# Kelley-Morse set theory does not prove the class Fodor principle

• V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem,” ArXiv e-prints, 2019. (manuscript under review)
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Abstract.
We show that Kelley-Morse (KM) set theory does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\newcommand\Ord{\text{Ord}}\Ord$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\Ord$ that there is a class function $F:\Ord\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a class $A\subseteq\omega\times\Ord$, such that each section $A_n=\{\alpha\mid (n,\alpha)\in A\}$ contains a class club, but $\bigcap_n A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.

The class Fodor principle is the assertion that every regressive class function $F:S\to\Ord$ defined on a stationary class $S$ is constant on a stationary subclass of $S$. This statement can be expressed in the usual second-order language of set theory, and the principle can therefore be sensibly considered in the context of any of the various second-order set-theoretic systems, such as Gödel-Bernays (GBC) set theory or Kelley-Morse (KM) set theory. Just as with the classical Fodor’s lemma in first-order set theory, the class Fodor principle is equivalent, over a weak base theory, to the assertion that the class club filter is normal. We shall investigate the strength of the class Fodor principle and try to find its place within the natural hierarchy of second-order set theories. We shall also define and study weaker versions of the class Fodor principle.

If one tries to prove the class Fodor principle by adapting one of the classical proofs of the first-order Fodor’s lemma, then one inevitably finds oneself needing to appeal to a certain second-order class-choice principle, which goes beyond the axiom of choice and the global choice principle, but which is not available in Kelley-Morse set theory. For example, in one standard proof, we would want for a given $\Ord$-indexed sequence of non-stationary classes to be able to choose for each member of it a class club that it misses. This would be an instance of class-choice, since we seek to choose classes here, rather than sets. The class choice principle $\text{CC}(\Pi^0_1)$, it turns out, is sufficient for us to make these choices, for this principle states that if every ordinal $\alpha$ admits a class $A$ witnessing a $\Pi^0_1$-assertion $\varphi(\alpha,A)$, allowing class parameters, then there is a single class $B\subseteq \Ord\times V$, whose slices $B_\alpha$ witness $\varphi(\alpha,B_\alpha)$; and the property of being a class club avoiding a given class is $\Pi^0_1$ expressible.

Thus, the class Fodor principle, and consequently also the normality of the class club filter, is provable in the relatively weak second-order set theory $\text{GBC}+\text{CC}(\Pi^0_1)$. This theory is known to be weaker in consistency strength than the theory $\text{GBC}+\Pi^1_1$-comprehension, which is itself strictly weaker in consistency strength than KM.

But meanwhile, although the class choice principle is weak in consistency strength, it is not actually provable in KM; indeed, even the weak fragment $\text{CC}(\Pi^0_1)$ is not provable in KM. Those results were proved several years ago by the first two authors, but they can now be seen as consequences of the main result of this article (see corollary 15. In light of that result, however, one should perhaps not have expected to be able to prove the class Fodor principle in KM.

Indeed, it follows similarly from arguments of the third author in his dissertation that if $\kappa$ is an inaccessible cardinal, then there is a forcing extension $V[G]$ with a symmetric submodel $M$ such that $V_\kappa^M=V_\kappa$, which implies that $\mathcal M=(V_\kappa,\in, V^M_{\kappa+1})$ is a model of Kelley-Morse, and in $\mathcal M$, the class Fodor principle fails in a very strong sense.

In this article, adapting the ideas of Karagila to the second-order set-theoretic context and using similar methods as in Gitman and Hamkins’s previous work on KM, we shall prove that every model of KM has an extension in which the class Fodor principle fails in that strong sense: there can be a class function $F:\Ord\to\omega$, which is not constant on any stationary class. In particular, in these models, the class club filter is not $\sigma$-closed: there is a class $B\subseteq\omega\times\Ord$, each of whose vertical slices $B_n$ contains a class club, but $\bigcap B_n$ is empty.

Main Theorem. Kelley-Morse set theory KM, if consistent, does not prove the class Fodor principle. Indeed, if there is a model of KM, then there is a model of KM with a class function $F:\Ord\to \omega$, which is not constant on any stationary class; in this model, therefore, the class club filter is not $\sigma$-closed.

We shall also investigate various weak versions of the class Fodor principle.

Definition.

