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"consider the set of numbers that cannot be described using any finite collection of symbols" - isn't that set empty? If "numbers" mean "real numbers" then I c
by Sankozi 4y ago
"consider the set of numbers that cannot be described using any finite collection of symbols" - isn't that set empty?
If "numbers" mean "real numbers" then I can describe easily "smallest positive number in the set" and "biggest negative number in the set" - so neither negative, 0 nor positive real numbers might be in the set.
- norrius 4y agoWhat is the "smallest positive number" in the set {x | x in reals and 0 < x < 1}?
- tsimionescu 4y agoIt's trivial to prove that the number you are describing doesn't exist: Let x be the smallest positive real greater than 0. Since x > 0, x/10 < x, and x/10 > 0. But now we have a contradiction, since x was supposed to be the smallest positive real number. So, x doesn't exist. Edit: note that this doesn't even require real numbers. Even for the rational numbers, with common definitions of less-than, there is no smallest positive rational number >0, per the same logic.