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In my lecture notes it explains the property of $(a,b) = (c,d) \iff a =c \wedge b =d $. It does so by evaluating $ (a,b)$ as $ \{a, \{a,b\}\} $ but I don't follow how this is done. How come $a$ needs only one element, and $b$ needs two? Would a third element, say $c$, need 3, such that $(a,b,c) = \{a,\{a,b\},\{a,b,c\}\}$?


I've tried to formalise my ideas as such;

$(a_1...,a_n) = \mathscr{C} := \{A_1, ..., A_n\} $

Taking $n \in \mathbb{N}$

$A_n \subsetneq \mathscr{C} $ where $A_{n} := \{\alpha_k \in A_n| k \in \mathbb{N} \leq n\} \,\ $

[N.B $|A_{n}| = n \iff a_n $]



In English;

An ordered tuple is represented by a collection of sets, e.g. C.

Each set is a proper subset of C.

Each set contains all the elements of the previous sets, plus a unique new element, which corresponds to the current ordered component.

e.g. $a_4 = A_3\cup\{\alpha_4\} $

The number of elements in a set corresponds to the place in the order tuple.

physicalcog
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  • By "standard set notation," I take it you mean ZFC notation. AFIAK, apart from set theorists, the ${a,{a,b}}$ notation for ordered pairs is not widely used (or understood?) by mathematicians. Another bizarre artifact of ZFC theory. – Dan Christensen Nov 03 '14 at 15:49

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Note that $(a,b,c)=((a,b),c)=\{(a,b),\{(a,b),c\}\}=\{\{a, \{a,b\}\},\{\{a, \{a,b\}\},c\}\}$

In general the idea is proceed recursively: $$a_0=(a_0)\\\\(a_0,a_1,a_2,a_3,\cdots,a_{n-1},a_n)=((a_0,a_1,a_2,a_3,\cdots,a_{n-1}),a_n)$$

AsdrubalBeltran
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