"Counterexamples" in *Constructive Analysis* (Bishop & Bridges) - Trouble with exercises

Question

In working through the exercises for Chapter 2 of Bishop & Bridges' Constructive Analysis, I'm having trouble dealing with the problems regarding "counterexamples". The authors say that for the purposes of exercises, the negation "not" should be understood in the spirit of an earlier remark. In this remark, the authors discuss what they call "fugitive properties" of $\mathbb{N}$, which is to say properties $P$ such that it is not known whether $P(n)$ holds for all $n \in \mathbb{N}$, nor is it known (at the present time) whether there exists $n \in \mathbb{N}$ such that $\neg P(n)$. He gives the example (at the time of writing) of Fermat's last theorem, although we might today replace it with Goldbach's strong conjecture (i.e. $P(n)$ is equivalent to there existing primes $p, q > 0$ such that $2n + 2 = p + q$), or similarly any conjecture of the sort about $\mathbb{N}$ which has not yet been proven or disproven.

The authors go on to say that we can "construct" examples of object which have neither one property nor another by designing them in such a way that if it had the former property, then one of these conjectures would be true, and if it had the latter property, the conjecture would be disproven. The authors provide the following example: Let $P$ be one of these fugitive properties of $\mathbb{N}$, and define a sequence $(a_{n})_{n \in \mathbb{N}} \in \{0, 1\}^{\mathbb{N}}$ be saying $a_{n} = 0$ if $P(n)$ holds, and $a_{n} = 1$ if $\neg P(n)$. So for example, we might set $P(n)$ to be the property that $2n + 2$ can be written as the sum of two positive primes. As far as we know on the early Texas afternoon of January 13th, year 2017 of the Common Era, there does not exist $n$ such that $\neg P(n)$, and for every $n$ tested we have $P(n)$, but it has not yet been proven that $P(n)$ holds for every $n$, nor that there exists $n$ for which $P(n)$.

Now, let $x = (x_{k})_{k \in \mathbb{N}}$ be a real number, where for given $k \in \mathbb{N}$, we write $x_{k} = 0$ if $a_{1} = a_{2} = \cdots = a_{k} = 0$, and if $\exists j \leq k$ such that $a_{j} = 1$, we write $x_{k} = 2^{-m}$, where $m = \min \{ j \leq k : a_{j} = 1 \}$. Per the book's definitions, clearly $x$ is nonnegative ($x_{k} \geq 0 \geq k^{-1}$), but it's not known whether $x = 0$ or $x > 0$, as in the former case we'd have proven the Goldbach conjecture, and in the latter case we'd have disproven it. So we say there exists an $x$ which is nonnegative, but neither positive nor zero, contenting ourselves with the hope that if someone tomorrow proves the Goldbach conjecture, then we can amend our construction by considering a different fugitive property, say one defined by the Collatz conjecture or the twin prime conjecture or something of the type.

My trouble is that (unless I'm missing some) that's the only example the book gives of such a statement in the chapter, but there are many problems in the chapter that ask for such constructions. As such, I have trouble knowing where to start with them, and furthermore am concerned that those I think I'm solving, I may be doing wrong, which I was hoping you might be able to help with. As best I can tell, my goal in these constructions is to describe an object that is wed to one of these fugitive properties in such a way that to demonstrate the object has some particular property would be tantamount to proving or disproving a conjecture of the form $(\forall n \in \mathbb{N})(P(n))$.

For instance, the first problem says to construct a set $A$ for which $(\forall x, y \in A)(x = y)$, but $A$ is neither subfinite nor void. So I want to describe $A$ in such a way that proving it's subfinite would mean proving our conjecture, and proving it's void would mean disproving the conjecture. So I say let $P$ be a fugitive property, and set $A = \left\{ \left( s_{k} / k! \right)_{k \in \mathbb{N}} \right\} \setminus \{ (0, 0, 0, 0, \ldots) \}$, where $s_{k} = 0$ if $P(k)$ and $s_{k} = 1$ if $\neg P(k)$. Evidently $(s_{k} / k!)_{k} = (0, 0, \ldots)$ in the sense of real numbers being equal, as $|0 - s_{k} / k!| \leq 1 / k! \leq 1 / k$. If the set $A$ is void, then $s_k = 0$ for all $k$, and if $A$ is subfinite, then $s_k = 1$ for some $k$. Is this correct?

As for the rest of the problems, I have no sense for how to go about them. Are there other sources which cover this topic, but give more examples of these counterexample constructions? I feel like I'd get more out of just looking at the worked solutions for the chater than working on these problems more, because I just don't have any intuition for them.

Answer 1

I don't have a copy of Bishop's book at hand - are there any other restrictions on the set $A$? Also, if I recall correctly, being subfinite simply means being a subset of a finite set.

If that is correct, and there are no other restrictions, perhaps we could make $A$ be a set of natural numbers which is empty if Goldbach's conjecture is true, and if Goldbach's conjecture is false contains only the least natural number $n$ such that $2n+2$ is not the sum of two primes. In other words, $n \in A$ if and only if $n \in \mathbb{N}$, and $2n+2$ is not the sum of two primes, and $2i+2$ is the sum of two primes for all $i < n$.

For this set, we can decide for each natural $m$ whether $m$ is in the set, by checking all primes less than $2m+2$. If we knew the set was empty, we would know that the conjecture if true. If we knew that the set was a subset of $\{0, \ldots, k\}$ for some $k$, we could effectively test whether the conjecture is true by checking whether $2i+2$ is the sum of two primes for $i \leq k$ -- so if we knew such a $k$ then we would also know whether the conjecture is true. In any case, we do not know a $k$ so that the conjecture is true if and only if it is true up to $k$, so we cannot assert that the set is subfinite.

There are other things we cannot assert about $A$. We can prove that it is impossible to have distinct $n,m \in A$. But we cannot prove that $|A| = 0 \lor |A| = 1$, using a similar argument as above.

In my mind, the key technique for these counterexamples is to leverage the quantifiers. In this case, the quantifier that is most useful is the one in the definition of subfinite. We have an infinite number of chances to make the set subfinite, by looking at values of $k$ one at a time.

Another viewpoint that may be helpful is to avoid thinking of these objects classically, as single pre-existing objects with well-determined properties. Instead, think of them in terms of their definition. We want to form a definition which might actually define several different objects, depending on whether some other statement holds. In many cases, this will mean that determining properties of the object obtained from the definition will also determine facts about the statement behind the definition.

Unfortunately, I don't think there are many "elementary" sources for constructive mathematics that work out numerous examples at length. Also, because there are so many kinds of constructive math, constructions that work in one setting may not work in another. If you do not have an experienced person to talk with, it really can be a difficult area to enter.