Maverick Philosopher

Infinity and Mathematics Education

Time for a re-post. This first appeared in these pages on 18 August 2010.

.........................

A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie); on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical.  If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory.  Cantor sought to achieve an exact mathematics of the actually infinite.  But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's Elements, David Hilbert's Foundations of Geometry, Richard Dedekind's Continuity and Irrational Numbers, Cantor's Contributions to the Founding of the Theory of Transfinite Numbers, etc.  Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc.  Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of constructing the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to supplant a textbook-driven approach, but that the latter ought to be supplemented by the foregoing.  I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No! 

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite.  The countably infinite has nothing to do with the potentially infinite.  I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity.    In so doing they took a lot of the excitement and wonder out of it. 

So what did you learn?  You learned how to solve problems and pass tests.  But how much actual understanding did you come away with?

Can a Mereological Sum Change its Parts?

This post is an attempt to understand and evaluate Peter van Inwagen's "Can Mereological Sums Change Their Parts," J. Phil. (December 2006), 614-630.  A preprint is available online here.

The Wise Pig and the Brick House: My Take

On Tuesday the Wise Pig  takes delivery of 10,000 bricks.  On the following Friday he completes construction of a house made of exactly these bricks and nothing else.  Call the bricks in question the 'Tuesday bricks.'  I would 'assay' the situation as follows.  On Tuesday there are some unassembled bricks laying about the building site.  By Unrestricted Composition, these bricks compose a classical mereological sum.  Call this sum 'Brick Sum.'  (To save keystrokes I will write 'sum' for 'classical mereological sum.' ) By Uniqueness of Composition, there is exactly one sum that the Tuesday bricks compose.  On Friday, both the Tuesday bricks and their (unique) sum exist.  But as I see it, the Brick House is identical neither to the Tuesday bricks nor to their sum.  Thus I deny that the Brick House is identical to the sum of the things that compose it. I give two arguments for this non-identity.

Nonmodal 'Historical' Argument:  Brick Sum has a property that Brick House does not have, namely the property of existing on Tuesday.  Therefore, by the Indiscernibility of Identicals, Brick Sum is not identical to Brick House.

Modal Argument:  Suppose that the actual world is such that Brick Sum and Brick House always existed, exist now, and always will exist:  every time t is such that both exist at t.  This does not alter the plain fact that the house depends for its existence on the bricks, while the bricks do not depend for their existence on the house.  Thus there are possible worlds in which Brick Sum exists but Brick House does not.  (Note that Brick Sum exists 'automatically' given the existence of the bricks.) These worlds are simply the worlds in which the bricks exist but in an unassembled state.  So Brick Sum has a property that Brick House does not have, namely, the modal property of being possibly such as to exist without composing a house.  Therefore, by the Indiscernibility of Identicals, Brick Sum is not identical to Brick House.

In sum (pardon the pun!), The Brick House is not a mereological sum.  (If it were, it would have existed on Tuesday as a load of bricks, which is absurd.)  This is not to say that there is no sum 'corresponding' to the Brick House: there is.  It is just that this sum -- Brick Sum -- is not identical to Brick House.  So what I am saying implies no rejection of Unrestricted Composition.  The point is rather that a material artifact such as a house cannot be identified with the mereological sum of the things it is made of.  This is because sums abstract or prescind from the mutual relations of parts in virtue of which parts form what we might call  'integral wholes' as opposed to a mere mereological sums.  Unassembled bricks do not a brick house make: you have to assemble them properly.  And the assembly, however you want to assay it, is an added ontological ingredient that escapes consideration by a general purely formal part-whole theory such as classical mereology.

I assume with van Inwagen that Brick House can lose a brick (or gain a brick)  without prejudice to its identity.  But, contra van Inwagen, I do not take this to imply that mereological sums can gain or lose parts.  And this for the simple reason that Brick House and things like it are not identical to sums of the things that compose them.  I would say, pace van Inwagen, that mereological sums can no more gain or lose parts than (mathematical) sets can gain or lose elements.

The Wise Pig and the Brick House: Van Inwagen's Take

I agree with van Inwagen that "The Tuesday bricks are all parts of the Brick House and every part of the Brick House overlaps at least one of the Tuesday bricks." (616-617)  But he takes this obvious truth to imply that " . . . 'a merelogical sum' is the obvious thing to call something of which the Tuesday Bricks are all parts and each of whose parts overlaps at least one of the Tuesday Bricks." (617)  Well, he can call it that but only if he uses 'mereological sum' in a way different that the way it is used in classical mereology.

Now if we acquiesce in van Inwagen's usage, and we grant that things like houses can change their parts, then it follows that mereological sums can change their parts.  But why should we acquiesce in van Inwagen's usage of 'mereological sum'?

Is Everything a Mereological Sum?

As I use 'mereological sum,' not everything is such a sum.  The Brick House is not a sum.  It is no more a sum than it is a set.  There are sums and there are sets, but not everything is a sum just as not everything is a set.  There is a set consisting of the Tuesday Bricks, and there is a singleton set of the Brick House.  But neither of these sets is identical to the Brick House.  Neither of them has anything to fear from the pulmonary exertions of the Big Bad Wolf -- not because they are so strong, but because they are abstract objects removed from the flux and shove of the causal order.  Sums of concreta, unlike sets of concreta,  are themselves concrete -- but the Brick House is not a sum.  Van Inwagen disagrees.  For him, "Everything is a mereological sum." (618)

His argument for this surprising claim is roughly as follows. PvI's presentation is tedious and technical but I think I will not be misrepresenting him if I sum up the gist of it as follows:

1. Everything, whether simple or composite, has parts.  (This is a consequence of the following definition: x is a part of y =_df x is a proper part of y or x = y.  Because everything is self-identical, everything has itself as a part, an improper part to be sure, but a part nonetheless. Therefore:

2. Everything is a mereological sum of its parts.  Therefore:

3. Everything is a mereological sum. Therefore:

4. ". . . mereological sums are not a special sort of object." (622)  In this respect they are unlike sets."'Mereological sum' is not a useful stand-alone general term." (622) 'Set' is.

