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Real Numbers in ACA 0 \text {ACA}_0 ACA0
ACA 0 \text {ACA}_0 ACA0 中的实数
Theorem I.6 ACA 0 \text {ACA}_0 ACA0 is strong enough to prove that R \mathbb {R} R is complete and that [ 0 , 1 ] [0, 1] [0,1] is sequentially compact.
定理 1.6 ACA 0 \text {ACA}_0 ACA0 的逻辑强度足以证明实数集 R \mathbb {R} R 是柯西完备的，且闭区间 [ 0 , 1 ] [0, 1] [0,1] 是列紧的。
Proof We will first prove the sequential compactness of [ 0 , 1 ] [0, 1] [0,1]. Let { x n ∈ [ 0 , 1 ] } n ∈ N \{x_{n} \in [0,1]\}{n \in \mathbb {N}} {xn∈[0,1]}n∈N be a sequence of real numbers in [ 0 , 1 ] [0, 1] [0,1]. We must show that there exists a subsequence that converges to a real number x in [ 0 , 1 ] [0, 1] [0,1].
证明 首先证明 [ 0 , 1 ] [0, 1] [0,1] 的列紧性。设 { x n ∈ [ 0 , 1 ] } n ∈ N \{x{n} \in [0,1]\}_{n \in \mathbb {N}} {xn∈[0,1]}n∈N 为闭区间 [ 0 , 1 ] [0, 1] [0,1] 中的一个实数序列，需证明该序列存在收敛子序列，且极限属于 [ 0 , 1 ] [0, 1] [0,1]。
Let 令 ϕ ( k , i ) = ∀ N ∃ n > N ( ( i < 2 k ) ∧ ( i ∗ 2 − k ≤ x n ≤ ( i + 1 ) ∗ 2 − k ) ) \phi (k, i)=\forall N \exists n>N\left (\left (i<2^{k}\right) \land\left (i * 2^{-k} \leq x_{n} \leq (i+1) * 2^{-k}\right)\right) ϕ(k,i)=∀N∃n>N((i<2k)∧(i∗2−k≤xn≤(i+1)∗2−k)) .
ϕ ( k , i ) \phi (k, i) ϕ(k,i) says that infinitely many x n x_{n} xn lie in the interval [ i ∗ 2 − k , ( i + 1 ) ∗ 2 − k ] ⊂ [ 0 , 1 ] [i * 2^{-k},(i+1) * 2^{-k}] \subset [0,1] [i∗2−k,(i+1)∗2−k]⊂[0,1] . Because 0 ≤ x n ≤ 1 0 ≤x_{n} ≤1 0≤xn≤1 for all n , infinitely many x n x_{n} xn must lie in at least one of these intervals. Thus ϕ ( k , i ) \phi (k, i) ϕ(k,i) for at least one i for each k .
公式 ϕ ( k , i ) \phi (k, i) ϕ(k,i) 表示有无限多个 x n x_n xn 落在区间 [ i ⋅ 2 − k , ( i + 1 ) ⋅ 2 − k ] ⊂ [ 0 , 1 ] [i \cdot 2^{-k},(i+1) \cdot 2^{-k}] \subset [0,1] [i⋅2−k,(i+1)⋅2−k]⊂[0,1] 内。由于对所有 n n n 都有 0 ≤ x n ≤ 1 0 ≤x_{n} ≤1 0≤xn≤1，因此必有至少一个这样的区间包含无限多个 x n x_n xn，即对每个 k k k，存在至少一个 i i i 使得 ϕ ( k , i ) \phi (k, i) ϕ(k,i) 成立。
Now let f = { ( k , i ) ∣ ϕ ( k , i ) ∧ ¬ ∃ j ( ( j > i ) ∧ ϕ ( k , j ) ) } f=\{(k, i) | \phi (k, i) \land \neg \exists j ((j>i) \land \phi (k, j))\} f={(k,i)∣ϕ(k,i)∧¬∃j((j>i)∧ϕ(k,j))}, which exists by arithmetical comprehension. f is a well defined function because ( k , i ) ∈ f (k, i) \in f (k,i)∈f exists and is unique for each k by definition.
令 f = { ( k , i ) ∣ ϕ ( k , i ) ∧ ¬ ∃ j ( ( j > i ) ∧ ϕ ( k , j ) ) } f=\{(k, i) | \phi (k, i) \land \neg \exists j ((j>i) \land \phi (k, j))\} f={(k,i)∣ϕ(k,i)∧¬∃j((j>i)∧ϕ(k,j))}，其存在性可由算术概括公理保证。由定义可知，对每个 k k k，存在唯一的 i i i 使得 ( k , i ) ∈ f (k, i) \in f (k,i)∈f，因此 f f f 是良定义函数。
Let x = { x k = f ( k ) ∗ 2 − k } k ∈ N x=\{x_{k}=f (k) * 2^{-k}\}{k \in \mathbb {N}} x={xk=f(k)∗2−k}k∈N . x is a real number because each rational number x k x{k} xk with k > N k>N k>N will lie in [ f ( N ) ∗ 2 − N , ( f ( N ) + 1 ) ∗ 2 − N ] [f (N) * 2^{-N},(f (N)+1) * 2^{-N}] [f(N)∗2−N,(f(N)+1)∗2−N] . 0 ≤ x k ≤ 1 0 ≤x_{k} ≤1 0≤xk≤1 for all k, so x ∈ [ 0 , 1 ] x \in [0,1] x∈[0,1] .
令 x = { x k = f ( k ) ⋅ 2 − k } k ∈ N x=\{x_{k}=f (k) \cdot 2^{-k}\}{k \in \mathbb {N}} x={xk=f(k)⋅2−k}k∈N，当 k > N k>N k>N 时，有理数 x k x_k xk 均落在区间 [ f ( N ) ⋅ 2 − N , ( f ( N ) + 1 ) ⋅ 2 − N ] [f (N) \cdot 2^{-N},(f (N)+1) \cdot 2^{-N}] [f(N)⋅2−N,(f(N)+1)⋅2−N] 内，因此 x x x 为实数；又因对所有 k k k 都有 0 ≤ x k ≤ 1 0 ≤x{k} ≤1 0≤xk≤1，故 x ∈ [ 0 , 1 ] x \in [0,1] x∈[0,1]。
Now define another function g ( k + 1 ) = least n such that ( n > g ( k ) ) ∧ ( ∣ x − x n ∣ ≤ 2 − k ) g (k+1)= \text {least } n \text { such that } (n>g (k)) \land\left (\left|x-x_{n}\right| \leq 2^{-k}\right) g(k+1)=least n such that (n>g(k))∧(∣x−xn∣≤2−k), which exists by Theorem I.1 and because infinitely many x n x_{n} xn lie within 2 − k 2^{-k} 2−k of x for all k by definition.
定义函数 g ( k + 1 ) = g (k+1)= g(k+1)= 满足 ( n > g ( k ) ) ∧ ( ∣ x − x n ∣ ≤ 2 − k ) (n>g (k)) \land\left (\left|x-x_{n}\right| \leq 2^{-k}\right) (n>g(k))∧(∣x−xn∣≤2−k) 的最小自然数 n n n，由定理 1.1 及定义可知，对所有 k k k，有无限多个 x n x_n xn 落在 x x x 的 2 − k 2^{-k} 2−k 邻域内，因此 g g g 的存在性可证。
The subsequence { x g ( k ) } k ∈ N \{x_{g (k)}\}_{k \in \mathbb {N}} {xg(k)}k∈N converges to x . Thus, [ 0 , 1 ] [0, 1] [0,1] is sequentially compact.
