(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation))
In axiomatic (ZFC) set theory, we define the ordered pair $(a,b)$ to equal $\{\{a\},\{a,b\}\}$. Then, we define the ordered triple in terms of the ordered pair: $(a,b,c):=((a,b),c)$. Similarly, we define the ordered quadruple as $(a,b,c,d)=((a,b,c),d)=(((a,b),c),d)$. In general, one can define the $n$-tuple using this nesting argument.
To define $(x)$, Herbert Enderton's 1977 Elements of Set Theory suggests the convention $(x)=x$. This seems reasonable: note that if $S=\{a,b,c,d\}$, then $(a,b,c) \in S^3$, $(a,b) \in S^2$, you would also expect $(a) \in S$, which would be true if you defined $(a)=a \in S$.
Finally, the "Empty Tuple" or "$0$-Tuple", $()$, is defined as
\begin{equation}
()=\{\}=\emptyset
\end{equation}
This is confirmed by The Wikipedia Tuple page as the correct choice of definition for $()$. Moreover, this agrees with the conclusion we discovered in the previous discussion, in which we decided that $S^0=\{()\}=\{\emptyset\}=1$. For consistency, then, it is pleasing to know that $()=\emptyset$.
My question, however, pertains to the properties of $()$. I have listed some which I believe true:
-
$(())=(\emptyset)=\emptyset$. Thus $(())=()$.
-
The "dissolution property": $((),s_1,\ldots,s_n)=(\emptyset,s_1,\ldots,s_n)=(s_1,\ldots,s_n)$
Any others?
- Although this doesn't have to do specifically with the empty tuple, as a follow-up to the dissolution property, it would be ideal if it were true that $((a,b),c)=(a,(b,c))$ (both suitable definitions for $(a,b,c)$), and possibly also $((a,b,c),d)=(a,(b,c,d))=((a,b),(c,d))$ (which are all suitable definitions for $(a,b,c,d)$), etc.
The reason I want (2) to be true is because certain definitions like the nullary operation require that
\begin{equation}
S^0 \times S^n = \{()\} \times S^n = \{\emptyset\} \times S^n = S^n.
\end{equation}
For this to be true, however, would require that $((),s_1,\ldots,s_n)=(\emptyset,s_1,\ldots,s_n)=(s_1,\ldots,s_n)$, i.e. property (2). However, I can't just "want" or "believe" these properties to be true; I need to prove them within ZFC set theory, using the set-theoretic definition of $n$-tuples. Any help with this?
Thanks for reading my trail of thoughts, and please let me know if you see anything anywhere which is incorrect. Thanks!
Edit 1: Looking at the "Tuples as Nested Ordered Pairs" section of the Wikipedia Tuple Page, I see they define $(a,b,c)$ not as $((a,b),c)$, but instead as $(a,(b,(c,\emptyset)))$. There are two thing strange about this to me: first why the nesting occurs on the right, and second why they choose to pair $c$ with $\emptyset$ instead of just writing it as $(a,(b,c))$. Any insight on this? I wish I could sort this all out. Thanks again!
Edit 2: Chris Culter's answer points out a contradiction which may stem from the definition of $(a)=a$. Perhaps if we define $(x)=(\emptyset,x)=\{\{\emptyset\},\{\emptyset,x\}\}$ then all the desired properties, including the dissolution property, fall into place. So far we haven't been successful in letting $(\emptyset, a) = (a)$, however. But this is the property we need, i.e. we need $((),s_1,\ldots,s_n)=(s_1,\ldots,s_n)$ if we want $S^0 \times S^n = S^n$!
Best Answer
There is a simple solution to the problem. It comes in two parts. And it says that what you're looking for cannot be done, in general.
The first part, is to see the larger picture, that we may want to extend the notion of tuple to infinite index sets, and iterated ordered pairs is not the way to go. Instead we define what an ordered pair is, and what is a function (using ordered pairs), then an $I$-tuple is a function whose domain is $I$. So an $n$-tuple is a function whose domain is $\{0,\ldots,n-1\}$, and in particular $0$-tuple is a function whose domain is the empty set, so a $0$-tuple is indeed $\varnothing$.
The more important part, as long as you define the product of $A\times B$ as the set of ordered pairs (or set of $2$-tuples) such that one coordinate is in $A$ and the other is in $B$, it will rarely be the case that $A^n\times A^m=A^{n+m}$.
This is because one is a set of ordered pairs, and the other $(n+m)$-tuples. If instead you recognize this issue, and decide to resolve it by relaxing the equality in favor of "canonical isomorphisms", then there is a unique way of concatenating $n$-tuples with $m$-tuples to form $(n+m)$-tuples.
We can acknowledge that this is what's going on, or we can decide to reject set theory outright for using equality rather than equivalences. I don't know how this sort of situation is handled in type-theoretic foundations, but I'm guessing that if you define the product of two collections to give you ordered pairs, then it can't be that different, and you'll just resort to an equivalence instead of equality (even if that equivalence is part of the language and so on).