What's At Issue Here?

I confess to not being clear about what exactly is at issue here.  One could of course use 'mereological sum' in the way that van Inwagen proposes, a way that implies that everything is a mereological sum, and that implies that there is no conceptual confusion in the notion of a mereological sum changing its parts.   But why adopt this usage?  How does it help us in the understanding of material composition?

What am I missing?

 

Infinity and Mathematics Education

A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie);
on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical.  If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory.  Cantor sought to achieve an exact mathematics of the actually infinite.  But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's Elements, David Hilbert's Foundations of Geometry, Richard Dedekind's Continuity and Irrational Numbers, Cantor's Contributions to the Founding of the Theory of Transfinite Numbers, etc.  Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc.  Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of construction the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to supplant a textbook-driven approach, but that the latter ought to be supplemented by the foregoing.  I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No! 

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite.  The countably infinite has nothing to do with the potentially infinite.  I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity.    In so doing they took a lot of the excitement and wonder out of it.  So what did you learn?  You learned how to solve problems and pass tests.  But how much actual understanding did you come away with?

Kline on Cantor on the Square Root of 2

Morris Kline, Mathematics: The Loss of Certainty, Oxford 1980, p. 200:

. . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the argument that the irrational numbers, such as the square root of 2, when expressed as decimals involved actually infinite sets because any finite decimal could only be an approximation.

Here may be one answer to the question that got me going on this series of posts. The question was whether one could prove the existence of actually infinite sets. Note, however, that Kline's talk of actually infinite sets is pleonastic since an infinite set cannot be anything other than actually infinite as I have already explained more than once.  Pleonasm, however, is but a peccadillo. But let me explain it once more.  A potentially infinite set would be a set whose membership is finite but subject to increase.  But by the Axiom of Extensionality, a set is determined by its membership: two sets are the same iff their members are the same.  It follows that a set cannot gain or lose members.  Since no set can increase its membership, while a potentially infinite totality can, it follows that that there are no potentially infinite sets.  Kline therefore blunders when he writes,

However, most mathematicians -- Galileo, Leibniz, Cauchy, Gauss, and others -- were clear about the the distinction between a potentially infinite set and an actually infinite one and rejected consideration of the latter. (p. 220)

Kline is being sloppy in his use of 'set.'  Now to the main point.  Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.4142136. . . . Despite the nonterminating decimal expansion, the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite? This is not an argument, of course, but a gesture in the direction of a possible argument.

If someone can put the argument rigorously, have at it.

Does Potential Infinity Rule Out Mathematical Induction?

In an earlier thread David Brightly states that "The position on potential infinity that he [BV] is defending is equivalent to the denial of the principle of mathematical induction."  Well, let's see.

1.  To avoid lupine controversy over 'potential' and 'actual,' let us see if we can avoid these words.  And to keep it simple, let's confine ourselves to the natural numbers (0 plus the positive integers).  The issue is whether or not the naturals form a set.  I hope it is clear that if the naturals form a set, that set will not have a finite cardinality!  Were someone to claim that there are 463 natural numbers, he would not be mistaken so much as completely clueless as to the very sense of 'natural number.'  But from the fact that there is no finite number which is the number of natural numbers, it does not follow that there is a set of natural numbers.

2.  So the dispute is between the Platonists -- to give them a name -- who claim that the naturals form a set and the Aristotelians -- to give them a name -- who claim that the naturals do not form a set.  Both hold of course that the naturals are in some sense infinite since both deny that the number of naturals is finite.  But whereas the Platonists claim that the infinity of naturals is completed, the Aristotelians claim that it is incomplete.  To put it another way, the Platonists -- good Cantorians that they are -- claim that  the naturals, though infinite,  are a definite totality whereas the Aristoteleans claim that the naturals are infinite in the sense of indefinite.  The Platonists are claiming that there are definite infinities, finite infinities -- which has an oxymoronic ring to it.  The Aristotelians stick closer to ordinary language.  To illustrate, consider the odds and evens.  For the Platonists, they are infinite disjoint subsets of the naturals.  Their being disjoint from each other and non-identical to their superset shows that for the Platonists there are definite infinities.

3.  Suppose 0 has a property P.  Suppose further that if some arbitrary natural number n has P, then n + 1 has P.  From these two premises one concludes by mathematical induction that all n have P.  For example, we know that 0 has a successor, and we know that if  arbitrary n has a successor, then n +1 has a successor.  From these premises we conclude by mathematical induction that all n have a successor.

4.  Brightly claims in effect that to champion the Aristotelean position is to deny mathematical induction.  But I don't see it.  Note that 'all' can be taken either distributively or collectively.  It is entirely natural to read 'all n have a successor' as 'each n has a successor' or 'any n has a successor.'  These distributivist readings do not commit us to the existence of a set of naturals.  Thus we needn't take 'all n have a successor' to mean that the set of naturals is such that each member of it has a successor.

5. Brightly writes, "My understanding of 'there is' and 'for all' requires a pre-existing domain of objects, which is why, perhaps, I have to think of the natural numbers as forming a set."  Suppose that the human race will never come to an end.  Then we can say, truly, 'For every generation, there will be a successor generation.'  But it doesn't follow that there is a set of all these generations, most of which have not yet come into existence.  Now if, in this example, the universal quantification does not require an actually infinite set as its domain, why is there a need for an actually infinite set  as the domain for the universal quantification, 'Each n has a successor'?

