John Edser wrote:

>>>>JE:-
>>>>You simply did not understand the definition you supplied.
>>>>Mathematics is applied logic. An application of logic within
>>>>mathematics produces a set property. Mathematics is therefore,
>>>>limited by the logic being applied to it. Mathematics cannot
>>>>determine which of many possible logical processes are being applied
>>>>within mathematics at any moment. Yes, the intersection has
>>>>"all elements that are in both A and B" but as we have seen
>>>>when set A = 3 and set B = 5 four different self exclusive set
>>>>intersections can exist and not just one, where in every
case it remains
>>>>true that the intersection contains "all elements that
are in both A
>>>>and B" depending entirely on how you define a minimum
of two points
>>>>of commonality.
> 
> 
>>>BOH:-
>>>Huh?  But which one of the intersections actually does exist is
>>>determined by which of the elements IS a member of both sets.
> 
> 
> 
>>>JE:-
>>>Yes, but this has 4 possible outcomes and not just
>>>one outcome given A=3 and B=5. _Each_ of the 4
>>>outcomes _satisfies_ the definition DEPENDING
>>>on "a minimum of two points of commonality".
> 
> 
>>BOH:-
>>
>>>snip<
>>
>>I assume you mean that A has 5 elements and B has 3.
> 
> 
>>JE:-
>>Of course.
> 
> 
>>BOH:-
>>But that is not
>>enough to define the sets A and B.
> 
> 
>>JE:-
>>It is if the set elements are just sets
>>of numbers.
> 
> 
> BOH:-
> Not if you don't define which numbers.
> 
> JE:-
> Yes, you must direct the numbers
> to do something otherwise they just
> sit there and no sets are processed
> (intersected or merged). 

Which numbers must you direct, John?  You still have to say what are the 
elements in A and B, not just how many there are.

>>>>snip<
> 
> 
>>>BOH
>>>Note that with your A's and B's, you have not defined what the elements
>>>are.
> 
> 
> 
>>>JE:-
>>>Mathematics only defines set elements as _numbers_.
> 
> 
>>BOH
>>Rubbish.  Set elements are set elements.
> 
> 
>>JE:-
>>There's a good tautology: " Set elements are set elements".
>>All the set elements within PURE MATHEMATICS are numbers.
> 
> 
> BOH:-
> No they're not.  They are objects, and will be defined to have certain
> properties (e.g. colour, size).  Of course, numbers can be set elements,
> if the sets are so defined.
> 
> JE:-
> You appear to be confusing propositions of mathematics
> such as sets of numbers with propositions that
> are not of mathematics e.g. “objects” that “will be defined
> to have certain  properties (e.g. colour, size)”.
> Mathematics only concerns itself with defined abstract
> concepts and not real world concepts. 

Fine.  We can define "colour" in an abstract way.  The point is that we 
can define elements as having certain properties.

Bob

-- 
Bob O'Hara
Department of Mathematics and Statistics
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
Fax:  +358-9-191 22 779
WWW:  http://www.RNI.Helsinki.FI/~boh/

Journal of Negative Results - EEB: www.jnr-eeb.org
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