The set of all finite sequences of members of a countable set is also countable

Question

While I was reading Enderton's "A mathematical introduction to Logic", I came across the proof of the following sentence: "The set of all finite sequences of members of the countable set A is also countable".

Proof: The set S of all such finite sequences can be characterized by the equation $$S=\bigcup_{n \in N} A^{n+1}$$ Since A is countable, we have a function f mapping A one-to-one into N. The basic idea is to map S one-to-one into N by assigning to $(a_0,a_1,...,a_m)$ the number $2^{f(a_0)+1}3^{f(a_1)+1}\cdot ... \cdot p_m^{f(a_m)+1}$, where $p_m$ is the $(m+1)$st prime. This suffers from the defect that this assignment might not be well-defined. For conceivably there could be $(a_0,a_1,...,a_m)=(b_0,b_1,...,b_n)$, with $a_i$ and $b_j$ in A but with $m\neq n$. But this is not serious; just assign to each member of S the smallest number obtainable in the above fashion. This gives us a well-defined map; it is easy to see that it is one-to-one.

Note: P is a finite sequence of members of A iff for some positive integer $n$, we have $P=(x_1,...,x_n)$, where each $x_i \in A$.

First of all, I cannot understand why the former assignment might not be well-defined and the latter assignment is well-defined. Secondly, I cannot understand what Enderton means by "just assign to each member of S the smallest number obtainable in the above fashion". By the way, is $(a,b,c,d) = ((a,b),(c,d))$ true? Also, in which cases can I omit/add parentheses in a tuple so as to have an equal tuple?

Answer 1

So maybe it is a good idea to move part of the comments here, since they are relevant to answering the question:

First of all, there is no unique definition of $(a,b,c,d)$, but the standard one is $(a,(b,(c,d)))$. My guess is that depending on the definition it may be impossible that a problem can arise, but using the definition I provided it is possible to have a conflict.

The problem arises in the case some tuple of elements of $A$ is also in $A$. Let for example let's assume that $a,b\in A$ but at the same time $(a,b)\in A$. Furthermore let's assume that $f(a)=1,f(b)=2,f((a,b))=3$. Then the triple $(a,a,b)$ by the standard definition is $(a,(a,b))$. Then it is not clear whether to send $(a,(a,b))$ to $2^2\cdot3^4$ or to $2^2\cdot 3^2\cdot 5^3$.

What Enderton suggests is to pick the least $n$ such that the sequence $P$ lies in $A^n$. Hence in our previous example we should choose to denote $(a,(a,b))$ with $2^2\cdot 3^4$ since this element is in $A^2$ and in $A^3$ but $2<3$.

Answer 2

I have already given this answer elsewhere. Let me repeat this here.

To prove the countability of $\bigcup _{n=1}^\infty A^n$, it suffices give an injective function $\phi$ from this union to $A$. Without loss of generalty take $A$ to be the set of positive integers and for the sake of this proof regard them as written out in base 10, which makes use of the symbols $0, 1,2$ upto $9$. Now let us bring in 3 more symbols namely the comma, the open and then the closed paraentheses which are to be interpreted as the (additional) symbols for the numbers ten, eleven and twelve respectively in base thriteen.

Now the injection $\phi$ takes a typical element such as $(3,8,17)$ and interprets them as an expression base thirteen: $(3,8,17)_{\rm thirteen}$ which when expressed in base ten is $(11\times 13^7) + (3\times 13^6) + (10\times 13^5) + (8\times 13^4) +(10\times 13^3) + (1\times 13^2) + (7\times 13^1) + (12\times 1).$

Answer 3

You don't define what $f$ is, but as long as it is injective and positive I don't see how you get a conflict. In particular, if $n \gt m$, one of the numbers will be divisible by $p_{m+1}$ and one will not. It looks simpler to me to take $(a_0,a_1,\ldots,a_m)$ to $2^{a_0}3^{a_1}\ldots p_m^{a_m}$. Unique factorization shows that each tuple goes to a distinct image.

For your last we haven't defined tuples of tuples, which is what $((a,b),(c,d))$ is. If we map it recursively using my function $(a,b)\to 2^a3^b$, so $(a,b,c,d) \to 2^a3^b5^c7^d$ while $((a,b),(c,d)) \to 2^{2^a3^b}3^{2^c3^d}$, which are not equal.

Answer 4

Just started that book and I'd like to add a little bit to @Apostolos answer (which actually helped me to understand the issue).

The issue of how the fact that two tuples are equal corresponds to the equality of their numbers of elements, and the elements itself is given in the book just before the Theorem in question (Lemma 0A) (Enderton's treatment of n-tuples is similar to @Apostolos' one with the only difference in the associativity, Enderton uses "left-associative" recursive definition of n-tuples).

I think that the source of the confusion here is that Enderton haven't quite focused reader's attention (in my opinion) on the issue that we face: to know how to make a choice of (one of the finite number of possible ways) how an element of $S$ could be constructed as a sequence of the elements of $A$.

The point is that we are just interested in the elements of $S$. So we aren't interested in enumerating all the ways of how a particular $s \in S$ can be constructed as a sequence of the elements of $A$.

Instead we start from an arbitrary $s \in S$, and then show how we can pick up a single chain from $A^n$, thus a single number from $\mathbb{N}$.