子序列 { x g ( k ) } k ∈ N \{x_{g (k)}\}_{k \in \mathbb {N}} {xg(k)}k∈N 收敛于 x x x，因此 [ 0 , 1 ] [0, 1] [0,1] 是列紧的。
Now we shall show that R \mathbb {R} R is complete. In other words, we must show that any Cauchy sequence, i.e. a sequence of reals { x n ∈ R } n ∈ N \{x_{n} \in \mathbb {R}\}{n \in \mathbb {N}} {xn∈R}n∈N such that ∀ ϵ > 0 ∃ m ∀ n ( m < n → ∣ x m − x n ∣ < ϵ ) \forall \epsilon>0 \exists m \forall n (m<n \to |x{m}-x_{n}|<\epsilon) ∀ϵ>0∃m∀n(m<n→∣xm−xn∣<ϵ) , converges to a real number x .
接下来证明实数集 R \mathbb {R} R 的柯西完备性，即证明任意柯西序列 ------ 满足 ∀ ϵ > 0 ∃ m ∀ n ( m < n → ∣ x m − x n ∣ < ϵ ) \forall \epsilon>0 \exists m \forall n (m<n \to |x_{m}-x_{n}|<\epsilon) ∀ϵ>0∃m∀n(m<n→∣xm−xn∣<ϵ) 的实数序列 { x n ∈ R } n ∈ N \{x_{n} \in \mathbb {R}\}_{n \in \mathbb {N}} {xn∈R}n∈N------ 都收敛于某个实数 x x x。
Now it is clear from the definition that every Cauchy sequence is bounded, so by linearly rescaling we may take the sequence to lie entirely in [ 0 , 1 ] [0, 1] [0,1].
由定义易知，所有柯西序列都是有界的，因此可通过线性缩放，将任意柯西序列映射到闭区间 [ 0 , 1 ] [0, 1] [0,1] 内。
The last result implies that a subsequence of this scaled sequence converges. However, if a subsequence of a Cauchy sequence converges, then the entire sequence converges to the same number.
由上述结论可知，该缩放后的序列存在收敛子序列，而柯西序列若有子序列收敛，则其本身也收敛于同一极限。
Thus, after reversing the scaling process, we have found an x ∈ R x \in \mathbb {R} x∈R such that { x n } → x \{x_{n}\} \to x {xn}→x .
□ \square □
还原缩放过程后，即可得到实数 x ∈ R x \in \mathbb {R} x∈R，使得 { x n } → x \{x_{n}\} \to x {xn}→x。
□ \square □
Theorem I.7 ACA 0 \text {ACA}_0 ACA0 is equivalent to the completeness of R \mathbb {R} R and the sequential compactness of [ 0 , 1 ] [0, 1] [0,1] over RCA 0 \text {RCA}_0 RCA0 .
定理 1.7 在 RCA 0 \text {RCA}_0 RCA0 上， ACA 0 \text {ACA}_0 ACA0 与实数集 R \mathbb {R} R 的柯西完备性、闭区间 [ 0 , 1 ] [0, 1] [0,1] 的列紧性均等价。
Proof We have already shown that these two theorems may be proven in ACA 0 \text {ACA}_0 ACA0 . Therefore, we need only find reversals. That is, we must prove that given RCA 0 \text {RCA}_0 RCA0 and either of these two theorems, we can show that arithmetical comprehension holds.
证明 前文已证明这两个结论均可在 ACA 0 \text {ACA}_0 ACA0 中得证，因此只需完成反向证明，即证明在 RCA 0 \text {RCA}_0 RCA0 中，若加上这两个结论中的任意一个，均可推出算术概括公理成立。
As we saw in the proof of the previous theorem, the sequential compactness of [ 0 , 1 ] [0, 1] [0,1] implies the completeness of R \mathbb {R} R . Therefore, it is sufficient to find a reversal for the completeness of R \mathbb {R} R .
由上一定理的证明可知， [ 0 , 1 ] [0, 1] [0,1] 的列紧性可推出 R \mathbb {R} R 的柯西完备性，因此只需对实数集的柯西完备性完成反向证明即可。
Working in RCA 0 \text {RCA}0 RCA0 , assume that R \mathbb {R} R is complete. Let f : N → N f: \mathbb {N} \to \mathbb {N} f:N→N be an arbitrary function. Let x n = ∑ i = 0 n 2 − f ( i ) x{n}=\sum_{i=0}^{n} 2^{-f (i)} xn=∑i=0n2−f(i) . { x n } \{x_{n}\} {xn} is a bounded, increasing sequence of real numbers, so it is Cauchy.
在 RCA 0 \text {RCA}0 RCA0 中，假设实数集 R \mathbb {R} R 是柯西完备的。设 f : N → N f: \mathbb {N} \to \mathbb {N} f:N→N 为任意函数，令 x n = ∑ i = 0 n 2 − f ( i ) x{n}=\sum_{i=0}^{n} 2^{-f (i)} xn=∑i=0n2−f(i)，则 { x n } \{x_{n}\} {xn} 是有界递增的实数序列，因此为柯西序列。
By the completeness of R \mathbb {R} R , { x n } → x \{x_{n}\} \to x {xn}→x for some x ∈ R x \in \mathbb {R} x∈R .
由 R \mathbb {R} R 的柯西完备性可知，存在 x ∈ R x \in \mathbb {R} x∈R 使得 { x n } → x \{x_{n}\} \to x {xn}→x。
Having this number x in hand allows us to effectively bound our search for a number i ∈ N i \in \mathbb {N} i∈N such that f ( i ) = k f (i)=k f(i)=k . That is, we know that for all k , ∃ i ( f ( i ) = k ) ↔ ∀ n ( ∣ x n − x ∣ < 2 − k → ∃ i ≤ n ( f ( i ) = k ) ) \exists i (f (i)=k) \leftrightarrow \forall n\left (\left|x_{n}-x\right|<2^{-k} \to \exists i \leq n (f (i)=k)\right) ∃i(f(i)=k)↔∀n(∣xn−x∣<2−k→∃i≤n(f(i)=k)) .
借助极限 x x x，可有效确定满足 f ( i ) = k f (i)=k f(i)=k 的自然数 i i i 的搜索范围，即对所有 k k k，有 ∃ i ( f ( i ) = k ) ↔ ∀ n ( ∣ x n − x ∣ < 2 − k → ∃ i ≤ n ( f ( i ) = k ) ) \exists i (f (i)=k) \leftrightarrow \forall n\left (\left|x_{n}-x\right|<2^{-k} \to \exists i \leq n (f (i)=k)\right) ∃i(f(i)=k)↔∀n(∣xn−x∣<2−k→∃i≤n(f(i)=k))。
Let X = { k ∣ ∃ i ( f ( i ) = k ) } X=\{k | \exists i (f (i)=k)\} X={k∣∃i(f(i)=k)} , the defining condition of X is Δ 1 0 \Delta^0_1 Δ10 (the left side is Σ 1 0 \Sigma^0_1 Σ10 , the right side is Π 1 0 \Pi^0_1 Π10 ), so X exists by Δ 1 0 \Delta^0_1 Δ10 comprehension.
令 X = { k ∣ ∃ i ( f ( i ) = k ) } X=\{k | \exists i (f (i)=k)\} X={k∣∃i(f(i)=k)}，其定义条件为 Δ 1 0 \Delta^0_1 Δ10 公式（左侧为 Σ 1 0 \Sigma^0_1 Σ10 公式，右侧为 Π 1 0 \Pi^0_1 Π10 公式），因此 X X X 的存在性可由 Δ 1 0 \Delta^0_1 Δ10 概括公理保证。
Thus, the range of any function f f f exists, which by Lemma I.4 implies arithmetical comprehension, i.e. ACA 0 \text {ACA}_0 ACA0 .