1. For a cardinal $\kappa$, the class $\kappa$-Fodor principle asserts that every class function $F:S\to\kappa$ defined on a stationary class $S\subseteq\Ord$ is constant on a stationary subclass of $S$.
2. The class ${<}\Ord$-Fodor principle is the assertion that the $\kappa$-class Fodor principle holds for every cardinal $\kappa$.
3. The bounded class Fodor principle asserts that every regressive class function $F:S\to\Ord$ on a stationary class $S\subseteq\Ord$ is bounded on a stationary subclass of $S$.
4. The very weak class Fodor principle asserts that every regressive class function $F:S\to\Ord$ on a stationary class $S\subseteq\Ord$ is constant on an unbounded subclass of $S$.

We shall separate these principles as follows.

Theorem. Suppose KM is consistent.

1. There is a model of KM in which the class Fodor principle fails, but the class ${<}\Ord$-Fodor principle holds.
2. There is a model of KM in which the class $\omega$-Fodor principle fails, but the bounded class Fodor principle holds.
3. There is a model of KM in which the class $\omega$-Fodor principle holds, but the bounded class Fodor principle fails.
4. $\text{GB}^-$ proves the very weak class Fodor principle.

Finally, we show that the class Fodor principle can neither be created nor destroyed by set forcing.

Theorem. The class Fodor principle is invariant by set forcing over models of $\text{GBC}^-$. That is, it holds in an extension if and only if it holds in the ground model.

Let us conclude this brief introduction by mentioning the following easy negative instance of the class Fodor principle for certain GBC models. This argument seems to be a part of set-theoretic folklore. Namely, consider an $\omega$-standard model of GBC set theory $M$ having no $V_\kappa^M$ that is a model of ZFC. A minimal transitive model of ZFC, for example, has this property. Inside $M$, let $F(\kappa)$ be the least $n$ such that $V_\kappa^M$ fails to satisfy $\Sigma_n$-collection. This is a definable class function $F:\Ord^M\to\omega$ in $M$, but it cannot be constant on any stationary class in $M$, because by the reflection theorem there is a class club of cardinals $\kappa$ such that $V_\kappa^M$ satisfies $\Sigma_n$-collection.

• V. Gitman, J. D. Hamkins, and A. Karagila, “Kelley-Morse set theory does not prove the class Fodor theorem,” ArXiv e-prints, 2019. (manuscript under review)
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# Kelley-Morse set theory does not prove the class Fodor Principle, CUNY Set Theory Seminar, March, 2019

This will be talk for the CUNY Set Theory seminar, Friday, March 22, 2019, 10 am in room 6417 at the CUNY Graduate Center.
Abstract. I shall discuss recent joint work with Victoria Gitman and Asaf Karagila, in which we proved that Kelley-Morse set theory (which includes the global choice principle) does not prove the class Fodor principle, the assertion that every regressive class function $F:S\to\text{Ord}$ defined on a stationary class $S$ is constant on a stationary subclass. Indeed, it is relatively consistent with KM for any infinite $\lambda$ with $\omega\leq\lambda\leq\text{Ord}$ that there is a class function $F:\text{Ord}\to\lambda$ that is not constant on any stationary class. Strikingly, it is consistent with KM that there is a sequence of classes $A_n$, each containing a class club, but the intersection of all $A_n$ is empty. Consequently, it is relatively consistent with KM that the class club filter is not $\sigma$-closed.
I am given to understand that the talk will be streamed live online. I’ll post further details when I have them.

# Must there be numbers we cannot describe or define? Definability in mathematics and the Math Tea argument, Norwich, February 2019

I shall speak for the Pure Mathematics Research Seminar at the University of East Anglia in Norwich on Monday, 25 February, 2019.

Abstract. An old argument, heard perhaps at a good math tea, proceeds: “there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions.” Does it withstand scrutiny? In this talk, I will discuss the phenomenon of pointwise definable structures in mathematics, structures in which every object has a property that only it exhibits. A mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it exhibit different properties. Is there a Leibnizian structure with no definable elements? Must indiscernible elements in a mathematical structure be automorphic images of one another? We shall discuss many elementary yet interesting examples, eventually working up to the proof that every countable model of set theory has a pointwise definable extension, in which every mathematical object is definable.

Lecture notes – Must every number be definable? Norwich Feb 2019

# Forcing as a computational process, Cambridge, Februrary 2019

This will be a talk for Set Theory in the United Kingdom (STUK 1), to be held in the other place, February 16, 2019.

