Bob Murphy is Wrong About Conditionally Convergent Series

Math, Math Proof, Set Theory / By /
Bob Murphy

Table of Contents


Context

I am a Bob Murphy fan. I loved his run on Contra Krugman, I currently enjoy The Bob Murphy Show, and I think his book ‘Choice’ is a must read for anyone wanting a solid foundation in economics. Another book he coauthored on the American medical care and insurance industry, ‘The Primal Prescription’, is best book on the topic I have read. I’ve never had the pleasure of meeting him in person, but from what I have seen he seems like a thoughtful considerate person.

All that is to say, this post isn’t intended as a “dunk” on Bob, mostly because I don’t have the ability to do that, but also because Bob does understand conditionally convergent series very well. I’m not claiming that he is out of his depth or the material is just too complex for him or anything like that.

Bob appears to have one slight misconception about infinity and infinite sets that has lead him to a false conclusion. This provides a great opportunity to give a detailed walk through of how countable infinity is constructed, both formally and intuitively, and use that to address Bob’s concerns.


What Are Conditionally Convergent Series?

Conditionally convergent series (abbreviated C.C.S.) are series (sums of infinite sets of values), that contain both negative and positive integers, where depending on how you arrange the the values the series can diverge (the sum tends toward negative or positive infinity) or converge (the sum tends toward) to any value.

If you’re unfamiliar, I strongly recommend listening to the first 30-45min of The Bob Murphy Show Ep. 229. Bob gives a great crash course on the topic.


What Did Bob Say About Conditionally Convergent Series?

On an episode of his podcast, Bob hosted philosopher and author, Steve Patterson. Steve believes that there is wide spread rot in American academia. I think in 2022 that goes without saying, but more interestingly Steve believes that this extends to high level mathematics, and not just the professors, but the content itself. Bob used this opportunity to share an apparent contradiction within Riemann’s Rearrangement Theorem.

I’m oversimplifying a bit, but Bob claims that the ability to “rearrange” the set of values in a C.C.S. and from that yield a different convergence for the series is a paradox. Essentially, Bob asks the question, how can you add up ALL of the same numbers, but in a different order, and get a different result?

The answer is you can’t. But Bob’s question reveals a misunderstanding about the subtler aspects of how infinity and infinite sets are defined, constructed, and used.


Where Did Bob Go Wrong?

I can sum up (pun intended) the exact nature of Bob’s mistake in one sentence: There is no “ALL” the numbers in an infinite set.

Said another way, there is no valid concept, completed (aka actual) infinity, its proponents not withstanding. Everything Bob brought up in regards to conditionally convergent series is a legitimate logical contradiction… IF you accept a completed infinity, a concept introduced through philosophical mathematics. However, using indefinite infinity, which is a valid concept that matches up to the pure mathematics, all of the contradictions raised by Bob are resolved.

Before proceeding, I want to make it clear that I am not claiming Steve or Bob are wrong in the central thrust of the episode. I don’t think there is rot in the actual content of mathematics, but I’m also not willing to hand waive it away like I might have 5 years ago. This post addresses only the specific claims made by Bob about C.C.S. and what that implies about the nature of series and infinite sets.


How to Construct Infinity

Nature of Infinity

NOTE: C.C.S. and this entire post deal entirely with COUNTABLE infinity, which is different from uncountable infinity. Every reference in this post to infinity or to infinite sets is a reference to countable infinity.

To not bury the lead, countable infinity is defined recursively and indefinitely, but before delving deeper into the formal construction I want to suggest an intuitive way to conceptualize this; it will help resolve paradoxes later.

Completed infinity is a logical trap, because it invites you to conceptualize infinity like a fixed space or a container from which you can remove objects. It gives the illusion that there is some static “all” the numbers. A more appropriate conceptualization of infinity is an unlimited, ordered, unique number (i.e. identity) generator.

