Let L denote the usual formal language used in mathematical literature. If a formula P in L is the definition of a mathematical construction O, then "O" denotes, simply, the corresponding symbol squence. For instance, the formula P={2n| n∈ ℕ} denotes the even integer set; while "P" denotes the symbol sequence
{, 2, n, |, n, ∈, ℕ,}.
Let F be the set of all finite definitions in L of subsets of ℕ, and "F"
the corresponding set of symbol sequences. Finally, let G:"F"→ℕ a Gödel numbering function. Take into account that F is countable because, by definition, every member of "F" is a finite symbol sequence. With these assumptions consider the following definition.
P = { G("X") | (X ∈ F) ∧ (G("X") ∉ X) }
This definition is paradoxical, becuase "P" is a finite symbol sequence, hence P ∈ F. In addition, if G("P") ∉ P, then satisfies the definition of P, and consequently G("P")∈P. Analogously if G("P")∈P, then, by the former definition, G("P") ∉ P. Notice that the former definition is a particular case
of Cantor's method.
What is wrong?: Cantor's method?. Göedel numbering?. The usual formal language?...
Dear Breuer
I can use the notation I like provided that I define it. I have defined the notation and the language as the usual one in mathematical literature.
The use of quotes is usual in programming languages like C, java or Haskell and I think that it is a notation that understands a wide audience. Nevertheless, changing the notation the paradox continues alive. Recall that Gödel numbering is defined even for infinite alphabets. Well, in muy introduction you can change L by any free monoid generated by an infinite alphabet containing every symbol used now, in the past or in the future. It does not matter the used alphabet, the paradox is the same, because Gödel numbering functions are defined for arbitrary large alphabets even infinite ones· Try changing L by the language you prefer in my introductory text, and you can see that the paradox again is alive.
Dear Breuer
I can use the notation I like provided that I define it. I have defined the notation and the language as the usual one in mathematical literature.
The use of quotes is usual in programming languages like C, java or Haskell and I think that it is a notation that understands a wide audience. Nevertheless, changing the notation the paradox continues alive. Recall that Gödel numbering is defined even for infinite alphabets. Well, in muy introduction you can change L by any free monoid generated by an infinite alphabet containing every symbol used now, in the past or in the future. It does not matter the used alphabet, the paradox is the same, because Gödel numbering functions are defined for arbitrary large alphabets even infinite ones· Try changing L by the language you prefer in my introductory text, and you can see that the paradox again is alive.
Dear Breuer
I can use the notation I like provided that I define it. I have defined the notation and the language as the usual one in mathematical literature.
The use of quotes is usual in programming languages like C, java or Haskell and I think that it is a notation that understands a wide audience. Nevertheless, changing the notation the paradox continues alive. Recall that Gödel numbering is defined even for infinite alphabets. Well, in muy introduction you can change L by any free monoid generated by an infinite alphabet containing every symbol used now, in the past or in the future. It does not matter the used alphabet, the paradox is the same, because Gödel numbering functions are defined for arbitrary large alphabets even infinite ones· Try changing L by the language you prefer in my introductory text, and you can see that the paradox again is alive.
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