Re: Coding of ordered pairs
- From: Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx>
- Date: Mon, 10 Sep 2007 10:58:14 +0200
lwalke3@xxxxxxxxx wrote:
It appears what HdB wants to define the ordered pair of
two natural numbers without relying on ZFC or any other
set theory at all. In other words, he wants to avoid
sets and set notation and instead define the ordered
pair only in the language of PA (with its primitives
zero, successor, and "is a natural number") rather than
the language of ZFC (with its primitive "is an element
of"), in order to show how many results of mathematics]
can be developed without set theory.
Quite close. My toy (bit mapped) set theory gives the following result
for the Kuratowski ordered pair:
{{a},{a,b}} = {2^a,2^a + 2^b} = 2^(2^a) + 2^(2^a + 2^b)
= 2^(2^a).(1 + 2^(2^b))
A very "sparse" result (as with the von Neumann ordinals in bit mapped
set theory). Clearly, much more "dense" representations are possible.
Han de Bruijn
.
- References:
- Coding of ordered pairs
- From: Han de Bruijn
- Re: Coding of ordered pairs
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- Re: Coding of ordered pairs
- From: lwalke3
- Coding of ordered pairs
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