6.  When we say that each human generation has a successor, we do not mean that each generation now has a successor; so why must we mean by 'every n has a successor' that each n now has a successor?  We could mean that each n is such that a successor for it can be constructed or computed.  And wouldn't that be enough to justify mathematical induction?

Addendum 8/15/2010  11:45 AM.  I see that I forgot to activate Comments before posting last night.  They are on now. 

It occurred to me this morning that  I might be able to turn the tables on Brightly by arguing that actual infinity poses a problem for mathematical induction.  If  the naturals are actually infinite, then each of them enjoys a splendid Platonic preexistence vis-a-vis our computational activities.  They are all 'out there' in Plato's heaven/Cantor's paradise.  Now consider some stretch of the natural number series so far out that it will never be reached by any computational process before the Big Crunch or the Gnab Gib, or whatever brings the whole shootin' match crashing down.  How do we know that the naturals don't get crazy way out there?   How can we be sure that the inductive conclusion For all n, P(n) holds?  Ex hypothesi, no constructive procedure can reach out that far.  So if the numbers exist out there, but we cannot reach them by computation, how do we know they behave themselves, i.e. behave as they behave closer to home?  This won't be a problem for the constructivist, but it appears to be a problem for the Platonist.

On Potential and Actual Infinity

Some remarks of Peter Lupu in an earlier thread suggest that he does not understand the notions of potential and actual infinity.  Peter writes:

(ii) An acorn has the potential to become an oak tree. But the acorn has this potential only because there are actual oak trees . . . .  If for some reason oak trees could not exist, then an acorn cannot be said to have the potential to become an oak tree. Similarly, if the proponent of syllogism S1 thinks that actual infinity is ruled out by some conceptual, logical, or metaphysical necessity, then he is committed to hold that there cannot be a potential infinity either. Thus, in order for something to have the potential to be such-and-such, it is required that it is at least possible for actual such-and-such to exist.

This is a very fruitful misunderstanding!  For it allows us to clarify the different senses of 'potential' and 'actual' as applied to the analysis of change and to the topic of infinity.  First of all, Peter is completely correct in what he says in the first two sentences of the above quotation.  The essence of what he is saying may be distilled in the following principle

If actual Fs are impossible, then potential Fs are also impossible.

But this irreproachable principle is misapplied if 'F' is instantiated by 'infinity.'  If an actual infinity is impossible, it does not follow that a potential infinity is impossible.  For when we say that a series, say, is potentially infinite we precisely mean to exclude its being actually infinite.  A potentially infinite series is not one that has the power or potency to develop over time into an actually infinite series, the way a properly planted acorn develops into an oak tree.   On the contrary, it is a series which, no matter how much time elapses, is never completed.  An actually infinite series, by contrast, is complete at every instant.

Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ).  If these numbers form a set, call it N, then N will of course be actually infinite.  A set is a single, definite object, a one-over-many, distinct from each of its members and from all of them.  N must be actually infinite because there is no greatest natural number, and because N contains all the natural numbers. 

It is worth noting, as I have noted before, that 'actually infinite set' is a pleonastic expression. It suffices to say 'infinite set.'  This is because the phrase 'potentially infinite set' is nonsense. It is nonsense because a set is a definite object whose definiteness derives from its having exactly the members it has.  In the case of the natural numbers, if they form a set, then that set will have a transfinite cardinality. Cantor refers to that cardinality as aleph-zero or aleph-nought.

But surely it is not obvious that the natural numbers form a set.  Suppose they don't.  Then the natural number series, though infinite, will be merely potentially infinite.  What 'potentially infinite' means in this case is that one can go on adding endlessly without ever reaching an upper bound of the series.  No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting.  The numbers are not 'out there' waiting to be counted; they are created by the counting.  In that sense, their infinity is merely potential.  But if the naturals are an actual infinity, then  they are not created but labeled.

Or consider a line segment. One can divide it repeatedly and in principle 'infinitely.'  But if one does so is one creating divisions  or recognizing  divisions that exist already?  If the former, then the infinity of divisions is merely potential; if the latter, it is actual. 

Peter seems worried by the fact that no human or nonhuman adding machine can enumerate all of the natural numbers.  But this is no problem at all.  If there is an actual infinity of natural numbers, then it is obvious that a complete enumeration is impossible:  the first transfinite ordinal omega has aleph-nought predecessors.  If there is only a potential infinity of naturals, then as many enumerations have taken place, that is the last number created.

Peter seems not to be taking seriously the notion of potential infinity by simply assuming that the naturals must form an infinite set.  He doesn't take it seriously because he confuses the use of 'potential' in the context of an analysis of change, where change is the reduction of potency to act, with the use of 'potential' in discussions of infinity.

But now I'm having second thoughts.  I want to say that from the fact that a line segment is infinitely divisible, it does not follow that it is actually divided into continuum-many points.  But  what about the number of possible dividings?  If that is a finite number, one that reflects the ability of some divider, then how can the segment be infinitely divisible?  But if the number of possible dividings  is a transfinite number, then it seems we have re-introduced an actual infinity, namely, an actual infinity of possible dividings.  In other words, infinite divisibility seems to require an actual infinity of possible dividings.  Or does it? 

Collective Inconsistency and Plural Predication

We often say things like

1. The propositions p, q, r are inconsistent.

Suppose, to keep things simple, that each of the three propositions is self-consistent.  It will then be false that each proposition is self-inconsistent. (1), then, is a plural predication that cannot be given a distributive paraphrase.  What (1) says is that the three propositions are collectively inconsistent.  This suggests to many of us  that there must be some one single entity that is the bearer of the inconsistency.  For if the inconsistency does not attach distributively to each of p, q, and r, then it attaches to something distinct from them of which they are members.  But what could that be?