□ \square □
由此可知，任意函数 f f f 的值域均存在，根据引理 1.4，这可推出算术概括公理成立，即 ACA 0 \text {ACA}_0 ACA0 得证。
□ \square □
Real Numbers in WKL 0 \text {WKL}_0 WKL0
WKL 0 \text {WKL}_0 WKL0 中的实数
Theorem I.8 WKL 0 \text {WKL}_0 WKL0 is strong enough to prove the Heine-Borel compactness of [ 0 , 1 ] [0, 1] [0,1]: every countable open cover of [ 0 , 1 ] [0, 1] [0,1] has a finite subcover.
定理 1.8 WKL 0 \text {WKL}_0 WKL0 的逻辑强度足以证明闭区间 [ 0 , 1 ] [0, 1] [0,1] 的海涅 - 博雷尔紧性： [ 0 , 1 ] [0, 1] [0,1] 的任意可数开覆盖都存在有限子覆盖。
Proof Working in WKL 0 \text {WKL}0 WKL0 , let { ( a i , b i ) } i ∈ N \{(a_i, b_i)\}{i\in\mathbb {N}} {(ai,bi)}i∈N be a countable open cover of [ 0 , 1 ] [0,1] [0,1] with a i , b i ∈ Q a_i, b_i \in \mathbb {Q} ai,bi∈Q for all i i i .
证明 在 WKL 0 \text {WKL}0 WKL0 中，设 { ( a i , b i ) } i ∈ N \{(a_i, b_i)\}{i\in\mathbb {N}} {(ai,bi)}i∈N 为 [ 0 , 1 ] [0,1] [0,1] 的一个可数开覆盖，且对所有 i i i，有 a i , b i ∈ Q a_i, b_i \in \mathbb {Q} ai,bi∈Q。
Define the Σ 1 0 \Sigma^0_1 Σ10 formula ϕ ( q , r ) = q ∈ Q ∧ r ∈ Q ∧ ∃ i ( a i < q < r < b i ) \phi (q,r)=q\in \mathbb {Q}\land r\in \mathbb {Q}\land \exists i\left ( a_i<q<r<b_i \right) ϕ(q,r)=q∈Q∧r∈Q∧∃i(ai<q<r<bi) , and let f : N → Q × Q f: \mathbb {N} \to \mathbb {Q} \times \mathbb {Q} f:N→Q×Q enumerate all pairs ( q , r ) (q, r) (q,r) satisfying ϕ ( q , r ) \phi (q, r) ϕ(q,r) (exists by Σ 1 0 \Sigma^0_1 Σ10 comprehension).
定义 Σ 1 0 \Sigma^0_1 Σ10 公式 ϕ ( q , r ) = q ∈ Q ∧ r ∈ Q ∧ ∃ i ( a i < q < r < b i ) \phi (q,r)=q\in \mathbb {Q}\land r\in \mathbb {Q}\land \exists i\left ( a_i<q<r<b_i \right) ϕ(q,r)=q∈Q∧r∈Q∧∃i(ai<q<r<bi)，令函数 f : N → Q × Q f: \mathbb {N} \to \mathbb {Q} \times \mathbb {Q} f:N→Q×Q 枚举所有满足 ϕ ( q , r ) \phi (q, r) ϕ(q,r) 的有序对 ( q , r ) (q, r) (q,r)，其存在性可由 Σ 1 0 \Sigma^0_1 Σ10 概括公理保证。
For s ∈ 2 < N s \in 2^{<\mathbb {N}} s∈2<N , define c s = ∑ i = 0 ℓ ( s ) − 1 s i 2 i + 1 c_{s}=\sum_{i=0}^{\ell (s)-1} \frac {s_{i}}{2^{i+1}} cs=∑i=0ℓ(s)−12i+1si , d s = c s + 1 2 ℓ ( s ) d_{s}=c_{s}+\frac {1}{2^{\ell (s)}} ds=cs+2ℓ(s)1 .
对 s ∈ 2 < N s \in 2^{<\mathbb {N}} s∈2<N，定义 c s = ∑ i = 0 ℓ ( s ) − 1 s i 2 i + 1 c_{s}=\sum_{i=0}^{\ell (s)-1} \frac {s_{i}}{2^{i+1}} cs=∑i=0ℓ(s)−12i+1si， d s = c s + 1 2 ℓ ( s ) d_{s}=c_{s}+\frac {1}{2^{\ell (s)}} ds=cs+2ℓ(s)1。
Define the binary tree T = { s ∈ 2 < N ∣ ¬ ∃ i ≤ ℓ ( s ) ( a i < c s < d s < b i ) } T=\left\{s \in 2^{<\mathbb {N}} | \neg \exists i \leq \ell (s)\left (a_i<c_s<d_s<b_i\right)\right\} T={s∈2<N∣¬∃i≤ℓ(s)(ai<cs<ds<bi)}, which exists by Σ 0 0 \Sigma^0_0 Σ00 comprehension.
定义二叉树 T = { s ∈ 2 < N ∣ ¬ ∃ i ≤ ℓ ( s ) ( a i < c s < d s < b i ) } T=\left\{s \in 2^{<\mathbb {N}} | \neg \exists i \leq \ell (s)\left (a_i<c_s<d_s<b_i\right)\right\} T={s∈2<N∣¬∃i≤ℓ(s)(ai<cs<ds<bi)}，其存在性可由 Σ 0 0 \Sigma^0_0 Σ00 概括公理保证。
Claim : There is no path through T.
断言 ：二叉树 T T T 上不存在路径。
Suppose for contradiction that f : N → { 0 , 1 } f: \mathbb {N} \to \{0,1\} f:N→{0,1} is a path through T. Associate f with a real number x f ∈ [ 0 , 1 ] x_f \in [0,1] xf∈[0,1] via its binary expansion: x f = { ∑ i = 0 n f ( i ) 2 i + 1 ∈ Q } n ∈ N x_f = \left\{\sum_{i=0}^{n} \frac {f (i)}{2^{i+1}} \in \mathbb {Q}\right\}_{n \in \mathbb {N}} xf={∑i=0n2i+1f(i)∈Q}n∈N .
反证法，假设存在路径 f : N → { 0 , 1 } f: \mathbb {N} \to \{0,1\} f:N→{0,1} 穿过 T T T，通过二进制展开将 f f f 对应到实数 x f ∈ [ 0 , 1 ] x_f \in [0,1] xf∈[0,1]： x f = { ∑ i = 0 n f ( i ) 2 i + 1 ∈ Q } n ∈ N x_f = \left\{\sum_{i=0}^{n} \frac {f (i)}{2^{i+1}} \in \mathbb {Q}\right\}_{n \in \mathbb {N}} xf={∑i=0n2i+1f(i)∈Q}n∈N。
By Σ 1 0 \Sigma^0_1 Σ10 induction, x f x_f xf is the unique real number such that x f ∈ [ c ( f ( 0 ) , . . . , f ( n ) ) , d ( f ( 0 ) , . . . , f ( n ) ) ] x_f \in [c_{(f (0), ..., f (n))}, d_{(f (0), ..., f (n))}] xf∈[c(f(0),...,f(n)),d(f(0),...,f(n))] for all n ∈ N n \in \mathbb {N} n∈N .
