Set Theory – Maverick Philosopher

Notes on Infinite Series

The resident nominalist writes,

Your post generated a lot of interest. What I have to say now is better put as a separate post, rather than a long comment. Feel free to post.

1) Plural reference provides a means of dealing with numbers-of-things without introducing extra unwanted entities such as sets. Even realists agree that we should not have more entities than necessary, the disagreement is about what is ‘necessary’.

BV: We agree that entities should not be multiplied beyond necessity, i.e., beyond what is needed for explanatory purposes. The disagreement, if any, will concern what is needed.

2) Using plural quantification we can postulate the existence of an infinite number-of-things. We simply postulate that for any number-of-things, there is at least one other thing. That gives a larger number-of-things, which itself is covered by the quantifier ‘any’, hence there must be a still larger number-of-things, etc.

BV: You give no example, so let me supply one. Consider the series of positive integers: 1, 2, 3 . . . n, n + 1, . . . . Given 1, we can generate the rest using the successor function: S(n) = n + 1. I used the word 'generate' since it comports well with your intuition that there are no actual infinities, and that therefore every infinity is merely potential.

3) In this way we neatly distinguish between actual and potential infinity. Using plural quantification, we can prove that there is no plural reference for ‘all the things’. For that would be a number-of-things, hence there must be an even larger number-of-things, which contradicts the supposition that we had all the things.

BV: Your argument is rather less than pellucid. Here is the best I can do by way of reconstructing your argument:

a) If the plural term, 'all the positive integers,' refers to something, then it refers to a completed totality of generated integers. But

b) There is no completed totality of generated integers.

Therefore, by modus tollens,

c) It is not the case that 'all the positive integers' refers to something.

Therefore

d) There is no actually infinite set of positive integers.

If that is your argument, then it begs the question at line (b). One man's modus tollens is another man's modus ponens. If the above is not your argument, tell me what your argument is. So far, then, a stand-off.

4) In this way we also avoid the pathological results of Cantorean set theory. If there is a set of natural numbers, then this is also a number, but it cannot itself be a natural number, so it is the first ‘transfinite number’. The nominalist approach avoids such weird numbers.

BV: But surely polemical verbiage is out of place in such serene precincts as we now occupy. You cannot shame Cantor's results out of existence by calling them 'pathological' or 'weird.' Most if not all working mathematicians accept them, no?

5) The problem for the nominalist arises when in trying to explain the sum of an infinite series, e.g. 1 + ½ + ¼ + ⅛ … The realist wants to argue that unless this series is ‘completed’, we don’t have all the members, so the sum will amount to less than 2.

BV: Note that the formula for the series is 1/2ⁿwhere n is a natural number with 0 being the first natural number. Recall that any number raised to the zeroth power = 1. (If you need to bone up on this, see here.)

Question for our nominalist: what does '1/2ⁿ' refer to? Can't be a set! And it can't be a property! Does it refer to nothing? Then so does '1-1/2n.' How then explain the difference between the two formulae (rules) for generating two different infinite series?

Or more simply, consider n. It is a variable. It has values and substituends. The values are the natural numbers. Only the ones we counted up to, or generated thus far? No, all of them. The ones we have actually counted up to in a finite number of countings, and the rest which are the possible objects of counting. The variable is a one-over-the-many of its values, and a one-over-the-many of its substituends, which are numerals, not numbers. Numerals bring in the type-token distinction. And so I will ask the nominalist what linguistic types are. Are they sets? No. Are they properties? No. What then?

6) It’s a difficult question for the nominalist, but here is my attempt to resolve it. Start with the notion of non-overlapping parts. Two non-overlapping parts have no part that is part of the other. Then there can be a number of non-overlapping parts such that there is no other such part, i.e. these are ‘all’ such parts.

BV: OK.

7) Then suppose we have a method of defining the parts. Start with a line of length 2. Note that the nominalist is OK here with the existence of lines, because lines are real things and not artificially constructed entities like ‘sets’. And suppose we can divide the line into two non-overlapping parts of equal length, i.e., a part of length 1, and another part of the same length.

BV: You shouldn't say that sets are artificially constructed. After all, you think numbers are artificially constructed, no? They are artifacts of counting. Your beef is with abstract objects, not artificial objects. Sets are abstract particulars. You oppose them for that reason. As a nominalist you hold that everything is a concrete particular. (Or am I putting words in your mouth?)