I realize this turns infinity into a quasi Schrödinger’s cat situation, where the numbers don’t exist until you generate them, but bear with me. As I detail the formal constructions, you will see how this conceptualization arises naturally and how it resolves Bob’s objections.


Infinity is Defined Recursively

Recursion is an iterative process, that can be repeated indefinitely, but not completed. A simple example is the the abbreviation G.N.U., which expands to “G.N.U. is not UNIX” (G = G.N.U., N = is not, and U = UNIX). This recursive abbreviation can be expanded to an indefinitely long sentence.

The myriad of very similar methods available to construct infinity ALL use a specific type of recursion called induction (not to be confused with “inductive reasoning,” which is an unrelated logic term). In math induction simply refers to a recursive process for defining the natural numbers and their properties.


Natural Numbers

The starting point for constructing infinity is the natural numbers: 0, 1, 2, etc. As mentioned above, this involves induction. All inductive math proofs require a seed case and then an inductive statement to begin the recursion from the “seed.”

I will demonstrate an inductive proof using the von Neumann system for integers. The seed case is, “0 is a natural number,” and in the von Neumann system 0 is encoded as the empty set.
\[
\begin{align}
\{\} = 0 && \small(\text{$0$ is a natural number})
\end{align}
\]
The inductive statement is, “the successor of any natural number is a natural number.” In set theory the successor function is the union of a natural number and its singleton (a set containing a single member, which can itself be set).
\[
\begin{align}
S(n) = n+1 = n\cup\{n\} && \small(\text{The successor of natural number $n$ is a natural number})
\end{align}
\]
Any number that can be produced via the inductive process described above is a natural number. See the first few below.

Natural NumberSet EncodingNumber Encoding
\(0\)\(0=\{\}\)\(0=\{\}\)
1 = \(S(0)\)\(1 = \{\}\cup\{\{\}\} = \{\{\}\}\)\(\begin{align}
1 &= 0\cup\{0\} \\
&= \{\}\cup\{0\} = \{0\}
\end{align}
\)
2 = \(S(1)\)\(2 = \{\{\}\}\cup\{\{\{\}\}\} = \{\{\},\{\{\}\}\}\)\(\begin{align}
2 &= 1\cup\{1\} \\
&= \{0\}\cup\{1\} = \{0,1\}
\end{align}
\)
3 = \(S(2)\)\(3 = \{\{\},\{\{\}\}\}\cup\{\{\{\},\{\{\}\}\}\} = \{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}\)\(\begin{align}
3 &= 2\cup\{2\} \\
&= \{0,1\}\cup\{2\} = \{0,1,2\}
\end{align}
\)

What’s interesting about the von Neumann construction is it relates numbers directly to sets. Every natural number is equal to the set of all natural numbers less than itself. Infinity is therefore an analogue for both the “largest” number and the set containing “all” the numbers (since sets and numbers are essentially the same thing under von Neumann).

Both concepts are illusions; there is no “all” there is no “largest,” but the indefinite construction of the natural numbers implicitly makes infinity a synthetic analogue for both.


The Set \(\mathbb{N}\)

**NOTE**: It is enough to understand how the natural numbers are defined to understand the concept of indefinite infinity. So feel free to skip to the next section. However, if you are interested in the formal construction of \(\mathbb{N}\), keep reading.

All inductive constructions of the natural numbers imply the existence of the set \(\mathbb{N}\) (the set containing only the natural numbers), but are not enough to define it formally. There are multiple methods to construct \(\mathbb{N}\); I will use Zermelo–Fraenkel set theory, which is an axiomatic system used to create a (purportedly) paradox free theory of sets.

Axioms are statements, accepted to be true, that cannot be proven by other axioms. You can think of them as to math what elements are to chemistry. The Z.F. axioms used to construct \(\mathbb{N}\) are as follows:

The Axiom of Infinity

\[
\begin{align}
\scriptsize\text{there exists a set $N$} && \scriptsize\text{containing the empty set AND} && \scriptsize\text{for all members of $N$} && \scriptsize\text{the successor member is in $N$} \\ \\
\exists N && (\emptyset \in N \;\land && \forall x\in N\;( && (x\cup\{x\})\in N))
\end{align}
\]
Plain English: There is an infinite set; it contains 0 as well as the successor of any natural number in the set.

