连续统与Turing度

Continuum Hypothesis and Turing Degrees

令Ｄ为所有Ｔｕｒｉｎｇ度的集合，≤为Ｄ上的图林化归关系．一函数ｆ：Ｄ→Ｄ称为前进函数如果对任何ａ∈Ｄ，ａ≤ｆ（ａ）。对于一个前进函数ｆ，我们说Ｄ中的两个度ａ，ｂ是ｆ－不可比较的，如果ａ≮ｆ（ｂ）且ｂ ≮ｆ（ａ），否则是ｆ－可比较的．本文的一个主要结果是：在ＺＦＣ中连续统假设成立当且仅当存在一个前进函数ｆ：Ｄ→Ｄ使得Ｄ中任何两个度都是ｆ－可比较的.

Let <D1≤> be the partially ordered set of all Turing degrees. A function j: D - D is said to be progressing if a ≤f(a) for all a ∈ D. For such a function f,we call a and b in D j-incomparable if a ≮ f(b) and b ≮ f(a), and f-comparable otherwise. The central result of the paper is that in ZFC the continuum Hypothesis 1 = 20 holds if and only if there exists a progressing function f: D → D such that any pair a and b in D is f-comparable.