由 Σ 1 0 \Sigma^0_1 Σ10 归纳法可知， x f x_f xf 是唯一满足对所有 n ∈ N n \in \mathbb {N} n∈N，有 x f ∈ [ c ( f ( 0 ) , . . . , f ( n ) ) , d ( f ( 0 ) , . . . , f ( n ) ) ] x_f \in [c_{(f (0), ..., f (n))}, d_{(f (0), ..., f (n))}] xf∈[c(f(0),...,f(n)),d(f(0),...,f(n))] 的实数。
Since { ( a i , b i ) } \{(a_i, b_i)\} {(ai,bi)} covers [ 0 , 1 ] [0,1] [0,1] , there exists i i i such that a i < x f < b i a_i < x_f < b_i ai<xf<bi .
由于 { ( a i , b i ) } \{(a_i, b_i)\} {(ai,bi)} 是 [ 0 , 1 ] [0,1] [0,1] 的开覆盖，因此存在 i i i 使得 a i < x f < b i a_i < x_f < b_i ai<xf<bi。
By the definition of x f x_f xf , there exists N ≥ i N ≥i N≥i such that a i < c ( f ( 0 ) , . . . , f ( N ) ) < d ( f ( 0 ) , . . . , f ( N ) ) < b i a_i < c_{(f (0),...,f (N))} < d_{(f (0),...,f (N))} < b_i ai<c(f(0),...,f(N))<d(f(0),...,f(N))<bi , which implies ( f ( 0 ) , . . . , f ( N ) ) ∉ T (f (0), ..., f (N)) \notin T (f(0),...,f(N))∈/T , a contradiction.
由 x f x_f xf 的定义可知，存在 N ≥ i N ≥i N≥i 使得 a i < c ( f ( 0 ) , . . . , f ( N ) ) < d ( f ( 0 ) , . . . , f ( N ) ) < b i a_i < c_{(f (0),...,f (N))} < d_{(f (0),...,f (N))} < b_i ai<c(f(0),...,f(N))<d(f(0),...,f(N))<bi，这意味着 ( f ( 0 ) , . . . , f ( N ) ) ∉ T (f (0), ..., f (N)) \notin T (f(0),...,f(N))∈/T，与 f f f 是 T T T 上的路径矛盾。
The claim is proven.
断言得证。
By Weak K¨onig's lemma, T is finite (infinite binary tree has a path). Let N be larger than the length of any s ∈ T s \in T s∈T .
由弱柯尼希引理可知， T T T 为有限集（无限二叉树必存在路径）。令 N N N 为大于所有 s ∈ T s \in T s∈T 长度的自然数。
By the definition of T,
由 T T T 的定义可得，
∀ s ∈ 2 < N ( ℓ ( s ) = N → ∃ i ≤ N ( a i < c s < d s < b i ) ) \forall s \in 2^{<\mathbb {N}}\left (\ell (s)=N \to \exists i \leq N\left (a_i<c_s<d_s<b_i\right)\right) ∀s∈2<N(ℓ(s)=N→∃i≤N(ai<cs<ds<bi)) .
The dyadic intervals { [ c s , d s ] } ℓ ( s ) = N \{[c_s, d_s]\}_{\ell (s)=N} {[cs,ds]}ℓ(s)=N cover [ 0 , 1 ] [0,1] [0,1] , so { ( a 0 , b 0 ) , . . . , ( a N , b N ) } \{(a_0, b_0), ...,(a_N, b_N)\} {(a0,b0),...,(aN,bN)} is a finite subcover of [ 0 , 1 ] [0,1] [0,1] .
□ \square □
二进区间族 { [ c s , d s ] } ℓ ( s ) = N \{[c_s, d_s]\}_{\ell (s)=N} {[cs,ds]}ℓ(s)=N 是 [ 0 , 1 ] [0,1] [0,1] 的覆盖，因此 { ( a 0 , b 0 ) , . . . , ( a N , b N ) } \{(a_0, b_0), ...,(a_N, b_N)\} {(a0,b0),...,(aN,bN)} 是 [ 0 , 1 ] [0,1] [0,1] 的一个有限子覆盖。
□ \square □
Definitions
定义
A real number x ∈ [ 0 , 1 ] x \in [0,1] x∈[0,1] is aCantor point if there exists a path f : N → { 0 , 1 } f: \mathbb {N} \to \{0,1\} f:N→{0,1} through 2 < N 2^{<\mathbb {N}} 2<N such that x = { ∑ i = 0 n 2 f ( i ) 3 i + 1 ∈ Q } n ∈ N x=\left\{\sum_{i=0}^{n} \frac {2 f (i)}{3^{i+1}} \in \mathbb {Q}\right\}_{n \in \mathbb {N}} x={∑i=0n3i+12f(i)∈Q}n∈N .
若存在路径 f : N → { 0 , 1 } f: \mathbb {N} \to \{0,1\} f:N→{0,1} 穿过 2 < N 2^{<\mathbb {N}} 2<N，使得 x = { ∑ i = 0 n 2 f ( i ) 3 i + 1 ∈ Q } n ∈ N x=\left\{\sum_{i=0}^{n} \frac {2 f (i)}{3^{i+1}} \in \mathbb {Q}\right\}_{n \in \mathbb {N}} x={∑i=0n3i+12f(i)∈Q}n∈N，则称实数 x ∈ [ 0 , 1 ] x \in [0,1] x∈[0,1] 为康托尔点。
If no such path exists, then we say that x ∈ [ 0 , 1 ] x \in [0,1] x∈[0,1] is anon-Cantor point .
若不存在这样的路径，则称实数 x ∈ [ 0 , 1 ] x \in [0,1] x∈[0,1] 为非康托尔点。
Additionally, for any binary tree T ⊂ 2 < N T \subset 2^{<\mathbb {N}} T⊂2<N , define the set of leaves of T to be L T = { s ∈ T ∣ s ⌢ ( 0 ) ∉ T ∧ s ⌢ ( 1 ) ∉ T } L_{T}=\{ s\in T | s\smallfrown (0)\notin T\land s\smallfrown (1)\notin T\} LT={s∈T∣s⌢(0)∈/T∧s⌢(1)∈/T}, which exists for any given T by Σ 0 0 \Sigma^0_0 Σ00 comprehension.
此外，对任意二叉树 T ⊂ 2 < N T \subset 2^{<\mathbb {N}} T⊂2<N，定义 T T T 的叶子集 为 L T = { s ∈ T ∣ s ⌢ ( 0 ) ∉ T ∧ s ⌢ ( 1 ) ∉ T } L_{T}=\{ s\in T | s\smallfrown (0)\notin T\land s\smallfrown (1)\notin T\} LT={s∈T∣s⌢(0)∈/T∧s⌢(1)∈/T}，对任意给定的 T T T，其叶子集的存在性可由 Σ 0 0 \Sigma^0_0 Σ00 概括公理保证。
Lemma I.9 Over RCA 0 \text {RCA}0 RCA0 for a given tree T , if L T L{T} LT is finite, then either T is finite or it has a path.
引理 1.9 在 RCA 0 \text {RCA}0 RCA0 上，对给定的二叉树 T T T，若其叶子集 L T L{T} LT 为有限集，则 T T T 要么是有限集，要么存在路径。
Proof Assume that T is infinite, but that L T L_{T} LT is finite. We shall show that T has a path.
