[offtopic] Re: Set of Dictionary

[James Dennett]

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).

> So
> can longs and strings. So I don't think the infinity issue is a big one.
> 
> 
>>>   sets of any unique type of objects including those
>>>of dictionaries, I should then be able to do:
>>>a_set.__contains__(a_dictionary) and things like that.
> 
> 
> 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