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Re: Set of Dictionary

 
 
Konstantin Veretennicov
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      06-16-2005
On 6/16/05, Vibha Tripathi <(E-Mail Removed)> 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.

> 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.


Maybe you can use a list for that:

>>> d1 = {1: 2}
>>> d2 = {3: 4}
>>> s = [d1, d2]
>>> {1: 2} in s

True
>>> {5: 6} in s

False

> Can sets in Python 2.4.1, be reimplemented from
> scratch to not have it work on top of dict?


Sure, why not?

- kv
 
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Steven D'Aprano
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      06-16-2005
On Thu, 16 Jun 2005 21:21:50 +0300, Konstantin Veretennicov wrote:

> On 6/16/05, Vibha Tripathi <(E-Mail Removed)> 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. 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.


> Maybe you can use a list for that:
>
>>>> d1 = {1: 2}
>>>> d2 = {3: 4}
>>>> s = [d1, d2]
>>>> {1: 2} in s

> True
>>>> {5: 6} in s

> False
>
>> Can sets in Python 2.4.1, be reimplemented from
>> scratch to not have it work on top of dict?

>
> Sure, why not?


Take a close look at the sets module, written in Python. You could copy
and modify the source code (taking care to obey whatever licencing
restrictions, if any, there are).



--
Steven

 
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James Dennett
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      06-26-2005
Steven D'Aprano wrote:

> On Thu, 16 Jun 2005 21:21:50 +0300, Konstantin Veretennicov wrote:
>
>
>>On 6/16/05, Vibha Tripathi <(E-Mail Removed)> 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
 
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Brian van den Broek
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      06-26-2005
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 <(E-Mail Removed)> 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


 
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