If you say that it is the set {p, q, r} that is inconsistent, then the response will be that a set is not the sort of entity that can be either consistent or inconsistent.  Note that it is not helpful to say

A set is consistent (inconsistent) iff its members are consistent (inconsistent).

For that leaves us with the problem of the proper parsing of the right-hand side, which is the problem with which we started.

And the same goes for the mereological sum (p + q + r).  A sum or fusion is not the sort of entity that can be either consistent or inconsistent.

What about the conjunction p & q & r?  A conjunction of propositions is itself a proposition.  (A set of propositions is not itself a proposition.) This seems to do the trick. We can parse (1) as

2. The conjunctive proposition p & q & r is (self)-inconsistent.

In this way we avoid construing (1) as an irreducibly plural predication.  For we now have a single entity that can serve as the logical subject of the predicate ' . . . is/are inconsistent.'  We can avoid saying, at least in this case, something that strikes me as only marginally intelligible, namely, that there are irreducible monadic non-distributive predicates.  My problem with irreducibly plural predication is that I don't know what it means to say of some things that they are F if that doesn't mean one of the following: (i) each of the things is F; (ii) there is a single 'collective entity' that is F; or (iii) the predicate 'is F'  is really relational. 

One could conceivably object that in the move from (1) to (2) I have 'changed the subject.'  (1) predicates inconsistency of some propositions, while (2) predicates (self)-inconsistency of a single conjunctive proposition.  Does this amount to a changing of thr subject?  Does (2) say something different about something different?

Sets and the Number of Objects: An Antilogism

Commenter Jan, the Polish physicist, gave me the idea for the following post.

An antilogism is an aporetic triad, an array of exactly three propositions which are individually plausible but collectively inconsistent.  For every antilogism, there are three corresponding syllogisms, where a syllogism is a deductive argument with exactly two premises and one conclusion.  Here is the antilogism I want to discuss:

1. Possibly, the number of objects is finite.
2. Necessarily, if sets exist, then the number of objects is not finite.
3. Sets exist.

The modality at issue is 'broadly logical' and sets are to be understood in the context of standard (ZFC) set theory. 'Object' here just means entity.  An entity is anything that is. (Latin ens, after all, is the present participle of the infinitive esse, to be.)

Corresponding to the above antilogism, there are three syllogisms. The first, call it S1, argues from the conjunction of (1) and (2) to the negation of (3).  The second, call it S2, argues from the conjunction of (2) and (3) to the negation of (1).  The third, call it S3, argues from (1) and (3) to the negation of (2). 

Note that each syllogism is valid, and that the validity of each reflects the logical inconsistency of the the antilogism. Note also that for every antilogism there are three corresponding syllogisms, and for every syllogism there is one corresponding antilogism.  A third thing to note is that S3 is uninteresting inasmuch as it is surely unsound.  It is unsound because (2) is unproblematically true. 

This narrows the field to S1 which argues to the nonexistence of (mathematical) sets and S2 which argues to the impossibility of the number of objects (entities) being finite.  Our question is which of these two syllogisms we should accept.  Obviously, both are valid, but both cannot be sound.  Do we have good reason to prefer one over the other?

Here are our choices.  We can say that there is no good reason to prefer S1 over S2 and vice versa; that there is good reason to prefer S1 over S2; or that there is good reason to prefer S2 over S1.

Being an aporetician, I incline toward the first option.  Peter Lupu, being less of an aporetician and more of dogmatist, favors the third option.  Thus he thinks that the antilogism is best solved by rejecting (1).  Peter writes:

(a) If there are infinitely many numbers, then (1) is false. Are there infinitely many numbers? Very few would deny this. How could they, for then they would have to reject most of mathematics. [. . .]

To keep it simple, let's confine ourselves to the natural numbers and the mathematics of natural numbers. (The naturals are the positive integers including 0.)  If there are infinitely many naturals, then there are infinitely many objects.  If so, then presumably this is necessarily so, whence it follows that (1) is false. 

I fail to see, however, why there MUST be infinitely many naturals.  I am of course not denying the obvious: for any n one can  add 1 to arrive at n + 1.  With a sidelong glance in the direction of Anselm of Canterbury: there is no n that fits the description 'that than which no greater can be computed.'   In plain English:  there is no greatest natural number.  But this triviality does not require that all of the results of possible acts of +1 computation actually be 'out there' in Plato's heaven.  When I drive along a road, I come upon milemarkers that are already out there before I come upon them.  But why must we think of that natural number series like this?  I don't bring the road and its milemarkers into being by driving.  But what is to stop us from viewing the natural number series along Brouwerian (intuitionistic) lines?  One can still maintain that the series is infinite, but the infinity is potential not actual or completed.  Peter's first argument, as it stands, is not compelling.  (Compare:  Everyone will agree that every line segment is infinitely divisible.  But it does not follow that every line segment is infinitely divided.)

(b) If propositions exist, then there are infinitely many propositions. Are there propositions? Kosher-nominalists obviously will have to deny that propositions exist. Sentences do not express propositions. But, then, what do they express?

I am on friendly terms with Fregean (not Russellian) propositions myself. And I grant that it is very plausible to say that if there is one proposition then there is an actual infinity of them.  Consider for example the proposition *p* expressed by 'Peter has a passion for philosophy.'  *P* entails *It is true that p* which entails *It is true that it is true that p,* and so on infinitely.  But again, why can't this be a potential infinity? 

The following three claims are consistent: (i) Declarative sentences express propositions; (ii)Propositions are abstract; (iii) Propositions are man-made. Karl Popper's World 3 is a world of abstracta.  It is a bit like Frege's Third Reich (as I call it), except that the denizens of World 3 are man-made.