Abstract. We investigate the senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model of set theory $\langle M,\in^M\rangle$, for example, we explain senses in which one may compute $M$-generic filters $G\subset P\in M$ and the corresponding forcing extensions $M[G]$. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory $M$ that lead by the computational process to non-isomorphic forcing extensions $M[G]\not\cong M[G’]$. Indeed, there is no Borel function providing generic filters that is functorial in this sense.

This is joint work with Russell Miller and Kameryn Williams.

# The rearrangement and subseries numbers: how much convergence suffices for absolute convergence? Mathematics Colloquium, University of Münster, January 2019

This will be a talk for the Mathematics Colloquium at the University of Münster, January 10, 2019.

Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The subseries number is defined similarly, as the smallest number of subseries whose convergence suffices to test a series for absolute convergence. The exact values of the rearrangement and subseries numbers turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement and subseries numbers into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on joint work with Andreas Blass, Joerg Brendle, Will Brian, myself, Michael Hardy and Paul Larson.

# An infinitary-logic-free proof of the Barwise end-extension theorem, with new applications, University of Münster, January 2019

This will be a talk for the Logic Oberseminar at the University of Münster, January 11, 2019.

Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of ZF has an end-extension to a model of ZFC + V=L. This theorem is both (i) a technical culmination of Barwise’s pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory. The new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal $\Sigma_1$-definable finite set, a set-theoretic version of Woodin’s universal algorithm.

# A new proof of the Barwise extension theorem, without infinitary logic, CUNY Logic Workshop, December 2018

I’ll be back in New York from Oxford, and this will be a talk for the CUNY Logic Workshop, December 14, 2018.

Abstract. I shall present a new proof, with new applications, of the amazing extension theorem of Barwise (1971), which shows that every countable model of ZF has an end-extension to a model of ZFC + V=L. This theorem is both (i) a technical culmination of Barwise’s pioneering methods in admissible set theory and the admissible cover, but also (ii) one of those rare mathematical results saturated with significance for the philosophy of set theory. The new proof uses only classical methods of descriptive set theory, and makes no mention of infinitary logic. The results are directly connected with recent advances on the universal $\Sigma_1$-definable finite set, a set-theoretic version of Woodin’s universal algorithm.

My lecture notes are available.

# Faculty respondent to paper of Ethan Jerzak on Paradoxical Desires, Oxford Graduate Philosophy Conference, November 2018

The Oxford Graduate Philosophy Conference will be held at the Faculty of Philosophy November 10-11, 2018, with graduate students from all over the world speaking on their papers, with responses and commentary by Oxford faculty.

I shall be the faculty respondent to the delightful paper, “Paradoxical Desires,” by Ethan Jerzak of the University of California at Berkeley, offered under the following abstract.

Ethan Jerzak (UC Berkeley): Paradoxical Desires
I present a paradoxical combination of desires. I show why it’s paradoxical, and consider ways of responding to it. The paradox saddles us with an unappealing disjunction: either we reject the possibility of the case by placing surprising restrictions on what we can desire, or we revise some bit of classical logic. I argue that denying the possibility of the case is unmotivated on any reasonable way of thinking about propositional attitudes. So the best response is a non-classical one, according to which certain desires are neither determinately satisfied nor determinately not satisfied. Thus, theorizing about paradoxical propositional attitudes helps constrain the space of possibilities for adequate solutions to semantic paradoxes more generally.

The conference starts with coffee at 9:00 am.  This session runs 11 am to 1:30 pm on Saturday 10 November in the Lecture Room.

Here are the notes I used for my response.

# Alan Turing, On computable numbers

I have been reading Alan Turing’s paper, On computable numbers, with an application to the entsheidungsproblem, an amazing classic, written by Turing while he was a student in Cambridge. This is the paper in which Turing introduces and defines his Turing machine concept, deriving it from a philosophical analysis of what it is that a human computer is doing when carrying out a computational task.

The paper is an incredible achievement. He accomplishes so much: he defines and explains the machines; he proves that there is a universal Turing machine; he shows that there can be no computable procedure for determining the validities of a sufficiently powerful formal proof system; he shows that the halting problem is not computably decidable; he argues that his machine concept captures our intuitive notion of computability; and he develops the theory of computable real numbers.