证明 假设 T T T 是无限集，但叶子集 L T L_{T} LT 为有限集，需证明 T T T 上存在路径。
Consider the set X = { s ∈ T ∣ ∃ t ∈ L T ( s ⊆ t ) } X=\left\{s \in T | \exists t \in L_{T}(s \subseteq t)\right\} X={s∈T∣∃t∈LT(s⊆t)}, which exists by Σ 0 0 \Sigma^0_0 Σ00 comprehension because L T L_T LT is finite.
考虑集合 X = { s ∈ T ∣ ∃ t ∈ L T ( s ⊆ t ) } X=\left\{s \in T | \exists t \in L_{T}(s \subseteq t)\right\} X={s∈T∣∃t∈LT(s⊆t)}，因 L T L_T LT 为有限集，其存在性可由 Σ 0 0 \Sigma^0_0 Σ00 概括公理保证。
For any given t ∈ L T t \in L_T t∈LT, there can only be finitely many subsequences of t. Because the finite union of finite sets is finite, this implies that X is finite.
对任意 t ∈ L T t \in L_T t∈LT，其初始段仅有有限个，而有限个有限集的并仍为有限集，因此 X X X 为有限集。
Because T is infinite, there must exist a binary sequence s 0 ∈ T s_0 \in T s0∈T such that s 0 ∉ X s_0 \notin X s0∈/X .
由于 T T T 是无限集，因此必存在二元序列 s 0 ∈ T s_0 \in T s0∈T 且 s 0 ∉ X s_0 \notin X s0∈/X。
By definition of X, there exists s 1 ∈ T s_1 \in T s1∈T such that s 1 s_1 s1 is either s 0 ⌢ ( 0 ) s_0 \smallfrown (0) s0⌢(0) or s 0 ⌢ ( 1 ) s_0 \smallfrown (1) s0⌢(1), and s 1 ∉ X s_1 \notin X s1∈/X .
由 X X X 的定义可知，存在 s 1 ∈ T s_1 \in T s1∈T，使得 s 1 s_1 s1 为 s 0 ⌢ ( 0 ) s_0 \smallfrown (0) s0⌢(0) 或 s 0 ⌢ ( 1 ) s_0 \smallfrown (1) s0⌢(1)，且 s 1 ∉ X s_1 \notin X s1∈/X。
By Σ 0 0 \Sigma^0_0 Σ00 induction there exists s n s_n sn for all n such that s n ∈ T s_n \in T sn∈T and s n + 1 s_{n+1} sn+1 is either s n ⌢ ( 0 ) s_n \smallfrown (0) sn⌢(0) or s n ⌢ ( 1 ) s_n \smallfrown (1) sn⌢(1) .
由 Σ 0 0 \Sigma^0_0 Σ00 归纳法可知，对所有 n n n，存在 s n ∈ T s_n \in T sn∈T，且 s n + 1 s_{n+1} sn+1 为 s n ⌢ ( 0 ) s_n \smallfrown (0) sn⌢(0) 或 s n ⌢ ( 1 ) s_n \smallfrown (1) sn⌢(1)。
Let f : N → { 0 , 1 } f: \mathbb {N} \to \{0, 1\} f:N→{0,1} be defined by f ( n ) = s 0 ( n ) f (n) = s_0 (n) f(n)=s0(n) if n < ℓ ( s 0 ) n < \ell (s_0) n<ℓ(s0) or f ( n ) = s n − ℓ ( s 0 ) + 1 ( n ) f (n) = s_{n-\ell (s_0)+1}(n) f(n)=sn−ℓ(s0)+1(n) otherwise. By the above, f is a path through T.
□ \square □
定义函数 f : N → { 0 , 1 } f: \mathbb {N} \to \{0, 1\} f:N→{0,1}：若 n < ℓ ( s 0 ) n < \ell (s_0) n<ℓ(s0)，则 f ( n ) = s 0 ( n ) f (n) = s_0 (n) f(n)=s0(n)，否则 f ( n ) = s n − ℓ ( s 0 ) + 1 ( n ) f (n) = s_{n-\ell (s_0)+1}(n) f(n)=sn−ℓ(s0)+1(n)。由上述构造可知， f f f 是 T T T 上的一条路径。
□ \square □
Theorem I.10 WKL 0 \text {WKL}_0 WKL0 is equivalent to the compactness of [ 0 , 1 ] [0, 1] [0,1] over RCA 0 \text {RCA}_0 RCA0 .
定理 1.10 在 RCA 0 \text {RCA}_0 RCA0 上， WKL 0 \text {WKL}_0 WKL0 与闭区间 [ 0 , 1 ] [0, 1] [0,1] 的海涅 - 博雷尔紧性等价。
Proof By Theorem I.8, Weak K¨onig's lemma implies the compactness of [ 0 , 1 ] [0,1] [0,1]. It is sufficient to find a reversal: assume the compactness of [ 0 , 1 ] [0,1] [0,1] in RCA 0 \text {RCA}_0 RCA0 , and prove Weak K¨onig's lemma.
证明 由定理 1.8 可知，弱柯尼希引理可推出 [ 0 , 1 ] [0,1] [0,1] 的海涅 - 博雷尔紧性，因此只需完成反向证明：在 RCA 0 \text {RCA}_0 RCA0 中假设 [ 0 , 1 ] [0,1] [0,1] 是海涅 - 博雷尔紧的，证明弱柯尼希引理成立。
Working in RCA 0 \text {RCA}_0 RCA0 , assume that [ 0 , 1 ] [0, 1] [0,1] is compact. Let T be a binary tree such that there are no paths through T. Our goal is to show that T must be finite.
在 RCA 0 \text {RCA}_0 RCA0 中，假设 [ 0 , 1 ] [0,1] [0,1] 是海涅 - 博雷尔紧的，设 T T T 为无路径的二叉树，需证明 T T T 必为有限集。
By Lemma I.9, it is sufficient to show that L T L_T LT is finite.
由引理 1.9 可知，只需证明 T T T 的叶子集 L T L_T LT 为有限集即可。
Define the following sequences of real numbers in [ 0 , 1 ] [0, 1] [0,1] indexed by binary sequences s ∈ 2 < N s \in2^{<\mathbb {N}} s∈2<N:
对每个二元序列 s ∈ 2 < N s \in2^{<\mathbb {N}} s∈2<N，定义 [ 0 , 1 ] [0,1] [0,1] 中的实数列：
a s = ∑ i = 0 ℓ ( s ) − 1 2 s i 3 i + 1 , b s = a s + 3 − ℓ ( s ) , c s = a s − 3 − ℓ ( s ) − 1 , d s = b s + 3 − ℓ ( s ) − 1 . \begin{aligned} a_s & = \sum_{i=0}^{\ell (s)-1} \frac{2s_i}{3^{i+1}}, \\ b_s & = a_s + 3^{-\ell (s)}, \\ c_s & = a_s - 3^{-\ell (s)-1}, \\ d_s & = b_s + 3^{-\ell (s)-1}. \end{aligned} asbscsds=i=0∑ℓ(s)−13i+12si,=as+3−ℓ(s),=as−3−ℓ(s)−1,=bs+3−ℓ(s)−1.
For any non-Cantor point x there exists s ∈ 2 < N s \in2^{<\mathbb {N}} s∈2<N such that x ∈ ( b s ⌢ ( 1 ) , a s ⌢ ( 0 ) ) x \in (b_{s\smallfrown (1)}, a_{s\smallfrown (0)}) x∈(bs⌢(1),as⌢(0)) while no Cantor points lie in any of these intervals (provable in RCA 0 \text {RCA}_0 RCA0 via standard Cantor set construction, see [3]).