Second, you are ignoring the difference between a geometrical line and a line drawn with pencil on paper, say. The latter is a physical line, which is actually a 3-D object with length, width and depth. In addition to its pure geometrical properties, it has physical and chemical properties. It is a physical line in physical space. The former is not a physical line, but an ideal line: it has length, but no width or depth. Ideal lines are not in physical space. Suppose physical space, the space of nature, is non-Euclidean. Then Euclidean lines are obviously not in physical space. But even if physical space is Euclidean, Euclidean lines would still not be in physical space.

8. So the proposition “2 = 1+1” says that a line of length two can be divided into two equal non-overlapping parts. Then suppose that we divide the second part into two equal parts. Thus “2 = 1 + ½ + ½” says that the line can be divided into three non-overlapping parts, one of length 1, and the other two equal. Do the same again, thus 2 = 1 + ½ + ¼ + ¼. And again and again!

BV: An obvious point is that the arithmetical proposition '2 = 1 + 1' is not about lines only. It could be about a two-degree linear cool-down of a poker. (I am thinking about Wittgenstein's famous poker-brandishing incident.) It could be about anything. Two pins. An angel on a pin joined by another.

Besides, "2 = 1 + 1" cannot be about the non-overlapping parts of a particular line, the one you drew in the sand. It is about a geometrical line, which is an ideal or abstract object. The theorem of Pythagoras is not about the right triangle you drew on the blackboard with chalk; it is about the ideal right triangle that the triangle you drew merely approximates to.

9. It is clear that for every such division, the parts ‘add up’ to the same number, i.e. 2.

10. Then consider what the proposition “2 = 1 + ½ + ¼ + ⅛ … “ expresses. Surely that every such series, however extended, has a sum of 2. Do we need the notion of a ‘set’? No.

BV: I don't see how this answers the question that you yourself raised in #5 above. What makes it the case that the series you mention actually has a sum of 2? The most you can say is that series potentially has a sum of 2. The Cantorean does not face this problem because he can say that there is an actual infinity of compact fractions that sums to 2. No endless task needs to be performed to get to the sum.

On Sets: A Response to Brightly

David Brightly in a comment far below writes:

Bill says, at 03:21 PM.

…a family cannot be reduced to a number of persons; it is not a mere manifold, or a mere many…

The same is true for a pair (of shoes) and other collective terms. They all imply relations of some sort between their members. But this is not so for a mathematical set. No such relations are implied. A set is a pure manifold. Bill also says, at 03:42 PM,

…it also makes no sense to say that an abstract object — which is what math sets are — has Gomer surrounded.

I dispute this: mathematical abstract objects have relations between their parts and sets qua sets do not, so sets cannot be abstract objects. These are the chief reasons for thinking that sets are pure plural referring terms. Artificial, structured, linguistic extensions engineered by us for clarity of mathematical thought and expression.

Respondeo

I am afraid I find most of the above unacceptable. First of all a pair of shoes is not a term, collective or singular. Terms are linguistic items; shoes are not. It is true, though, that mathematical sets abstract from any relation their members bear to one another. But this is not something I denied.

It is also not the case that "a set is a pure manifold." That cannot be right because a set can have sets as elements (members). The Power Set axiom of axiomatic set theory states that for any set S, there exists a set P such that X is an element of P if and only if X is a subset of S. Thus the power set P of a set S is the set of all of S's subsets. So the power set of {1,2} = {{1}, {2}, {1,2}, { }}.

The subsets of S are elements of the power set which fact shows that sets are distinct from their members and are therefore not pure manifolds. Sets are distinct from their members in that they count as objects in their own right. So if there is a set of Ed's dancing shoes, then this set is distinct from the left shoe, the right shoe, and the two taken collectively. This is one of the ways a set differs from a mereological sum/fusion. The sum of Ed's shoes is just those shoes, not an object distinct from them.

Mereology is "ontologically innocent" as David Lewis puts it, whereas set theory is not:

Mereology is innocent . . . we have many things, we do mention one thing that is the many taken together, but this one thing is nothing different from the many. Set theory is not innocent. . . . when we have one thing, then somehow we have another wholly distinct thing, the singleton. And another, and another . . . ad infinitum. (Parts of Classes, Basil Blackwell 1991, p. 87)

Brightly argues: "mathematical abstract objects have relations between their parts and sets qua sets do not, so sets cannot be abstract objects." This makes no sense to me since the mereological notion of parthood does not belong in standard axiomatic set theory. It is at best analogous to the elementhood and subset relations which are essential to set theory. The sum/fusion of Max and Manny has them and their cat-parts as parts. The set {Max, Manny} does not have as elements their whiskers, claws, tails, etc.