What I was extremely surprised to find, however, and what I want to tell you about today, is that despite the title of the article, Turing adopts an incorrect approach to the theory of computable numbers. His central definition is what is now usually regarded as a mistaken way to proceed with this concept.

Let me explain. Turing defines that a computable real number is one whose decimal (or binary) expansion can be enumerated by a finite procedure, by what we now call a Turing machine. You can see this in the very first sentence of his paper, and he elaborates on and confirms this definition in detail later on in the paper.

He subsequently develops the theory of computable functions of computable real numbers, where one considers computable functions defined on these computable numbers. The computable functions are defined not on the reals themselves, however, but on the programs that enumerate the digits of those reals. Thus, for the role they play in Turing’s theory, a computable real number is not actually regarded as a real number as such, but as a program for enumerating the digits of a real number. In other words, to have a computable real number in Turing’s theory is to have a program for enumerating the digits of a real number. And it is this aspect of Turing’s conception of computable real numbers where his approach becomes problematic.

One specific problem with Turing’s approach is that on this account, it turns out that the operations of addition and multiplication for computable real numbers are not computable operations. Of course this is not what we want.

The basic mathematical fact in play is that the digits of a sum of two real numbers $a+b$ is not a continuous function of the digits of $a$ and $b$ separately; in some cases, one cannot say with certainty the initial digits of $a+b$, knowing only finitely many digits, as many as desired, of $a$ and $b$.

To see this, consider the following sum $a+b$
\begin{align*} &0.343434343434\cdots \\ +\quad &0.656565656565\cdots \\[-7pt] &\hskip-.5cm\rule{2in}{.4pt}\\ &0.999999999999\cdots \end{align*}
If you add up the numbers digit-wise, you get $9$ in every place. That much is fine, and of course we should accept either $0.9999\cdots$ or $1.0000\cdots$ as correct answers for $a+b$ in this instance, since those are both legitimate decimal representations of the number $1$.

The problem, I claim, is that we cannot assign the digits of $a+b$ in a way that will depend only on finitely many digits each of $a$ and $b$. The basic problem is that if we inspect only finitely many digits of $a$ and $b$, then we cannot be sure whether that pattern will continue, whether there will eventually be a carry or not, and depending on how the digits proceed, the initial digits of $a+b$ can be affected.

In detail, suppose that we have committed to the idea that the initial digits of $a+b$ are $0.999$, on the basis of sufficiently many digits of $a$ and $b$. Let $a’$ and $b’$ be numbers that agree with $a$ and $b$ on those finite parts of $a$ and $b$, but afterwards have all $7$s. In this case, the sum $a’+b’$ will involve a carry, which will turn all the nines up to that point to $0$, with a leading $1$, making $a’+b’$ strictly great than $1$ and having decimal representation $1.000\cdots00005555\cdots$. Thus, the initial-digits answer $0.999$ would be wrong for $a’+b’$, even though $a’$ and $b’$ agreed with $a$ and $b$ on the sufficiently many digits supposedly justifying the $0.999$ answer. On the other hand, if we had committed ourselves to $1.000$ for $a+b$, on the basis of finite parts of $a$ and $b$ separately, then let $a”$ and $b”$ be all $2$s beyond that finite part, in which case $a”+b”$ is definitely less than $1$, making $1.000$ wrong.

Therefore, there is no algorithm to compute the digits of $a+b$ continuously from the digits of $a$ and $b$ separately. It follows that there can be no computable algorithm for computing the digits of $a+b$, given the programs that compute $a$ and $b$ separately, which is how Turing defines computable functions on the computable reals. (This consequence is a subtly different and stronger claim, but one can prove it using the Kleene recursion theorem. Namely, let $a=.343434\cdots$ and then consider the program to enumerate a number $b$, which will begin with $0.656565$ and keep repeating $65$ until it sees that the addition program has given the initial digits for $a+b$, and at this moment our program for $b$ will either switch to all $7$s or all $2$s in such a way so as to refute the result. The Kleene recursion theorem is used in order to know that indeed there is such a self-referential program enumerating $b$.)

One can make similar examples showing that multiplication and many other very simple functions are not computable, if one insists that a computable number is an algorithm enumerating the digits of the number.