The Axiom Schema of Specification

**NOTE**: An axiom schema simply means an undefined formula is part of the axiom therefore the axiom can define many sets.

\[
\begin{align}
{\scriptsize\text{for all sets $A$ and}\\ \scriptsize\text{variables $w_1,…,w_k$}\\ \scriptsize\text{there exists set B}} && {\scriptsize\text{for all $x$, $x$ is member}\\ \scriptsize\text{of $B$ ONLY IF}} && {\scriptsize\text{$x$ is member}\\ \scriptsize\text{of $A$ AND}} && {\scriptsize\text{$x$ satisfies formula $\phi$}\\ \scriptsize\text{w/o $B$ as a “free variable”}}\\ \\
\forall w_1,…,w_k\forall A \exists B && \forall x\;(x \in B \iff && (x\in A \;\land && \phi(x,w_1,…,w_k,A)))
\end{align}
\]
Plain English: For any set \(A\) there exists a subset \(B\) where the members of \(B\) are members of \(A\) and satisfy some formula \(\phi\) whose “free variables” do not include \(B\).

The Axiom of Extensionality

\[
\begin{align}
{\scriptsize\text{for all set pairs}\\ \scriptsize\text{$A$ & $B$}}&& {\scriptsize\text{IF for all}\\ \scriptsize\text{sets $X$}} && {\scriptsize\text{$X$ is member of $A$}\\ \scriptsize\text{ONLY IF}} && {\scriptsize\text{$X$ is member of $B$}} && {\scriptsize\text{THEN $A$ is}\\ \scriptsize\text{equal to $B$}} \\ \\
\forall A \forall B\;( && \forall X\; ( && X\in A \iff && X\in B) && \Longrightarrow A = B)
\end{align}
\]
Plain English: If two sets have exactly the same catalogue of members, they are the same set.

Constructing \(\mathbb{N}\) from the Zermelo-Fraenkel Axioms

The axiom of infinity defines an infinite set using the same inductive process that defines the natural numbers. It does not say that there is ONLY one infinite set nor that the set contains ONLY natural numbers. You might be rolling your eyes at this point, but if you consider that a set containing -1 and the natural numbers would fulfill every statement in the A.O.I., you’d see that we haven’t yet defined \(\mathbb{N}\).

The axiom schema of specification defines a subset built from a superset and a formula. Defining a formula that holds true for natural numbers allows it to whittle down any inductive set (any set that satisfies the A.O.I.) to only the natural numbers.

The A.O.I. never guaranteed there was only one set containing the natural numbers. Let’s assume there are exactly two such sets. Again, the A.S.O.S. can be used to extract the natural numbers from both. So which subset is \(\mathbb{N}\)? According to the axiom of extensionality they both are. They have only the same values, therefore they are the same set.

To review, the axiom of infinity defines an infinite set, the axiom schema of specification allows for a subset of any infinite set based on a formula that holds true only for natural numbers, and the axiom of extensionality says any such subset is \(\mathbb{N}\).

What I’ve presented here is a very bare bones construction of \(\mathbb{N}\). For a more detailed overview, see these lecture notes from a Prof. Bartlett. I like what he has done, because in addition to a more detailed version of the Z.F. construction he also provides an example of a paradox that arises from taking short cuts (i.e. not following the seemingly unnecessary steps of the Z.F. axioms).


Common “Paradoxes”

How does indefinite infinity resolve paradoxes introduced by completed infinity? By rejecting the concept that “all” or quantity having any meaning in relation to infinity. There is no “all.” There is no quantity. There is only a process to create more numbers directly following other numbers.