I am agreeing with Peter and against the illustrious William that there are (Fregean) propositions, understood as the senses of context-free declarative sentences.  I simply do not understand how a declarative sentence-token could be a vehicle of a truth-value.  But why can't I say that propositions are mental constructs?  (This diverges from Frege, of course.)

(c) Are there sentence types? A nominalist will have to deny the existence of sentence types. But, then, it is difficult to see how any linguistic analysis can be done.

Peter may be conflating two separate questions.  The first is whether there are any abstract objects, sentence types for example. The second is whether there is an actual infinitity of them.  He neeeds the latter claim as a countrerexample of (1).  So again I ask:  why couldn't there be a finite number of abstract objects:  a finite number of sets, propositions, numbers, sentence types, etc.  This would make sense if items of this sort were Popperian World 3 items.

I conclude that, so far, there is no knock-down refutation of (1).  But there is also no knock-down refutation of (3) either, as Peter will be eager to concede.  So I suggest that the rational course is to view my (or my and Jan's) antilogism as a genuine intellectual knot that so far has not been definitively solved.

 

The Hatfields and the McCoys

Whether or not it is true, the following  has a clear sense:

1. The Hatfields outnumber the McCoys.

(1) says that the number of Hatfields is strictly greater than the number of McCoys.  It obviously does not say, of each Hatfield, that he outnumbers some McCoy.  If Gomer is a Hatfield and Goober a McCoy, it is nonsense to say of Gomer that he outnumbers Goober. The Hatfields 'collectively' outnumber the McCoys. 

It therefore seems that there must be something in addition to the individual Hatfields (Gomer, Jethro, Jed, et al.) and something in addition to the individual McCoys (Goober, Phineas, Prudence, et al.) that serve as logical subjects of number predicates.  In

2. The Hatfields are 100 strong

it cannot be any individual Hatfield that is 100 strong.  This suggests that there must be some one single entity, distinct but not wholly distinct from the individual Hatfields, and having them as members, that is the logical subject or bearer of the predicate '100 strong.'

So here is a challenge to William the nominalist.  Provide analyses of (1) and (2) that make it unnecessary to posit a collective entity (whether set, mereological sum, or whatever) in addition to individual Hatfields and McCoys.

Nominalists and realists alike agree that one must not "multiply entities beyond necessity."   Entia non sunt multiplicanda praeter necessitatem!  The question, of course, hinges on what's necessary for explanatory purposes.  So the challenge for William the nominalist is to provide analyses of (1) and (2) that capture the sense of the analysanda and obviate the felt need to posit entities in addition to concrete particulars.

Now if such analyses could be provided, it would not follow that there are no 'collective entities.'  But a reason for positing them would have been removed.

I Need to Study Plural Predication

Here is a beautiful aphorism from Nicolás Gómez Dávila (1913-1994), in Escolios a un Texto Implicito (1977), II, 80, tr. Gilleland: 

Stupid ideas are immortal. Each new generation invents them anew.

Clearly this does not mean:

1. Each stupid idea is immortal and is invented by each new generation anew.

So we try:

2. The set of stupid ideas is immortal in the sense that every new generation invents some stupid idea or other.

(2) is much closer to the intended meaning. The idea is that there are always stupid ideas around, not that any one stupid idea is always around. (2) seems to capture this notion. But (2) presents its own puzzles. A set is a collection, and a collection is not the mere manifold of its members: it is "a further entity over and above them" as Michael Potter puts it in Set Theory and its Philosophy (Oxford 2004, p. 22).

Potter speaks of collections versus fusions. The distinction emerges starkly when we consider that there is a distinction between a singleton collection and its member, but no distinction between a 'singleton' fusion and its member. Thus Quine is distinct from {Quine}, the set consisting of Quine and nothing else. But there is no distinction between Quine and the sum or fusion, (Quine). {Quine}, unlike Quine, has a member; but neither (Quine) nor Quine have members. A second difference is that, while it makes sense to speak of a set with no members, the celebrated null set, it makes no sense to speak of a null fusion. The set consisting of nothing, the null set { } is something; the fusion of nothing is nothing.

Getting back to stupid ideas, what I want to say is that 'stupid ideas are immortal' can be understood neither along the lines of (1) nor along the lines of (2). The generality expressed is quite obviously not distributive, but it is not quite collective either. We are not expressing the idea that there is some one entity "over and above" its members to which immortality is being ascribed. 'Stupid ideas' seems to pick out a fusion; but if a fusion is a pure manifold, how can it be picked out?  

The puzzle is that immortality is not being predicated of each stupid idea, but it is also not being predicated of some one item distinct from stupid ideas which has them as members, whether this one item be a mathematical set or a mereological sum.

We know what we mean when we say that stupid ideas are immortal, but we cannot make it precise, or at least I can't make it precise given my present level of logical acumen.

So rather than contribute any stupid ideas of my own, I will go to the library and check out Thomas McKay's Plural Predication.  

 

The Axiom of Infinity as Easy Way Out?

I posed the question, Can one prove that there are infinite sets?  Researching this question, I consulted the text I studied when I took a course in set theory in a mathematics department quite a few years ago. The text is Karl Hrbacek and Thomas Jech, Introduction to Set Theory (Marcel Dekker, 1978). On pp. 53-54  we read:

It is useful to formulate Theorem 2.4 a little differently. We call a set A inductive if (a) 0 is an element of A; (b) if x is an element of A, then S(x) is an element of A. [The successor of a set  x is the set S(x) = x U {x}.]