If there are sets, they are abstract objects, which is to say that they do not exist in space or time. I can trip over my cat, but I can't trip over my cat's singleton. Where in space is the null set? When did it come to be? How long will it last? Is it earlier or later than the null set's singleton? Is the nondenumerably infinite set of real numbers somewhere in spacetime? These questions involve category mistakes.

I said that if the Hatfields have Gomer surrounded, that cannot be taken to mean that the set having all and only the members of the Hatfield clan as its members has poor Gomer surrounded. What I said is obviously true.

So I conjecture that David is using 'abstract' in some idiosyncratic way. I await his clarification.

Nominalism Presupposes What it Denies

What makes a pair of shoes a pair and not just two physical artifacts? Nominalist answer: nothing in reality. Our resident nominalist tells us that it is our use of 'a pair' that imports a unity, conventional and linguistic in nature, a unity that does not exist in reality apart from our conventional importation. We are being told that out there in the world there are no ones-in-many, let alone any ones-over-many. If that is right, then there are no sets. For a set is a one-over-many in this sense: it is one item distinct from its many members. (Let's not worry about the null set, which has no members and unit-sets or singletons which have exactly one member each. Here lies yet another rich source of aporiai, but one problem at a time.)

If there are no sets, then there are neither finite sets nor infinite sets. There are just pluralities, and all grouping, collecting, subsuming under common rubrics, unifying, etc. is done in language by language-users. What I will try to show is that if you think carefully about all of this you will have to make distinctions that are inconsistent with nominalism.

My aim is purely negative: to show that the nominalism of the resident nominalist is untenable. If you have read a good amount of what I have written you will recall that I am a solubility skeptic, which in this instance means that I am not endorsing any realist solution of the problem. I am not pushing an opposing theory.

I will start with some data that I find 'Moorean,' i.e., rationally indisputable and pre-theoretical. (Unfortunately, one man's datum is another man's theory.) The phrase 'a pair' has a sense that remains the same over time and space, a sense that is the same for all competent speakers of English whether here or abroad. The same holds for ein Paar in German, and similarly for all languages. The sense or meaning of an expression, whether word, phrase, sentence, etc. must be distinguished from the expression. An expression is something physical and thus sensible. The sensible is that which is able to be sensed via one of our senses. I hear the sound that conveys to me the meaning of 'cat,' say, or I see the marks on paper. Hearing and seeing are outer senses that somehow inform us or, more cautiously, purport to inform us of the existence and properties of physical or material things that exist whether or not we perceive them. But I don't hear or see the meaning conveyed to me by your utterance of a sentence such 'The cats are asleep.' The sentence, being a physical particular, is sensible; the meaning is intelligible. That's just Latin for understandable. I hear the words you speak, and if all goes well, I understand their meaning or sense, thereby understanding the proposition you intend to convey to me, namely, that the cats are asleep. Note that while one can trip over sleeping cats, one cannot trip over that the cats are asleep.

There are two distinctions implicit in the above that need to be set forth clearly. I argue that neither is compatible with nominalism

A. The distinction between the sense/meaning of a linguistic expression and the expression. Why must we make this distinction? (a) Because the same sense can be expressed at different times by the same person using the same expression. (b) Because the same sense can be expressed at the same and at different times by different people using the same expression. (c) Because the same sense can be expressed in different languages using different expressions by the same and different people at the same and at different times. For example the following sentences express, or rather can be used to express, the same sense (meaning, proposition):

The cat is black.
Il gatto è nero.
Die Katze ist schwarz.
Kedi siyah.
Kočka je černá.

So the sense of a word or phrase or sentence is a one-in-many in that each tokening of the word or phrase expresses numerically the same sense. A tokening, by definition, is the production of a token, in this case, a linguistic token. One way a speaker can produce such a token is by uttering the word or phrase in question. Another way is by writing the word or phrase down on a piece of paper. (There are numerous other ways as well.) This production of tokens therefore presupposes a further distinction:

B. The distinction between linguistic types and linguistic tokens. In the following array, how many words are there?

cat
cat
cat

Three or one? Is the same word depicted three times? Or are there three words? Either answer is as good as the other but they contradict each other. So we need to make a distinction: there are three tokens of the same type. We are forced by elementary exegesis of the data to make the type-token distinction. If you don't make it, then you will not be able to answer my simple question: three words or one?

