Originally posted by FabianFnas
Statement (p): Every positive integer has an important property.
Proof by contradiction:
Anti-statement (-p): There is a number that has no important property.
The lowest number of all non-important number must be important being the lowest one. Then there is another non-important number. Apply this number to (-p).
This contradicts the anti-statement (-p).
Therefore the statement (p) holds.
Conclusion: Every number
i do love this, but i was looking forward to contributing a great wikipedia article i read once about mathematical paradoxes and systems of reference... but i can't seem to find it! pretty sure it was in reading about Hilbert/Russel/Godel, etc. if anyone finds it it's a great link.
that being said, the main issue with this proof (i think) is not only in its self-reference - the assumption that the lowest of a group is thereby important - but also in it's CHANGE of system of reference. this is similar to the (in)famous proof that "all numbers are interesting" in that we seek to prove that all numerical
identities have a property that is defined only in language
and has no definable mathematical meaning... and then in order to prove that all numbers share that property, we apply the linguistic notion of "importance" to a numerical
property of an element of a list of numbers. it reminds me of the grammatical error of comparisons in the sentence: "Mick Jagger's voice is the better than all the singers in America." it is intuitively understood by the reader, but is incorrect and logically false to compare a voice to a group of people (instead it should be voice -> voice, or person -> people). similarly, a numerically defined concept, such as "lowest," should not be related to a linguistically defined concept like "important."
i.e. it is inappropriately (though naturally) assumed that the smallest number in a list is also thereby important. this of course speaks to each reader as an intuitive idea.. but if you disregard that assumption the paradox breaks down.