Re: Axiom of union

Hero wrote:

Han wrote:

Hero wrote:

So with this interpretation of 0-1-strings You can only represent
naturals. What about 2D and it's ordered pairs?

_All_ quantities in a digital computer are represented by bits: strings
as well as naturals as well as floating point numbers, etcetera. It is
not possible to distinguish an integer from a float by looking at their
bit patterns. So yes, ordered pairs are just .. naturals, in my theory.

All the 0-1-strings in computers are build according to CPU's which
interpretes certain parts as commands, other parts as a hint, that
the following byte is a character (letter) and only parts as numbers.

Your coding is a bijection between naturals, expressed in all 0-1-
strings and the sets, build from the empty set.
So, You do not have any 0-1-strings left to represent any thing else.

Correct. Because my set theory is just a theory of sets & nothing else.
I don't want my set theory to become a foundation of whole mathematics.

In my proposal it is, following Kuratowski:
{{ a} , { a , b } } := ( a, b)
translated
11 a 01 a, b 00

Sure, and for a <> b : {{ a } , { a , b } } = { 2^a , 2^a + 2^b } =
2^(2^a) + 2^(2^a + 2^b) = natural. But there are more efficient / dense
representations of ordered pairs in the naturals. My favorite is the one
presented by Daryl McCullough in the thread "Coding of ordered pairs":

http://groups.google.nl/group/sci.math/msg/12773e8474efe750

So You can not differ between ordered pairs of two naturals (2, 1) and
a third natural 80.

Not as far as the (static) bit patterns are concerned, indeed. It's here
where (dynamic) time and programming become involved, for the purpose of
adressing these issues. Everything quite close to computing practice.

Han de Bruijn

.

Relevant Pages

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  • Re: Axiom of union
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  • Re: Axiom of union
    ... What about 2D and it's ordered pairs? ... as well as naturals as well as floating point numbers, etcetera. ...
    (sci.math)
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