对任意非康托尔点 x x x，存在 s ∈ 2 < N s \in2^{<\mathbb {N}} s∈2<N 使得 x ∈ ( b s ⌢ ( 1 ) , a s ⌢ ( 0 ) ) x \in (b_{s\smallfrown (1)}, a_{s\smallfrown (0)}) x∈(bs⌢(1),as⌢(0))，且所有康托尔点都不在这类区间中（可通过经典康托尔集构造在 RCA 0 \text {RCA}_0 RCA0 中证明，参见文献 [3]）。
Thus, X = { ( b s ⌢ ( 1 ) , a s ⌢ ( 0 ) ) } s ∈ 2 < N X = \{(b_{s\smallfrown (1)}, a_{s\smallfrown (0)})\}_{s\in2^{<\mathbb {N}}} X={(bs⌢(1),as⌢(0))}s∈2<N is a countable open cover of all non-Cantor points of [ 0 , 1 ] [0,1] [0,1] .
因此， X = { ( b s ⌢ ( 1 ) , a s ⌢ ( 0 ) ) } s ∈ 2 < N X = \{(b_{s\smallfrown (1)}, a_{s\smallfrown (0)})\}_{s\in2^{<\mathbb {N}}} X={(bs⌢(1),as⌢(0))}s∈2<N 是 [ 0 , 1 ] [0,1] [0,1] 中所有非康托尔点的一个可数开覆盖。
We claim that Y = { ( c s , d s ) } s ∈ L T Y = \{(cs, ds)\}_{s\in L_T} Y={(cs,ds)}s∈LT is a set of pairwise disjoint intervals covering all Cantor points of [ 0 , 1 ] [0,1] [0,1] .
断言： Y = { ( c s , d s ) } s ∈ L T Y = \{(c_s, d_s)\}_{s\in L_T} Y={(cs,ds)}s∈LT 是由两两不交的区间构成的集合，且覆盖 [ 0 , 1 ] [0,1] [0,1] 中的所有康托尔点。
-
Pairwise disjoint : For s , t ∈ L T s,t \in L_T s,t∈LT , s ⊈ t s \nsubseteq t s⊈t and t ⊈ s t \nsubseteq s t⊈s , so ( c s , d s ) ∩ ( c t , d t ) = ∅ (c_s, d_s) \cap (c_t, d_t) = \emptyset (cs,ds)∩(ct,dt)=∅ .
两两不交 ：对任意 s , t ∈ L T s,t \in L_T s,t∈LT， s ⊈ t s \nsubseteq t s⊈t 且 t ⊈ s t \nsubseteq s t⊈s，因此 ( c s , d s ) ∩ ( c t , d t ) = ∅ (c_s, d_s) \cap (c_t, d_t) = \emptyset (cs,ds)∩(ct,dt)=∅。 -
Covers Cantor points : Suppose there is a Cantor point x not in any ( c s , d s ) (c_s, d_s) (cs,ds) for s ∈ L T s \in L_T s∈LT . Then the path f associated with x is a path through T, a contradiction.
覆盖康托尔点 ：若存在康托尔点 x x x 不在任意 s ∈ L T s \in L_T s∈LT 对应的 ( c s , d s ) (c_s, d_s) (cs,ds) 中，则与 x x x 关联的路径 f f f 是 T T T 上的路径，与 T T T 无路径矛盾。
Thus, X ∪ Y X \cup Y X∪Y is a countable open cover of [ 0 , 1 ] [0,1] [0,1] . By compactness, there exists a finite subcover X ′ ∪ Y ′ X' \cup Y' X′∪Y′ with X ′ ⊂ X X' \subset X X′⊂X , Y ′ ⊂ Y Y' \subset Y Y′⊂Y .
因此， X ∪ Y X \cup Y X∪Y 是 [ 0 , 1 ] [0,1] [0,1] 的一个可数开覆盖，由海涅 - 博雷尔紧性可知，存在有限子覆盖 X ′ ∪ Y ′ X' \cup Y' X′∪Y′，其中 X ′ ⊂ X X' \subset X X′⊂X， Y ′ ⊂ Y Y' \subset Y Y′⊂Y。
Since the intervals in Y are pairwise disjoint, Y ′ = Y Y' = Y Y′=Y (each interval in Y contains a unique Cantor point not covered by X), so Y is finite.
由于 Y Y Y 中的区间两两不交，每个区间都包含一个不被 X X X 覆盖的唯一康托尔点，因此 Y ′ = Y Y' = Y Y′=Y，即 Y Y Y 为有限集。
This implies that L T L_{T} LT is finite, which by Lemma I.9 implies that T is finite.
□ \square □
由此可知 L T L_{T} LT 为有限集，根据引理 1.9， T T T 必为有限集。
□ \square □
Proof of Main Theorem I
主定理 1 的证明
The theorem follows directly from Theorems I.5, I.7, and I.10.
□ \square □
本定理可直接由定理 1.5、定理 1.7 和定理 1.10 推出。
□ \square □
Further Results
进一步的结论
In this section we will list several other standard reverse mathematical results and briefly discuss the remaining standard subsystems of second order arithmetic used in reverse mathematics, ATR 0 \text {ATR}_0 ATR0 and Π 1 1 -CA 0 \Pi^1_1\text {-CA}_0 Π11-CA0 . Proofs of all of these theorems may be found in [1].
本节将列出若干其他经典的逆数学结论，并简要介绍逆数学中用到的另外两个标准二阶算术子系统： ATR 0 \text {ATR}_0 ATR0 和 Π 1 1 -CA 0 \Pi^1_1\text {-CA}_0 Π11-CA0。所有定理的证明均可参见文献 [1]。
Theorem I.11 The following theorems can be proven in RCA 0 \text {RCA}_0 RCA0:
定理 1.11 下述定理均可在 RCA 0 \text {RCA}_0 RCA0 中证明：
(i) The real line satisfies the intermediate value property.
实数轴满足介值定理；
(ii) The Baire category theorem.
贝尔纲定理；
(iii) A version of the Tietze extension theorem for complete separable metric spaces.
完备可分度量空间上的泰茨扩张定理（某一版本）；
(iv) A strong version of the soundness theorem in mathematical logic.
数理逻辑中可靠性定理的强版本；
(v) The algebraic closure for any countable field exists.
任意可数域的代数闭包存在；
(vi) The Banach/Steinhaus theorem.
巴拿赫 - 施坦豪斯定理。
Theorem I.12 The following are equivalent to WKL 0 \text {WKL}_0 WKL0 over RCA 0 \text {RCA}_0 RCA0:
定理 1.12 在 RCA 0 \text {RCA}_0 RCA0 上，下述结论均与 WKL 0 \text {WKL}_0 WKL0 等价：
(i) The Heine/Borel theorem for compact metric spaces.
紧度量空间上的海涅 - 博雷尔定理；
(ii) Several properties of continuous functions on compact metric spaces including uniform continuity, the maximum principle, Riemann integrability, and Weierstrass approximation.
紧度量空间上连续函数的若干性质，包括一致连续性、最大值原理、黎曼可积性和魏尔斯特拉斯逼近定理；
(iii) The completeness and compactness theorems of mathematical logic.
数理逻辑中的完备性定理和紧致性定理；
(iv) The uniqueness of the algebraic closure for countable fields.
可数域代数闭包的唯一性；
(v) The Brouwer and Schauder fixed point theorems.
布劳威尔不动点定理和绍德尔不动点定理；
(vi) The Peano existence theorem for solutions of ordinary differential equations.
常微分方程的皮亚诺存在性定理；
(vii) The separable Hahn/Banach theorem.
