``...Have you a reference re trying to prove a negative is a fallacy...''
Cheers, Ken Hanly
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Begin with the proposition that there exists a set X that contains all numbers not equal to 7.
We know among the cardinals that at least one set does contain elements that equal 7, namely the cardinal set 7. The set of all numbers not equal to 7 then is at least X = C - 1, where C is the set of all cardinals. But, a finite set subtracted from an infinite set has an infinite set as a remainder, C - 1 = C, so X = C. Therefore there are an infinite collection of numbers in the set X, that do not equal 7.
The problem in this case is that the proof that the negative proposition is true, requires an unexhaustible enumeration to demonstrate that truth. However the converse, does not.
Begin with the proposition that there exists some set of cardinals that are equal to the cardinal set 7. There maybe no such sets or many such sets, but only the first demonstration is necessary and sufficient to prove the proposition is true. For example the additive partitions of 7 all equal 7, 1 + 6 = 7, so the partition {1, 6} is such a set.
Chuck Grimes