The set \(\mathbb{N}\) can be used to construct other countable infinite sets. One method is to simply map one set to another via a bijective function (this is not a formal set theory construction, but is not incorrect and works well for illustration). Below \(\mathbb{N}\) is mapped to the set of all integers (\(\mathbb{Z}\)) and also to only even natural numbers.

\(n\in\mathbb{N}\)\(f(n) = \lceil \frac{n}{2}\rceil({-1})^{n+1}\)\(g(n)=2n\)
000
112
2-14
326
4-28

Under completed infinity we have run into a paradox. How can the set of only even natural numbers have a 1 to 1 correspondence with the set of “all” natural numbers? Aren’t there half as many even numbers? NO! There is no “all” the numbers. There is no quantity to be halved in the first place. As many natural numbers as can be created an equal number of even natural numbers can be created, because there is no upper bound to infinity.

This works in the other direction as well. A common construction for \(\mathbb{Z}\) is the union of \(\mathbb{N}\) and its negative.
\[
\begin{align}
\mathbb{Z} = \mathbb{N}\cup\mathbb{N}^{-} && \small (\text{integers are the union of natural numbers and their negative})
\end{align}
\]
So \(\mathbb{N}\), being a subset of \(\mathbb{Z}\), has fewer members than \(\mathbb{Z}\), right? WRONG! Look at the chart above. \(\mathbb{N}\) and \(\mathbb{Z}\) share a 1 to 1 relationship. Also again, there is no quantity to even make the comparison of one being “fewer” and the other “greater.” When people loosely use the phrase “larger” (or “smaller”) infinity they are either referring to countable vs uncountable infinity or simply an expansion of the class of values that can be generated in a countable infinity (positives, negatives, rational numbers, etc.).


Summary

To be sure that my point is not getting lost in the barrage of formal constructions above, or if you just don’t want to read the formal constructions above (understandable), I think it’s useful for me summarize here.

Infinity is NOT a completed static object. The best way to make the formal set theory constructions of infinity more intuitive is to imagine infinity as an unlimited, ordered, identity generator. This framing flows naturally from the formal constructions, which are all based on iterative recursion. Since the generators are iterative and unlimited, you can configure one to spit out more classes of identities (e.g. negative v.s. positive numbers), but not more or fewer quantity of identities than any other infinite machine.

In functional terms this implies…

  1. You cannot have a countable infinity that is larger than another countable infinity (in the sense of quantity of members)

  2. You cannot remove a subset (even an infinite subset) of an infinity to produce something that is smaller than infinity

  3. You cannot combine two countable infinities to get a larger infinity (e.g. \(\mathbb{N}\) has 1 to 1 correspondence with \(\mathbb{Z}\))

  4. Every action described in the first 3 points (expanding, combining, subdividing infinities) can be framed as “re-configuring” the infinite generator with no issues, but any framing involving fixed spaces, sets, or any other static object produces paradoxes.


Rearranging a Series Is a Trick

Rearranging a series to affect the convergence value is mathematical sleight of hand. It looks mind bending at first glance, but if you know how infinite sets are built and you pick apart the mechanics of “rearrangement,” you can see the trick.


How to Rearrange a Sequence

A series is a sum evaluated over an infinite sequence. Rearranging the series really means rearranging the sequence underneath it.

