties of the form
where
The notion of cardinality is not a generalization of the notion of finite size, since the collection of all cardinalities does not contain the finite sizes. This is allegedly established by the argument from
(discussed in section 3 above), as that argument is supposed to show that finite sizes, unlike finite cardinalities, cannot be understood in terms of bijections.Cardinalities are “just a completely different sort of property from sizes” (Whittle 2018, 864). For, sizes have nothing in particular to do with functions; they are, rather, intrinsic: they can be accounted for only in terms of the things that constitute the set, such as its elements, their parts, or their elements. This is supposed to be established by an argument similar to the argument from
.
There are, however, two serious problems with this line of argument. First, if my criticism of Whittle’s argument from an impossible situation (see section 3 above) is correct, then that argument fails to establish that finite sizes are distinct from finite cardinalities, and it is unclear how a similar argument could establish the supposed intrinsic nature of size properties.
Second, even if the argument from
(i) Finite cardinalities are at least co-extensive with finite sizes (Whittle concedes this).
(ii) If a subset
(iii) The following version of
Hume’s principle (which Whittle accepts for finite sizes) holds: Sets
(iv) Assuming the axiom of choice, sizes* (i.e., cardinalities) are well-ordered by the relevant smaller-than* relation.
Given these problems, I submit, Whittle’s points against Thesis 2 do not suffice as a refutation of it.
5 Conclusion
Whittle puts forward four arguments against theses 1 and 2; these, as I explained in the introduction, constitute an essential component of his objection to Cantor’s conception of infinite size. As I hope to have established, however, none of Whittle’s arguments against those two theses succeed.
Ran Lanzet
University of Haifa
lanzetr@gmail.com
Acknowledgements
For helpful comments and discussions I thank Balthasar Grabmayr, David Kashtan, Aviv Keren, Eli Pitcovski, Gil Sagi, and four anonymous referees. The research leading to this paper was supported by post-doc fellowships at the University of Haifa and at the Hebrew University of Jerusalem.
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