可分版本的哈恩 - 巴拿赫定理。
Theorem I.13 The following are equivalent to ACA 0 \text {ACA}_0 ACA0 over RCA 0 \text {RCA}_0 RCA0:
定理 1.13 在 RCA 0 \text {RCA}_0 RCA0 上，下述结论均与 ACA 0 \text {ACA}_0 ACA0 等价：
(i) Every countable vector space over a countable field has a basis.
可数域上的任意可数向量空间都有基；
(ii) Every countable commutative ring has a maximal ideal.
任意可数交换环都有极大理想；
(iii) The divisible closure of an arbitrary countable Abelian group is unique.
任意可数交换群的可除闭包是唯一的；
(iv) K¨onig's lemma for subtrees of N < N \mathbb {N}^{<\mathbb {N}} N<N.
N < N \mathbb {N}^{<\mathbb {N}} N<N 子树的柯尼希引理（全柯尼希引理）；
(v) Ramsey's theorem for colorings of [ N ] 3 [\mathbb {N}]^{3} [N]3.
N \] 3 \[\\mathbb {N}\]\^{3} \[N\]3 着色的拉姆齐定理。
#### Arithmetical Transfinite Recursion: ATR 0 \\text {ATR}_0 ATR0
算术超穷递归： ATR 0 \\text {ATR}_0 ATR0
ATR 0 \\text {ATR}_0 ATR0 , or arithmetical transfinite recursion with limited induction, is a subsystem of second order arithmetic that is logically stronger than ACA 0 \\text {ACA}_0 ACA0 and the other systems we discussed in this chapter.
ATR 0 \\text {ATR}_0 ATR0 即带有限归纳的算术超穷递归系统，是二阶算术的子系统，其逻辑强度高于 ACA 0 \\text {ACA}_0 ACA0 及本章讨论的其他系统。
The system may be described informally as taking ACA 0 \\text {ACA}_0 ACA0 and allowing for the transfinite iteration of the Turing jump operator along any countable well-ordering.
该系统可直观描述为：在 ACA 0 \\text {ACA}_0 ACA0 的基础上，允许图灵跳跃算子沿任意可数良序进行超穷迭代。
**Theorem I.14** The following are equivalent to ATR 0 \\text {ATR}_0 ATR0 over RCA 0 \\text {RCA}_0 RCA0:
**定理 1.14** 在 RCA 0 \\text {RCA}_0 RCA0 上，下述结论均与 ATR 0 \\text {ATR}_0 ATR0 等价：
(i) Lusin's separation theorem.
卢津分离定理；
(ii) The Borel domain theorem.
博雷尔定义域定理；
(iii) The perfect set theorem.
完美集定理；
(iv) The existence of Ulm resolutions.
乌尔姆分解的存在性；
(v) The comparability of countable well-orderings.
可数良序的可比较性；
(vi) The open and clopen Ramsey theorems.
开集和闭开集的拉姆齐定理。
#### Π 1 1 \\Pi\^1_1 Π11-CA 0 _0 0
Π 1 1 \\Pi\^1_1 Π11 概括公理系统： Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0
Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 , or Π 1 1 \\Pi\^1_1 Π11 comprehension with limited induction, is the strongest subsystem of second order arithmetic considered in standard reverse mathematics.
Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 即带有限归纳的 Π 1 1 \\Pi\^1_1 Π11 概括公理系统，是标准逆数学研究中考虑的逻辑强度最强的二阶算术子系统。
It is defined similarly to RCA 0 \\text {RCA}_0 RCA0 and ACA 0 \\text {ACA}_0 ACA0 except recursive and arithmetical comprehension are replaced with Π 1 1 \\Pi\^1_1 Π11 comprehension, i.e. the comprehension schema applies to formulae of the form ∀ X ( ϕ ( X ) ) \\forall X (\\phi (X)) ∀X(ϕ(X)) where X is a set variable and ϕ ( X ) \\phi (X) ϕ(X) is arithmetical.
其定义方式与 RCA 0 \\text {RCA}_0 RCA0 和 ACA 0 \\text {ACA}_0 ACA0 类似，区别在于将递归概括和算术概括替换为 Π 1 1 \\Pi\^1_1 Π11 概括，即概括公理模式适用于形如 ∀ X ( ϕ ( X ) ) \\forall X (\\phi (X)) ∀X(ϕ(X)) 的公式，其中 X X X 为集合变元， ϕ ( X ) \\phi (X) ϕ(X) 为算术公式。
**Theorem I.15** The following are equivalent to Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 over RCA 0 \\text {RCA}_0 RCA0:
**定理 1.15** 在 RCA 0 \\text {RCA}_0 RCA0 上，下述结论均与 Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 等价：
(i) The Cantor/Bendixson theorem for closed sets.
闭集的康托尔 - 本迪克松定理；
(ii) Kondo's theorem on coanalytic uniformization.
余解析集一致化的近藤定理；
(iii) Silver's theorem on Borel equivalence relations.
博雷尔等价关系的西尔弗定理；
(iv) Every countable Abelian group is a direct sum of a divisible group and a reduced group.
任意可数交换群都是一个可除群和一个既约群的直和；
(v) The Δ 2 0 \\Delta\^0_2 Δ20 Ramsey theorem.
Δ 2 0 \\Delta\^0_2 Δ20 拉姆齐定理。
As can be seen from this list of results, reverse mathematics provides a wealth of information about the equivalencies and relative logical strengths of many of the theorems of mathematics.
从上述结论可以看出，逆数学为众多数学定理的等价性和相对逻辑强度研究提供了丰富的结论。
It is surprising that all of these theorems can be seen as equivalent up to constructive mathematics to basic axioms that describe what counts as a set.
令人意外的是，在构造性数学的框架下，所有这些定理都可与描述集合存在性的基本公理等价。
Furthermore, it is quite intriguing that all of these theorems can be arranged in a linear ordering of logical strength starting from RCA 0 \\text {RCA}_0 RCA0 and working up to Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 .
更有趣的是，这些定理对应的公理系统可从 RCA 0 \\text {RCA}_0 RCA0 到 Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 按逻辑强度形成线性排序。
Finally, it may be interesting to note that each of the five standard subsystems of second order arithmetic that arise in the study of reverse mathematics can be seen as corresponding to different philosophical approaches to the foundations of mathematics.
值得一提的是，逆数学研究中出现的这五个标准二阶算术子系统，分别对应数学基础研究的不同哲学流派：
* RCA 0 \\text {RCA}_0 RCA0 can be associated with the constructivism of Bishop.
RCA 0 \\text {RCA}_0 RCA0 对应毕晓普的构造主义；
* WKL 0 \\text {WKL}_0 WKL0 with the finitistic reductionism of Hilbert.
WKL 0 \\text {WKL}_0 WKL0 对应希尔伯特的有限主义还原论；
* ACA 0 \\text {ACA}_0 ACA0 with the predicativism of Weyl and Feferman.
ACA 0 \\text {ACA}_0 ACA0 对应外尔和费弗曼的直谓主义；
* ATR 0 \\text {ATR}_0 ATR0 with the predicative reductionism of Friedman and Simpson.
ATR 0 \\text {ATR}_0 ATR0 对应弗里德曼和辛普森的直谓主义还原论；
* Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 with the impredicativity of Feferman and others.
Π 1 1 -CA 0 \\Pi\^1_1\\text {-CA}_0 Π11-CA0 对应费弗曼等人的非直谓主义。
These connections are explored in more detail in \[1\].