Sequences are like sets, they have members, but they allow order and repeat membership. This is accomplished by mapping an index set (usually the natural numbers) to a range set. The index set provides the order and when two or more index members are mapped to the same range member, repeat sequence membership occurs.
\[
\begin{align}
\stackrel{\normalsize\text{Index}}{\begin{bmatrix} 0 \\ 1 \\ 2 \\ 3\end{bmatrix}} && \stackrel{\normalsize\text{Sequence}}{\begin{bmatrix} 0 & C \\ 1 & B \\ 2 & A \\ 3 & A \end{bmatrix}} && \stackrel{\normalsize\text{Set}}{\begin{bmatrix} A \\ B \\ C \end{bmatrix}}
\end{align}
\]
Rearranging a sequence, and this is crucial to understand, means changing the index assignments such that the total number of appearances for any member does not change (i.e. if a member appears once, twice, or infinitely in the the original arrangement, it must maintain those appearances in the rearrangement).
\[
\begin{align}
\stackrel{\normalsize\text{Index}}{\begin{bmatrix} 0 \\ 1 \\ 2 \\ 3\end{bmatrix}} && \stackrel{\normalsize\text{Sequence}}{\begin{bmatrix} 0 & C \\ 1 & B \\ 2 & A \\ 3 & A \end{bmatrix}} && \stackrel{\normalsize\text{Rearranged Seq}}{\begin{bmatrix} 0 & A \\ 1 & B \\ 2 & C \\ 3 & A \end{bmatrix}} && \stackrel{\normalsize\text{Set}}{\begin{bmatrix} A \\ B \\ C \end{bmatrix}}
\end{align}
\]
Infinite sequence are best conceptualized the same way as infinite sets, with the two exceptions: the number generator doesn’t have to produce only unique values (though almost any C.C.S. will only have unique values) and the order is explicitly, not implicitly defined.


The Trick

Bob stated (paraphrasing), you can’t add up the same numbers in a different order and get a different result. I fully agree. However, Bob concluded that because every value in the original must appear in the rearranged sequence, that rearranging a series involves “all” the same values in a different order. And that false assumption is where the trick happens. Just as magic tricks are made possible by audience members’ false assumption that a magician’s hat is empty, the false assumption of “all” the numbers makes Reimann’s Rearrangement Theorem look fantastical.

For demonstration purposes we can represent rearrangement by mapping an index to a sequence and its rearrangement via functions. See the example below. The right hand sequence will generate positive numbers at \(1.\overline{3}\) times the rate of the left hand. The left hand will generate negative numbers at \(1.5\) times the rate of the right hand.

\(n\in\mathbb{N}\)\(f(n)=\frac{(-1)^{n-1}}{\lceil (n+1) / 2 \rceil}\)\(g(n)=\frac{\lceil \frac{(n+1)\text{ mod }3}{3}\rceil}{(n+1)\;-\;\lfloor (n+1)/3\rfloor}\;+\;\frac{-\lfloor \frac{3\;-\;(n+1)\text{ mod }3}{3}\rfloor}{\lceil (n+1)/3\rceil}\)
011
1-11/2
21/2-1
3-1/21/3
41/31/4
5-1/3-1/2
61/41/5
7-1/41/6

Where is the paradox here? Series built on these two sequences would converge on different values based on the different rates of positive and negative number generation. While it’s true that any value generated by one arrangement will eventually be generated by the other that isn’t the whole story.

Every value after the first will appear at a different index position between sequences. The values generated between sequences prior to any given index position will be very different and there is an inexhaustible supply of negative and positive numbers. That last point means that unlike with a finite set if you begin placing positive numbers in the sequence at an increased rate, you will never “run out” and be forced to only place negatives thereafter.

So while you can change the order of a finite sequence to temporarily change the summation, eventually the sum will move back to its ultimate value. There is nothing temporary about the reconfiguration of the an infinite number generator. Would you expect a number generator configured to produce positives to negatives at a ratio of 20,000,000:1 to converge to the same value as a generator with a 1:1 ratio simply because they share the same catalogue of what they can generate?


Conclusion

I admit the title of this post is a little misleading. Bob is both wrong and right about conditionally convergent series. What he outlined is a very real contradiction and problem for proponents of completed infinity (and there are proponents of that concept). Where Bob seems to have gone wrong is he failed to understand the implicit assumption he was using and also that there was an alternative assumption (a more common and functionally correct one) that resolves the contradictions.

Because infinity is defined recursively and iteratively it is not a completed object. Even without trudging through the formal constructions, conceptualizing infinity as an ordered number generator can help avoid the pitfalls of treating it like a finite or static object.

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