I'm looking into nonstandard analysis, and am in a chapter which introduces the whole load of basic terms they'll use.
One of this is a proof for ordered pairs (Kuratowski definition) by induction. The ordered pairs are defined like this:
\begin{equation}
\begin{aligned}
(a)_k :&=\{a\} \\
(a,b)_{k} :&= \{\{a\},\{a,b\}\} \\
(a_1,\,…\,,a_n)_k :&= ((a_1,\,…\,,a_{n-1}),a_n)
\end{aligned}
\end{equation}
The theorem to show is :
$(a_1,\,…\,,a_n) = (b_1,\,…\,,b_n) \Rightarrow a_k = b_k \text{ for k = 1, … , n}$
They do it by induction: Case n = 1 is trivial, and case n = 2 (the part I don't understand) goes like this:
It is $(a_1 , a_2) = (b_1,b_2) $. This is per definition equal to $\{\{a_1\},\{a_1,a_2\}\} = \{\{b_1\},\{b_1,b_2\}\} $.
Now the following cases are possible:
$
\begin{align}
\{a_1\} &= \{b_1\} &\text{and}&\quad\quad \{a_1,a_2\} &= \{b_1,b_2\} ,\\
\{a_1\} &= \{b_1,b_2\} &\text{and}& \quad\quad\{b_1\} &= \{a_1,a_2\}
\end{align}
$
First case seems simple enough, but I don't understand how a set with one element can be equal to a set with two elements. Even worse, they say for both cases follows
$ a_1 = b_1 $ and $ a_2 = b_2$
… but why?
Best Answer
Hint: the set $\{1, 1\}$ does not have two elements.