inhabited set
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inhabited set
A set is called inhabited, if there exists an element . Note that in classical mathematics this is equivalent to (i.e. being nonempty), yet in intuitionistic mathematics we actually have to find an element . For example the set, which contains if Goldbach’s conjecture is true and if it is false is certainly nonempty, yet by today’s state of knowledge we cannot say if is inhabited, since we do not know an element of .
Type of Math Object:
Definition
Major Section:
Reference
Mathematics Subject Classification
03F55 Intuitionistic mathematics
Comments
Formal Definition
Hi,
I appreciate the difference between intuitionistic maths and classical maths, and can see roughly what you mean by an 'inhabited' set. However, I was wondering if you could make the definition a bit clearer. In particular, what are you taking as the definition of nonempty? How formally does this differ from an inhabited set?
Thanks,
Grayum
Re: Formal Definition
I don't know, but I suspect it means that an explicit constructive proof exists of this fact. (That's not a particularly formal definition either.)
Re: Formal Definition
Hi,
I suspect that Mr. mathwizard takes the definition of nonempty in the
classic sense. Furthermore, if one demonstrates that the Goldbach's conjecture turns out to be true or false, the set stops of being inhabited. Although I think that such definition lacks much utility.
Pedro
Re: Formal Definition
By nonempty I mean being not equal to the empty set (i.e. the set with no elements.
The point is that in intuitionistic mathematics a proof of a statement with the existential quantifier has to contain an algorithm by which I can construct an object which satisfies these conditions. Therefore I cannot prove the statement "there is an a in A" in the example by saying "Goldbach's conjecture is either true or false, therefore A cannot be the empty set". This only changes if Goldbach's conjecture turns out to be either true or false.
--
"Do not meddle in the affairs of wizards for they are subtle and quick to anger."
Re: Formal Definition
"I suspect that Mr. mathwizard takes the definition of nonempty in the classic sense"
I was wondering what he took to be the 'classic sense' - his reply suggests not being equal to the empty set.
"Furthermore, if one demonstrates that the Goldbach's conjecture turns out to be true or false, the set stops of being inhabited."
I'm not sure this is the case - the set will be inhabited once Goldbach's conjecture is proved or disproved.
"Although I think that such definition lacks much utility."
I think it is the sort of thing that allows people who are (philosophically) intuitionistically inclined to sleep at night, but doesn't add a great deal more.
Re: Formal Definition
Can you say that either A is inhabited or A is not inhabited?
It is easy to see that constructing a member of A proves that A is inhabited, but how would one show that A is not inhabited?
Can the truth value of Goldbach's conjecture be ascertained in the absence of a valid intuitionistic proof? Does "the truth value of Goldbach's conjecture" have any meaning apart from the existence of such a proof?
> By nonempty I mean being not equal to the empty set (i.e.
> the set with no elements.
> The point is that in intuitionistic mathematics a proof of a
> statement with the existential quantifier has to contain an
> algorithm by which I can construct an object which satisfies
> these conditions. Therefore I cannot prove the statement
> "there is an a in A" in the example by saying "Goldbach's
> conjecture is either true or false, therefore A cannot be
> the empty set". This only changes if Goldbach's conjecture
> turns out to be either true or false.
>
>
>
> --
> "Do not meddle in the affairs of wizards for they are subtle
> and quick to anger."