Let f1:ℕ⭢ℕ be the identity, that is, ∀n∈ℕ: f1(n)=n. For each n in ℕ, let fn:ℕ⭢ℕ be the map defined recursively as follows.
fn(m) = fn-1(m+1) if fn-1(m) = 2
fn(m+1) = fn-1(m) = 2 if fn-1(m) = 2
fn(m) = fn-1(m) otherwise.
Since every map fn is defined recursively with a transposition in the image
of fn-1, the map fn is again a bijection that satisfies the following properties.
1) ∀n ≤ m: fm(n)≠ 2.
2) ∀n ≤ m the map fn:ℕ⭢ℕ is a bijection.
If the recursion process never ends, both properties are compatible.
However, assuming the actual infinity existence, every infinite process can be completed. Under this assumption the recursive proces give rise to the following properties for m =∞.
1) ∀n∈ℕ: f∞(n)≠ 2.
2) The map f∞:ℕ⭢ℕ is a bijection.
which is a contradiction, unless we assume that every inifinity is potential, and
n keeps always finite.
Taneli Huuskonen
Please, can you point out the mistake and prove it? Otherwise somebody can think that you do not understand the paradox.
The result of completing the infinite process you describe is an infinite sequence of bijections from N to N, not a single bijection with impossible properties.
Taneli Huuskonen
I think that your answer is true in the scope of potential infinity, but my question requires the actual infinity axiom. Recall that, according to actual infinity concept, every process can be ended. Thus, actual infinity is not synonimous of endless. To avoid any wrong interpretation of my words, I write both axioms below.
Axiom 1. Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.
Henri Pointcaré (1854-1912).
Axiom 2. Every mathematical construction can form an actual and completed totality.
G. Cantor (1845-1918)
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