So what is the right definition of computable number? Turing was right that in working with computable real numbers, we want to be working with the programs that compute them, rather than the reals themselves somehow. What is needed is a better way of saying that a given program computes a given real.

The right definition, widely used today, is that we want an algorithm not to compute exactly the digits of the number, but rather, to compute approximations to the number, as close as desired, with a known degree of accuracy. One can define a computable real number as a computable sequence of rational numbers, such that the $n^{th}$ number is within $1/2^n$ of the target number. This is equivalent to being able to compute rational intervals around the target real, of size less than any specified accuracy. And there are many other equivalent ways to do it. With this concept of computable real number, then the operations of addition, multiplication, and so on, all the familiar operations on the real numbers, will be computable.

But let me clear up a confusing point. Although I have claimed that Turing’s original definition of computable real number is incorrect, and I have explained how we usually define this concept today, the mathematical fact is that a real number $x$ has a computable presentation in Turing’s sense (we can compute the digits of $x$) if and only if it has a computable presentation in the contemporary sense (we can compute rational approximations to any specified accuracy). Thus, in terms of which real numbers we are talking about, the two approaches are extensionally the same.

Let me quickly prove this. If a real number $x$ is computable in Turing’s sense, so that we can compute the digits of $x$, then we can obviously compute rational approximations to any desired accuracy, simply by taking sufficiently many digits. And conversely, if a real number $x$ is computable in the contemporary sense, so we can compute rational approximations to any specified accuracy, then either it is itself a rational number, in which case we can certainly compute the digits of $x$, or else it is irrational, in which case for any specified digit place, we can wait until we have a rational approximation forcing it to one side or the other, and thereby come to know this digit. (Note: there are issues of intuitionistic logic occurring here, precisely because we cannot tell from the approximation algorithm itself which case we are in.) Note also that this argument works in any desired base.

So there is something of a philosophical problem here. The issue isn’t that Turing has misidentified particular reals as being computable or non-computable or has somehow got the computable reals wrong extensionally as a subset of the real numbers, since every particular real number has Turing’s kind of representation if and only if it has the approximation kind of representation. Rather, the problem is that because we want to deal with computable real numbers by working with the programs that represent them, Turing’s approach means that we cannot regard addition as a computable function on the computable reals. There is no computable procedure that when given two programs for enumerating the digits of real numbers $a$ and $b$ returns a program for enumerating the digits of the sum $a+b$. But if you use the contemporary rational-approximation representation of computable real number, then you can computably produce a program for the sum, given programs for the input reals. This is the sense in which Turing’s approach is wrong.

# On set-theoretic mereology as a foundation of mathematics, Oxford Phil Math seminar, October 2018

This will be a talk for the Philosophy of Mathematics Seminar in Oxford, October 29, 2018, 4:30-6:30 in the Ryle Room of the Philosopher Centre.

Abstract. In light of the comparative success of membership-based set theory in the foundations of mathematics, since the time of Cantor, Zermelo and Hilbert, it is natural to wonder whether one might find a similar success for set-theoretic mereology, based upon the set-theoretic inclusion relation $\subseteq$ rather than the element-of relation $\in$.  How well does set-theoretic mereological serve as a foundation of mathematics? Can we faithfully interpret the rest of mathematics in terms of the subset relation to the same extent that set theorists have argued (with whatever degree of success) that we may find faithful representations in terms of the membership relation? Basically, can we get by with merely $\subseteq$ in place of $\in$? Ultimately, I shall identify grounds supporting generally negative answers to these questions, concluding that set-theoretic mereology by itself cannot serve adequately as a foundational theory.

This is joint work with Makoto Kikuchi, and the talk is based on our joint articles:

The talk will also mention some related recent work with Ruizhi Yang (Shanghai).

Slides

# Parallels in universality between the universal algorithm and the universal finite set, Oxford Math Logic Seminar, October 2018

This will be a talk for the Logic Seminar in Oxford at the Mathematics Institute in the Andrew Wiles Building on October 9, 2018, at 4:00 pm, with tea at 3:30.

Abstract. The universal algorithm is a Turing machine program $e$ that can in principle enumerate any finite sequence of numbers, if run in the right model of PA, and furthermore, can always enumerate any desired extension of that sequence in a suitable end-extension of that model. The universal finite set is a set-theoretic analogue, a locally verifiable definition that can in principle define any finite set, in the right model of set theory, and can always define any desired finite extension of that set in a suitable top-extension of that model. Recent work has uncovered a $\Sigma_1$-definable version that works with respect to end-extensions. I shall give an account of all three results, which have a parallel form, and describe applications to the model theory of arithmetic and set theory.

