[offtopic] Re: Set of Dictionary

[Brian van den Broek]

James Dennett said unto the world upon 26/06/2005 03:51:
> Steven D'Aprano wrote:
> 
> 
>>On Thu, 16 Jun 2005 21:21:50 +0300, Konstantin Veretennicov wrote:
>>
>>
>>
>>>On 6/16/05, Vibha Tripathi <vibtrip at yahoo.com> wrote:
>>>
>>>
>>>>I need sets as sets in mathematics:
>>>
>>>That's tough. First of all, mathematical sets can be infinite. It's
>>>just too much memory :)
>>>Software implementations can't fully match mathematical abstractions.
>>
>>
>>:-)
>>
>>But lists can be as long as you like, if you have enough memory. 
> 
> 
> But you never have enough memory to store, for example,
> a list of all the prime integers (not using a regular list,
> anyway).


An even better example is the set of reals in the interval (0, 1). 
Even an idealized Turing machine with (countably) infinite memory will 
choke on that :-)

<snip>


>>Standard Set Theory disallows various constructions, otherwise you get
>>paradoxes.
>>
>>For example, Russell's Paradox: the set S of all sets that are not an
>>element of themselves. Then S should be a set. If S is an element of
>>itself, then it belongs in set S. But if it is in set S, then it is an
>>element of itself and it is not an element of S. Contradiction.
>>
>>The price mathematicians pay to avoid paradoxes like that is that some
>>sets do not exist. For instance, there exists no universal set (the set
>>of all sets), no set of all cardinal numbers, etc.
>>
>>So even in mathematics, it is not true that sets can contain anything.
> 
> 
> See "Set Theory With a Universal Set" by T. Forster, which covers
> some set theories in which there *is* a set of all things, and
> in which Russell's paradox is avoided in other ways (such as by
> restricting the comprehension axioms).
> 
> (Sorry for drifting offtopic, I happen to find non-standard
> set theories interesting and thought that some others here
> might too.)
> 
> -- James


So do I :-)

Do you know of non-well-founded set theory (non-standard set theory 
which allows sets A, such that A is in A)? Not really on point for any 
of the above, but being on topic is in the rear view mirror, anyway :-)

<http://cslipublications.stanford.edu/site/1575860082.html>
<http://cslipublications.stanford.edu/site/0937073229.html>

Best,

Brian vdB