RE: 1/[0,2]=NaI

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I think there is a logical confusion here. An element x belonfgs to the set of all a for which P(a) is true if and only if P(x) holds.

The word "all" in Sunaga's definition does not mean "for all".

The statement (for all x \in [0,2]) y = 1/x is clearly false for all y, since it means that for all x, including x=1 and x=2, we must have y=1/x. In particular, this means that this must be true for x=1, which leads to y=1, and for x=2, which means y=1/2. Since no number can be equal to 1 and to 1/2, this mean no real number satisfies the above condition.

Ralph Baker Kearfott wrote:
> Nate Hayes wrote:
>> Arnold Neumaier wrote:
>>> Nate Hayes wrote
>>> (in: Re-submission of motion 5: multiple-format arithmetic):
>>>
>>>> Arnold Neumaier wrote:
>>>>> Nate Hayes schrieb:
>>>>>> Arnold Neumaier wrote:
>>>>>>> Nate Hayes schrieb:
> .
> .
> .
>
>> My position on this subject therefore allows you to compute
>> 1/[0,2]=[1/2,Inf) in your range enclosure example, i.e., it allows
>> to avoid NaI in this case because it implies
>>
>>     1/[0,2] = 1/[0,0] \union 1/(0,2]
>>         = {empty} \union [1/2,Inf)
>>         = [1/2,Inf)
>>
>
> According to my understanding, THE ABOVE (taking the union of limiting
> of point values) is the basic idea underlying csets, for what it's
> worth.
>> In the context of predicate logic, though, such an interpretation is
>> not correct. For example, if I seek values of y such that
>>
>>     (for all x \in [0,2]) y = 1/x
>>
>> is true, there is no value of y when x=0 to make the conditional
>> equation true.
>
> But doesn't classical interval arithmetic (which I thought is the
> basic thing we are standardizing) seek
>
>      (the set of all y such that there exists an x \in [0,2] with
> y=1/x) ??
> That is different from what you have above.

I probably should have added they are different but the "for all" predicate
is the classical definition, e.g., Sunaga provides this defintion: "By the
quotient of X to Y, provided that zero does not fall in Y, we mean the
interval consisting of the set Z of all { x/y | x in X, y in Y }."

Nate