Slides

# The rearrangement number: how many rearrangements of a series suffice to validate absolute convergence? Warwick Mathematics Colloquium, October 2018

This will be a talk for the Mathematics Colloquium at the University of Warwick, to be held October 19, 2018, 4:00 pm in Lecture Room B3.02 at the Mathematics Institute. I am given to understand that the talk will be followed by a wine and cheese reception.Abstract. The Riemann rearrangement theorem asserts that a series $\sum_n a_n$ is absolutely convergent if and only if every rearrangement $\sum_n a_{p(n)}$ of it is convergent, and furthermore, any conditionally convergent series can be rearranged so as to converge to any desired extended real value. How many rearrangements $p$ suffice to test for absolute convergence in this way? The rearrangement number, a new cardinal characteristic of the continuum, is the smallest size of a family of permutations, such that whenever the convergence and value of a convergent series is invariant by all these permutations, then it is absolutely convergent. The exact value of the rearrangement number turns out to be independent of the axioms of set theory. In this talk, I shall place the rearrangement number into a discussion of cardinal characteristics of the continuum, including an elementary introduction to the continuum hypothesis and an account of Freiling’s axiom of symmetry.

This talk is based in part on joint work with Andreas Blass, Will Brian, myself, Michael Hardy and Paul Larson.

# The propagation of error in classical geometry constructions

I’d like to discuss the issue of error and error propagation in the constructions of classical geometry. How does error propagate in these constructions? How sensitive are the familiar classical constructions to small errors in the use of the straightedge or compass?

Let me illustrate what I have in mind by considering the classical construction of Apollonius of the perpendicular bisector of a line segment $AB$.  One forms two circles, centered at $A$ and $B$ respectively, each with radius $AB$. These arcs intersect at points $P$ and $Q$, respectively, which form the perpendicular bisector, meeting the original segment at the midpoint $C$.

That is all fine and good, and one can easily prove that indeed $PQ$ is perpendicular to $AB$ and that $C$ is the midpoint of $AB$, as desired.

When carrying out such a construction in practice, however, there will inevitably be some small errors. We do not expect to implement it exactly, with infinite precision, but rather, we expect some small errors in the placement of the compass or straightedge, and perhaps these errors may accumulate and they propagate through the construction. What I would like to discuss is the sensitivity of this construction and the other classical constructions to these small errors.

For example, suppose that we are given points $A$ and $B$. When we seek to construct the circle centered at $A$ with radius $AB$, we place the point of the compass at $A$, and this placement may have some small error deviating from $A$, landing somewhere in the blue circle. Similarly, the writing (or etching) implement end of the compass is placed at $B$, with its own possibly different error, landing the orange circle at $B$. The arc actually resulting will be some arc arising with some such small errors, like this:

We may represent the space of all arcs that could arise in conformance with those error bounds as the blurred orange arc below. This image was created simply by drawing many dozens of such arcs in orange, with various choices for the center and radius within the error circles, and blending the results together.

We carry out the same construction with similar errors for the other arc, centered at $B$ and passing through $A$. These arcs overlap in the darker orange regions above and below, determining the points $P$ and $Q$.

The actual arcs we draw and the corresponding vertical will land somewhere inside these blurred regions, perhaps like this:

Note that in this particular case, the resulting line $PQ$ is noticably non-perpendicular to $AB$, and the resulting point $C$ is noticably not the midpoint. Consider the space of all the bisectors $PQ$ that might arise in conformance with our errors on $A$ and $B$, showing the result as the vertical red shaded region.   The darker red region is the space of possible points $C$ that we might have constructed as the “midpoint” $C$, in conformance with the error estimates.

Given the size of the original error bounds on the points $A$ and $B$, it may be surprising that even such a standard simple construction as this — constructing the perpendicular bisector and midpoint of a segment — appears to have comparatively large error propagation, since the shaded red region $C$ is quite large and includes many points that one would not say are close to being the midpoint.  In this sense, the Apollonius perpendicular bisector construction appears to be sensitive to the errors of compass placement.

