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Topic: Re: Set Theorists use EXTENSIONALITY ..... Logic Programmers UNIFY!
Replies: 4   Last Post: Jun 13, 2013 4:47 AM

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Re: Set Theorists use EXTENSIONALITY ..... Logic Programmers UNIFY!
Posted: Jun 12, 2013 6:20 AM
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Graham Cooper <> writes:

> On Jun 12, 9:57 am, Jan Burse <> wrote:
>> Graham Cooper schrieb:

>> > I thought it was a good time for  blurb!
>> > Elements of Sets occur in groups!
>> > e( A , setname )  :-  predicate(A).
>> > its exactly the topic around sci.logic for the last 20 years!
>> Do you mean the comprehension axiom? Something
>> along the following schema:
>>    forall x exists y forall z(z in y <-> z in x & A(z))
>> Abbreviated as:
>>    forall x exists y (y = {z in x | A(z)})
>> Problem is that comprehension doesn't work without
>> negative information. Take the following Horn clauses:
>>     even(n).
>>     forall X (even(s(s(X))) <- even(X)).
>> Then we cannot derive:
>>     T |- s(n) in {z in nat | even(z)} /* not derivable */
>> The reason is that Horn clauses don't prevent the predicate
>> even to have odd extra elements in an interpretation. So
>> that the collection {.|.} varies, and a definite statement
>> about a membership of s(n) is not possible.

> Yes it is.

Not in general, the way you go about it.

What happens in the example you were just asked about?
(rather than in the ones you have already worked out, where
the set of positive solutions is finite).

Alan Smaill

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