In this terminology, Theorem 2. 4 is asserting that the set of natural numbers is inductive. There is only one difficulty with this reformulation: We have not yet proved that the set of all natural numbers exists. There is a good reason for it: It cannot be done, axioms adopted so far do not imply existence of infinite sets. Yet the possibility of collecting infinitely many objects into a single entity is the essence of set theory and the main reason for its usefulness in many branches of abstract mathematics.  We, therefore, extend our axiomatic system by adding to it the following axiom.

The Axiom of Infinity. An inductive set exists.

Intuitively, the set of all natural numbers is such a set.

Therefore, if we turn to the mathematicians for help in answering our question, we get the following. There are infinite (inductive) sets because we simply posit their existence! Thus their existence is not proven, but simply assumed. Philosophically, this leaves something to be desired. For it is not self-evident that there should be any infinite sets.  If there are infinite sets, then they are actually, not potentially, infinite.  (The notion of a potentially infinite mathematical set is senseless.)  And it is not self-evident that there are actual infinities.

I will be told that there is no necessity that an axiom be self-evident.  True: axiomhood does not require self-evidence.  But if an axiom is an arbitrary posit, then I am free to reject it.  Being a cantankerous philosopher, however, I demand a bit more from a decent axiom.  I suppose what I am hankering after is a compelling reason to accept the Axiom of Infinity.

A comparison with complex (imaginary) numbers occurs to me.  They are strange animals.  But however strange they are, there is a sort of argument for them in the fact that they 'work,' i.e. they find application in alternating current theory the implementation of which is in devices all around us. But can a similar argument be made for the denizens of Cantor's Paradise?  I don't know, but I have my doubts.  Nature is finite and so not countably infinite let alone uncountably infinite.  But caveat lector:  I am not a philosopher of mathematics; I merely play one in the blogosphere.  What you read here are jottings in an online notebook.  So read critically.

 

Sets, Pluralities, and the Axiom of Pair

In a thread from the old blog, resident nominalist gadfly 'Ockham'/'William' made the fascinating double-barreled claim that:

. . . (a) there are such things as sets and (b) the axiom of pairs is false. Briefly, I claim that 'a set of x's' is just another way of saying 'those x's'. The fundamental error of set theory is using a logically singular expression {a, b} to refer to what in ordinary language a plural term refers to, using an expression such as 'a and b' or similar.

I take O to be saying that there are sets, but they are not the sets we read about in standard treatments of axiomatic set theory, and whose properties are all and only the properties ascribed to them in axiomatic set theory, Zermelo-Fraenkel with Choice, to be specific. Suppose we call the latter mathematical sets, and the former ordinary language (commonsense) sets. Then what O is claiming is that there are ordinary language (OL) sets, but there are no mathematical sets. That there are no mathematical sets on O's view follows from O's denial of the Axiom of Pair, a crucial ingredient of ZFC. Here is a formulation of the latter:

PAIR. Given any x and y, there is a set {x, y} the members of which are exactly x and y.

X and y can be either sets or nonsets. So given that Socrates exists and that Plato exists, it follows by PAIR that a third item exists, namely, {Socrates, Plato}. (I use 'there is' and 'there exists' interchangeably.) That a third item exists is what I affirm and what O denies. For O, the plural term 'Socrates and Plato' does not refer to a single third item, the set consisting of Socrates and Plato; and yet it does refer to something, a thing that is an ordinary language set. For O, there are exactly two items in our example, Socrates and Plato, and not three, as I claim.

Let us say that the referent of a plural term such as 'Socrates and Plato' or 'the British Empiricists' or 'the Hatfields' is a plurality. A plurality is an ordinary language set. A gaggle of geese, a pride of lions, a coven of witches, a bunch of grapes, a pack of wolves -- these are all pluralities or OL sets. That there are OL sets, or pluralities, is presumably not in dispute. Nor, I think, could anyone rationally dispute their existence. That there is such a thing as a pair of shows cannot be reasonably denied; that the two shoes form a mathematical set can be reasonably denied at least prima facie.

If I understand O, he is saying that all reference to sets is via plural referring expressions such as 'these books,' 'Dick Dale and the Deltones,' 'the barristers of London,' etc. There is no reference to any set via a singular referring device such as the singular definite description, 'the set consisting of these books.'

Now consider the question whether there are sets of sets. I claim that it is a fact that there are sets of sets, and that this fact causes trouble for O's nominalist view that all sets are pluralities. Consider the Hatfields and the McCoys. These are two famous feuding Appalachian families, and therefore two pluralities or OL sets. But there is also the two-membered plurality of these pluralities to which we refer with the phrase 'the Hatfields and the McCoys' in a sentence like 'The Hatfields and the McCoys are feuding families.'

If, however, a plurality of pluralities has exactly two members, as in the case of the Hatfields and the McCoys, then the latter cannot themselves be pluralities, but must be single items, albeit single items that have members. That is to say: In the sentence, 'The Hatfields and the McCoys are two famous feuding Appalachian families,' 'the Hatfields' and 'the McCoys' must each be taken to be referring to a single item, a family, and not to a plurality of persons. For if each is taken to refer to a plurality of items, then the plurality of pluralities could not have exactly two members but would many more than two members, as many members as there are Hatfields and MCoys all together. Compare the following two sentences:

1. The Hatfields and the McCoys number 100 in toto.

2. The Hatfields and the McCoys are two famous feuding Appalachian families.

In (1),'the Hatfields and the McCoys' can be interpreted as referring to a plurality of persons as opposed to a mathematical set of persons. But in (2), 'the Hatfields and the McCoys' cannot be taken to be referring to a plurality of pluralities; it must be taken to be referring to a plurality of two single items.

Or consider the following said to someone who mistakenly thinks that the Hatfields and the McCoys are one and the same family under two names:

3. The Hatfields and the McCoys are two, not one.

Clearly, in (3) 'the Hatfields and the McCoys' refers to a two-membered plurality of single items, each of which has many members, and not to a plurality of pluralities. And so we must introduce mathematical sets into our ontology.