Is there a better construction? For example, in terms of improving the accuracy at least of the perpendicularity of $PQ$ to $AB$, it would seem to help to use a much larger circle, which would lower the variation in the resulting “right” angle.  But this is partly because we have so far assumed that compass error arises only with the placement of the points of the compass, and not during the course of actually drawing the arc. But of course, one can imagine that errors arise from a flexing of the compass during use, causing it to deviate from circularity, or from slippage, which might reasonably be expected to cause increasing error with the length or degree of the arc, and so on, and such a model of error might have greater errors with large circles.

One could in principle carry out similar analyses for any geometric construction, and use the corresponding results to compare the sensitivity of various methods for constructing the same object, as well as modeling different sources of error.  The goal might be to mount a precise analysis of all the standard constructions and compare competing constructions for accuracy.

There is a literature of papers doing precisely this, and I will try to post some references later (or please do so in the comments, if you have some good ones).

Another approach to error estimation would be to think of the errors at points $A$ and $B$ as probability distributions, centered at $A$ and $B$ and with a certain variation; and one then gets corresponding distributions for the points $P$ and $Q$, which are not rigid shapes as in my diagrams, but qualitatively similar distributions spread out in that region, and a resulting probability distribution for the point $C$.

Finally, let me mention that one might hope to improve the accuracy of a construction, simply by repeating it and averaging the result, or by some other convergence algorithm. For example, as a first step, we might simply perform the Apollonius bisector construction twice, producing midpoint candidates $C_0$ and $C_1$, and we could proceed simply find the midpoint of $C_0C_1$ as a further presumably more accurate midpoint. Or we could iterate in some other manner and hope to converge to the actual midpoint. For example, we could produce seven midpoint candidates, and take the resulting median point.

# Draw an infinite chessboard in perspective, using straightedge only

I’d like to explain to you how to draw chessboards by hand in perfect perspective, using only a straightedge.  In this post, I’ll explain how to construct chessboards of any size, starting with the size of the basic unit square.

This post follows up on the post I made yesterday about how to draw a chessboard in perspective view, using only a straightedge.  That method was a subdivision method, where one starts with the boundary of the desired board, and then subdivides to make a chessboard. Now, we start with the basic square and build up. This method is actually quite efficient for quickly making very large boards in perspective view.

I want to emphasize that this is something that you can actually do, right now. It’s fun! All you need is a piece of paper, a pencil and a straightedge. I’ll wait right here while you gather your materials. Use a ruler or a chop stick (as I did) or the edge of a notebook or the lid of a box. Sit at your table and draw a huge chessboard in perspective. You can totally do this.

Start with a horizon, having two points at infinity (orange), at left and right, and a third point midway between them (brown), which we will call the diagonal infinity. Also, mark the front corner of your chess board (blue).

Extend the front corner to the points at infinity. And then mark off (red) a point that will be a measure of the grid spacing in the chessboard. This will the be size of the front square.

You can extend that point to infinity at the right. This delimits the first rank of the chessboard.

Next, extend the front corner of the board to the diagonal infinity.

The intersection of that diagonal with the previous line determines a point, which when extended to infinity at the left, produces the first square of the chessboard.

And that line determines a new point on the leading rank edge. Extend that point up to the diagonal infinity, which determines another point on the second rank line.

Extend that line to infinity at the left, which determines another point on the leading rank edge.

Continuing in this way, one can produce as many first rank squares as desired. Go ahead and do that. At each step, you extend up to the diagonal infinity, which determines a new point, which when extended to infinity at the left determines another point, and so on.

If you should now reflect on the current diagram, you may notice that we have actually determined many further points in the grid than we have mentioned — and thanks to my daughter Hypatia for noticing this simplification — for there is a whole triangle of further intersection points between the files and the diagonals.

We can use these points (and we do not need them all) to construct the rest of the board, by drawing out the lines to infinity at the right. Thus, we construct the whole chessboard:

One can construct a perspective chessboard of any size this way, and one can simply continue with the construction and make it larger, if desired.

It will look a little better if you add a point at infinity down below (and do so directly below the diagonal point at infinity, but a good distance down below the board), and extend the board downward one level. The corresponding diagram on yesterday’s post might be helpful.

You can now color the tile pattern, and you’ll have a chessboard in perfect perspective view.

If you keep going, you can make extremely large chessboards. In time, I hope that you will come to learn how to complete an infinite chess board in finite time.