You see (using the optical transducers in your head, and not by any visio intellectualis) the three tokens. And note that the tokens you now see are not the tokens I saw when I wrote this entry. Those were different tokens of the same type, tokens which, at the time of your reading are wholly past. Linguistic tokens are in time, and in space, which is not obviously the case for linguistic types. I said: not obviously the case, not: obviously not the case. You see the three tokens, but do you see the type of which they are the tokens? If you do, then you have powers I lack. And yet the tokens are tokens of a type. No type, no tokens. So types exist. How will our nominalist accommodate them? He cannot reduce types to sets of tokens since he eschews sets. No sets, no sets of linguistic tokens. Linguistic types are multiply instantiable. That makes them universals. But no nominalist accepts universals. Nominalists hold that everything is a particular. I grant that the rejection of sets and the rejection of universals are different rejections. But if one rejects sets because they are abstract objects, one ought also to reject universals for the same reason.

Now glance back at the first array. What we have there are five different sentence tokens from five different languages. Each is both token – and type-distinct from the other four.

To conclude, I present our nominalist with two challenges. The first is to give a nominalist account of linguistic types without either reducing them to sets or treating them as ones-in-many or ones-over-many. The second challenge is to explain the distinction between the sense or meaning of an expression, which is not physical/material and the expression which is.

Suppose he responds to the second challenge by embracing conceptualism according to which meanings are mental. Conceptualism is concept-nominalism, as D. M. Armstrong has maintained. My counterargument would be that the meaning/sense expressed by a tokening of 'The cats are asleep' is objectively either true or false, and thus either true or false for all of us, not just for the speaker. Sentential meanings are not private mental contents. Fregean Gedanken, for example, are not dependent for their existence or truth-value on languages or language-users.

Why Not be a Nominalist about Sets?

The resident nominalist comments:

Nominalists say that the conception of an actual infinity of natural numbers depends on there being a set of all such numbers. But Ockhamists do not believe in sets. They say that the term ‘a pair of shoes’ is a collective noun which deceives by the singular expression ‘a pair’. Deceives, because it means no more than ‘two shoes’, and if there is only a pair of shoes, then there are only two things. But if a ‘pair’ of two things is a single thing, there are three things, the two things and the pair. Ergo etc.

I agree that there cannot be an actual infinity of natural numbers unless there is a (mathematical as opposed to commonsense) set of all such numbers. But of course this holds for all numbers, rational, irrational, transcendental, etc. Indeed, it holds for any category of item that is actually infinite. If there is an actual infinity of propositions, for example, then there must be a set of all propositions. I would point out however that there is nothing nominalistic about our friend's opening remark.

Nominalism kicks in with the claim that there are no sets. What there are are plural referring devices such as 'a pair of shoes' which fools us into thinking that in reality, i.e., extralinguistically, there are three things, a left shoe, a right shoe, and the pair, when there are only two things, the two shoes. The same goes for the following seemingly singular but really plural phrases: a gaggle of geese, a pride of lions, a parliament of owls, a coven of witches, etc.

This all makes good sense up to a point. When I put on my shoes, I put on one, then the other. It would be a lame joke were you to say to me, "You put on the left shoe and then the right one; when are you going to put on the pair?" To eat a bunch of grapes is to eat each grape in the bunch; after that task is accomplished there is nothing left to do. The bunch is not something 'over and above' the individual grapes that I still need to eat.

Consider now the Hatfields and the McCoys. These are two famous feuding Appalachian families, and therefore two pluralities. They cannot be (mathematical) sets on the nominalist view. 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 families feuding with each other.'

If, however, a plurality of pluralities has exactly two members, as in the case of the Hatfields and the McCoys — taking those two collections collectively — then the latter cannot themselves be mere pluralities, but must be single items, albeit single items that have members. They must be both one and many. 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 persons; 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.

My conclusion, contra the resident nominalist, is that we cannot scrape by on pluralities alone. (Man does not live by manifold alone! He needs unity!) We need mathematical sets or something like them: entities that are both one and many. A set, after all, is a one-in-many. It is not a mere many, and it is not a one 'over and above' a many. The nominalist error is to recoil from the latter absurdity and end up embracing the former. The truth is in the middle.

What I have given is an argument from ordinary language to mathematical sets. But there are also mathematical arguments for sets. Here is a very simple one. The decimal expansion of the fraction 1/3 is nonterminating: .33333333 . . . . But if I trisect a line, i.e., divide it into three equal lengths, I divide it into three quite definite actual lengths. This can be the case only if the the decimal expansion is a completed totality, an actual infinity, not a merely potential one. An even better example is that of the irrational number, the square root of 2 — it is irrational because it cannot be expressed as a ratio of two numbers, the numerator and the denominator of a fraction as in the case of of the rational 1/3. If the hypotenuse of a right triangle is ${\sqrt {2}}$

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.

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".

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

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.

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".

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.