This is connected with the fact that '___ is an element of . . .' in axiomatic set theory does not pick out a transitive relation: If x is an element of y, and y is an element of z, it does not follow that x is an element of z. Socrates, a nonset, is an element of various sets; but he is clearly not a member of any of these set's power sets. (The power set P(S) is the set of all of S's subsets. Clearly, no nonset can be a member of any power set.) But if there are no mathematical sets, and every set is a plurality, then it seems that the elementhood or membership relation would be transitive. A set of sets would be a plurality of pluralities such that if x is an element of S and S an element of S *, then x is an element of S*. My conclusion, contra 'Ockham,' is that we cannot scrape by on OL sets, or pluralities, alone. We need mathematical sets or something like them: entities that are both one and many.

REFERENCES

Max Black, "The Elusiveness of Sets," Review of Metaphysics, vol. XXIV, no. 4 (June 1971), 614-636.

Stephen Pollard, Philosophical Introduction to Set Theory, University of Notre Dame Press, 1990.

A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

A commenter in the 'Nothing' thread spoke of possible worlds as sets.  What follows is a reposting from 1 March 2009 which opposes that notion.

................

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.

In general, if a set S has n members, then P(S) has 2ⁿ members. Hence the name power set. Cantor's Theorem that the power set of a set S is always strictly larger that S is easily proven. But the proof needn't concern us.  It is available in any standard book on set theory.

2. Suppose there is a set T of all truths, {t₁, . . . , t_i, t_{i + 1}, . . .}. Consider the power set P(T) of T. The truth t₁ in T will be a member of some of T's subsets but not of others. Thus, t₁ is an element of {t₁, t₂}, but is not an element of { }. In general, for each subset s in the power set P(T) there will be a truth of the form t₁ belongs to s or t₁ does not belong to s. But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all truths.

3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths.  This implies that possible worlds cannot be maximally consistent sets of propositions.   I learned the Cantorian argument that there  is no set of all truths from Patrick Grim. I don't know whether he applies it to the question whether worlds are sets.

4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from from being maximally consistent conjunctive propositions.

A commenter in the 'Nothing' thread spoke of possible worlds as sets.  What follows is a reposting from 1 March 2009 which opposes that notion.

................

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.

In general, if a set S has n members, then P(S) has 2ⁿ members. Hence the name power set. Cantor's Theorem that the power set of a set S is always strictly larger that S is easily proven. But the proof needn't concern us.  It is available in any standard book on set theory.

2. Suppose there is a set T of all truths, {t₁, . . . , t_i, t_{i + 1}, . . .}. Consider the power set P(T) of T. The truth t₁ in T will be a member of some of T's subsets but not of others. Thus, t₁ is an element of {t₁, t₂}, but is not an element of { }. In general, for each subset s in the power set P(T) there will be a truth of the form t₁ belongs to s or t₁ does not belong to s. But according to Cantor's Theorem, the power set of T is strictly larger than T. So there will be more of those truths than there are truths in T. It follows that T cannot be the set of all truths.

3. Given that there cannot be a set of all truths, the actual world cannot be the set of all truths.  This implies that possible worlds cannot be maximally consistent sets of propositions.   I learned the Cantorian argument that there  is no set of all truths from Patrick Grim. I don't know whether he applies it to the question whether worlds are sets.

4. As far as I can see, the fact that possible worlds cannot be maximally consistent sets does not prevent them from from being maximally consistent conjunctive propositions.

On the Abstractness of Mathematical Sets

Let us agree that x is concrete iff x is causally/active passive and abstract otherwise.  Many say that mathematical sets ('sets' hereafter: 'mathematical' as opposed to 'commonsense') are abstract objects, abstract entities, abstracta.  Why?

Argument One:  In set theory there are singleton sets, e.g. {Quine}.  Obviously, Quine is not identical to {Quine}.  The second is a set, the first is not.  Yet the difference cannot be the difference between two concreta.  Quine is a concretum.  Therefore, {Quine} is an abstractum.  This is of course meant generally: singletons are abstracta.  Now if singletons are abstracta, then all sets are. 

Argument Two:  In set theory there is a null set.  It is not nothing, but something despite having no members. Yet it cannot be a concrete something.  Therefore, it is an abstract something.  And if one set is abstract, all are.

Contra Argument One:  A statue and the lump of clay that constitute it are numerically distinct.  (For the one has properties the other doesn't have, e.g., the lump, but not the statue, can exist without having the form of a statue.)  And yet both are concrete, i.e., both are causally active/passive.  If this is possible, why should it not also be possible that Quine and {Quine} both be concrete?  One could say that Quine and {Quine} occupy the same 'plime' to borrow a term form D. C. Williams, the same place-time, in the way statue and lump do.

Contra Argument Two:  Possibly, there is a concrete atomic entity. Being atomic, it has no parts.  So why should a set's having no members rule out its being concrete?

Are any of these arguments compelling?

Another Example of a Necessary Being Depending for its Existence on a Necessary Being

The Father and the Son are both necessary beings.  And yet the Father 'begets' the Son.  Part, though not the whole, of the notion of begetting here must be this: if x begets y, then y depends for its existence on x.  If that were not part of the meaning of 'begets'' in this context, I would have no idea what it means.  But how can a necessary being depend for its existence on a necessary being?  I gave a non-Trinitarian example yesterday, but it was still a theological example. Now I present a non-theological example.

I assume that there are mathematical (as opposed to commonsense) sets.  And I assume that numbers are necessary beings.  (There are powerful arguments for both assumptions.) Now consider the set S = {1, 3, 5} or any set, finite or infinite, the members of which are all of them necessary beings, whether numbers, propositions, whatever.  Both S and its membership are necessary beings.  If you are worried about the difference between members and membership, we can avoid that wrinkle by considering the singleton set T = {1}.

T and its sole member are both necessary beings.  And yet it seems obvious to me that one depends on the other for its existence:  the set is existentially dependent on the member, and not vice versa.  The set exists because -- though this is not an empirically-causal use of 'because' -- the members exist, and not the other way around.   Existential dependence is an asymmetrical relation.  I suppose you either share this intuition or you don't.  In a more general form, the intuition is that collections depend for their existence on the things collected, and not vice versa.  This is particularly obvious if the items collected can also exist uncollected.  Think of Maynard's stamp collection.  The stamps in the collection will continue to exist if Maynard sells them, but then they will no longer form Maynard's collection. The point is less obvious if we consider the set of stamps in Maynard's collection.  That set cannot fail to exist as long as all the stamps exist.  Still, it seems to me that the set exists because the members exist and not vice versa.

And similarly in the case of T.  {1} depends existentially on 1 despite the fact that there is no possible world in which the one exists without the other.  If, per impossibile, 1 were not to exist, then {1} would not exist either. But it strikes me as false to say: If, per impossibile, {1} were not to exist, then 1 would not exist either.  These counterfactuals could be taken to unpack the sense in which the set depends on the member, but not vice versa.

It therefore is reasonable to hold that two necessary beings can be such that one depends for its existence on the other.  And so one cannot object to the notion of the Father 'begetting'  the Son by saying that no necessary being can be existentially dependent upon a necessary being.  Of course, this is not to say that other objections cannot be raised.

Trinity and Set Theory

Let S and T be mathematical sets. Now consider the following two propositions:

1. S is a proper subset of T.

2. S and T have the same number of elements.

Are (1) and (2) consistent? That is, can they both be true? If yes, explain how.

If you think (1) and (2) are consistent, then consider whether there is anything to the following analogy. If there is, explain the analogy. There is a set G. G has three disjoint proper subsets, F, S, H. All four sets agree in cardinality: they have the same number of elements.

A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

Transitive Sets and the Distinctness of Sets From Their Members

Vlastimil asked for examples of transitive sets.  A transitive set is a set every element of which is a subset of it.  (Hrbacek and Jech, Introduction to Set Theory, p. 50) There is no lack of examples.  The null set vacuously satisfies the condition 'if x is an element of S, then x is a subset of S.'  The set consisting of the null set -- {{ }} -- is also transitive:  it has exactly one element, the null set, and that element is a subset of it because the null set is a subset of every set.

Now consider the set consisting of the foregoing two sets, the null set and the set consisting of the null set:  {{ }, {{ }}}.  This set has two elements and both are subsets of it.  The null set is a subset of every set, and the set consisting of the null set is also a subset of it in virtue of the fact that the null set is an element of it.

If we identify 0 with the null set, and 1 with the set consisting of the null set, and 2 with the set consisting of the null set and the set consisting of the null set, then 3 will be the set whose elements are the elements of 0, 1, and 2 which is:  {{ }, {{ }}, {{ }, {{ }}}}.  This last set has three elements and each is a subset of it.  One can continue like this and generate as many transitive sets as one likes.  For each natural number there is a corresponding transitive set.

Now how does all this bear upon my assertion that a (mathematical) set is an entity 'over and above' its members (elements)?  That sets are treated in set theory as single items 'over and above' their members can be seen from the fact that some sets have sets as members without having their members as members.   The power set of {Socrates, Plato} has {Socrates} and {Plato} as members, but it does not have Socrates and Plato as members. Therefore, {Socrates} is distinct from Socrates, and {Plato} from Plato.  For if these singletons were identical to their members, then the power set would have Socrates and Plato as members.

Vlastimil seems to think that the existence of transitive sets is somehow at odds with the claim that sets are distinct from their members.  Or perhaps he thinks that some sets are distinct from their members and some are not.  So consider {{ }, {{ }}}.  This is a transitive set since every member of it is a subset of it, which is equivalent to saying that every member of a member of it is a member of it.  Thus { } is a member of {{ }}, which is a member of {{ }, {{ }}}.  But although every member of the set in question is a subset of it, this does not alter the fact that the set is distinct from its members.

So I'm not sure what Vlastimil is driving at.

Note that if every member of a set is a subset of it, this is not to say that every subset of it is a member of it.  {{ }, {{ }}} has itself and {{{ }}} as subsets but not as elements.  Only if there were a set all of whose members are subsets of it and all of whose subsets are members of it could one argue that there are sets for which the membership and subset relations collapse, and with it the distinction between a set and its members.

On the Elusive Notion of a Set: Sets as Products of Collectings

In an important article, Max Black writes:

Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)

1. A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.   A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set  is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many.  A set is one in that it is a single item 'over and above' its members or elements as I have just explained.  It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.)  The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect.  So it seems there is a genuine puzzle here.  How remove it?

Non-Empty Thoughts About the Empty Set

1. The empty or null set is a strange animal. It is a set, but it has no members. This is of course not a contingent fact about it, but one bound up with its very identity: the null set is essentially null. Intuitively, however, one might have thought that a set is a group of two or more things. Indeed, Georg Cantor famously defines a set (Menge) as "any collection into a whole (Zusammenfassung zu einem Ganzen) of definite and separate objects of our intuition and thought." (Contributions to the Founding of the Theory of Transfinite Numbers, Dover 1955, p. 85) In the case of the null set, however, there are no definite objects that it collects. So in what sense is the null set a set? One might ask a similar question about singletons, sets having exactly one member. But I leave this for later.