这些关联的详细研究可参见文献 \[1\]。
Reverse mathematics, thus, gives deep insight into the nature of many theorems of ordinary mathematics, into the relations and equivalencies between these theorems, into the axioms that we may choose to consider and the nature of the set, and into several of the philosophical approaches to the foundations of mathematics.
由此可见，逆数学不仅深刻揭示了经典数学中诸多定理的本质、定理间的关联与等价性，还为集合存在性公理的选择、集合的本质，以及数学基础的多种哲学研究路径提供了重要洞见。
In the following two chapters we will explore some interesting theorems of reverse mathematics that go beyond the standard results and that serve to further illustrate these themes.
在接下来的两章中，我们将探讨一些超出经典结论的有趣逆数学定理，进一步阐释上述研究主题。
## Chapter II - The Reverse Mathematics of Hilbert's Basis Theorem
第二章 希尔伯特基定理的逆数学
In this chapter we will prove a unique theorem of reverse mathematics. Namely, we will show that Hilbert's basis theorem is equivalent to the well-ordering of ω ω \\omega\^\\omega ωω over RCA 0 \\text {RCA}_0 RCA0 .
本章将证明一个独特的逆数学定理：在 RCA 0 \\text {RCA}_0 RCA0 上，希尔伯特基定理与 ω ω \\omega\^\\omega ωω 的良序性等价。
This chapter is based on the proof by Simpson in \[4\], although the presentation and structure have been significantly altered.
本章内容基于辛普森在文献 \[4\] 中的证明，对表述和结构进行了大幅调整。
### Definitions
定义
A**countable ring** A is a tuple ( ∣ A ∣ , + A , ∗ A , 0 A , 1 A ) (\|A\|, +_A, \*_A, 0_A, 1_A) (∣A∣,+A,∗A,0A,1A) where ∣ A ∣ \|A\| ∣A∣ is a set of natural numbers - the set of codes for elements of A - + A +_A +A and A _A A are functions from ∣ A ∣ × ∣ A ∣ \|A\|×\|A\| ∣A∣×∣A∣ to ∣ A ∣ \|A\| ∣A∣, and 0 A 0_A 0A and 1 A 1_A 1A are distinct distinguished elements of ∣ A ∣ \|A\| ∣A∣ such that these objects obey the usual axioms for a commutative ring with unit.
**可数环** A A A 是一个五元组 ( ∣ A ∣ , + A , ∗ A , 0 A , 1 A ) (\|A\|, +_A, \*_A, 0_A, 1_A) (∣A∣,+A,∗A,0A,1A)，其中 ∣ A ∣ \|A\| ∣A∣ 是自然数的子集（作为 A A A 中元素的编码集）， + A +_A +A 和 ∗ A \*_A ∗A 是从 ∣ A ∣ × ∣ A ∣ \|A\|×\|A\| ∣A∣×∣A∣ 到 ∣ A ∣ \|A\| ∣A∣ 的函数， 0 A 0_A 0A 和 1 A 1_A 1A 是 ∣ A ∣ \|A\| ∣A∣ 中相异的特殊元，且这些对象满足有单位元交换环的所有经典公理。
Standard ring notation will be freely used in place of the explicit notation using ∣ ・ ∣ \|・\| ∣・∣ and the subscript A.
本文将自由使用环的标准符号，替代带 ∣ ・ ∣ \|・\| ∣・∣ 和下标 A A A 的显式符号。
The **polynomial ring in m variables** associated with a countable ring A, denoted A \[ x 1 , . . . , x m \] A \[x_1,...,x_m\] A\[x1,...,xm\], is a countable ring whose elements are (codes for) finite sums of the form a i 1 , . . . , i m x 1 i 1 ・・・ x m i m a_{i_1,...,i_m} x_1\^{i_1}・・・x_m\^{i_m} ai1,...,imx1i1・・・xmim.
与可数环 A A A 关联的 **m 元多项式环** 记为 A \[ x 1 , . . . , x m \] A \[x_1,...,x_m\] A\[x1,...,xm\]，是一个可数环，其元素为形如 a i 1 , . . . , i m x 1 i 1 ・・・ x m i m a_{i_1,...,i_m} x_1\^{i_1}・・・x_m\^{i_m} ai1,...,imx1i1・・・xmim 的有限和（的编码）。
Addition, multiplication and the additive and multiplicative identities are defined as usual for polynomials.
多项式的加法、乘法以及加法单位元、乘法单位元均按经典方式定义。
A **monomial** is an element of A \[ x 1 , . . . , x m \] A \[x_1,...,x_m\] A\[x1,...,xm\] that consists of only one term and does not include the coefficient of this term, i.e., an expression of the form x 1 i 1 ・・・ x m i m x_1\^{i_1}・・・x_m\^{i_m} x1i1・・・xmim.
**单项式** 是 A \[ x 1 , . . . , x m \] A \[x_1,...,x_m\] A\[x1,...,xm\] 中仅含一项且不含系数的元素，即形如 x 1 i 1 ・・・ x m i m x_1\^{i_1}・・・x_m\^{i_m} x1i1・・・xmim 的表达式。
The**monomial ordering** is a total-ordering on the monomials given by first ordering by total degree (the sum i 1 + ・・・ + i m i_1 +・・・+ i_m i1+・・・+im) and then ordering lexicographically.
**单项式序** 是单项式上的全序，先按总次数（ i 1 + ・・・ + i m i_1 +・・・+ i_m i1+・・・+im）排序，次数相同时按字典序排序。
For any polynomial P ∈ A \[ x 1 , . . . , x m \] P \\in A \[x_1,...,x_m\] P∈A\[x1,...,xm\], the monomial that appears in P and which is greatest under this ordering is called the**leading monomial** of P.
对任意多项式 P ∈ A \[ x 1 , . . . , x m \] P \\in A \[x_1,...,x_m\] P∈A\[x1,...,xm\]，其在单项式序下最大的单项式称为 P P P 的**首单项式**。
A monomial M = x 1 i 1 ・・・ x m i m M = x_1\^{i_1}・・・x_m\^{i_m} M=x1i1・・・xmim is said to **divide** another monomial N = x 1 j 1 ・・・ x m j m N = x_1\^{j_1}・・・x_m\^{j_m} N=x1j1・・・xmjm if i k ≤ j k i_k ≤j_k ik≤jk for all k from 1 to m.
若对所有 1 ≤ k ≤ m 1≤k≤m 1≤k≤m，有 i k ≤ j k i_k ≤j_k ik≤jk，则称单项式 M = x 1 i 1 ・・・ x m i m M = x_1\^{i_1}・・・x_m\^{i_m} M=x1i1・・・xmim **整除** 单项式 N = x 1 j 1 ・・・ x m j m N = x_1\^{j_1}・・・x_m\^{j_m} N=x1j1・・・xmjm。
A countable ring A is said to be**Hilbertian** if it possesses the following property: For every sequence of elements { a k ∈ A } k ∈ N \\{a_k \\in A\\}_{k\\in\\mathbb {N}} {ak∈A}k∈N there exists a natural number N such that for all k there exist f 0 , . . . , f N ∈ A f_0,..., f_N \\in A f0,...,fN∈A such that a k = f 0 ∗ a 0 + ・・・ + f N ∗ a N a_k = f_0 \* a_0 +・・・+ f_N \* a_N ak=f0∗a0+・・・+fN∗aN.
若可数环 A A A 满足下述性质，则称其为**希尔伯特环** ：对任意元素序列 { a k ∈ A } k ∈ N \\{a_k \\in A\\}_{k\\in\\mathbb {N}} {ak∈A}k∈N，存在自然数 N N N，使得对所有 k k k，存在 f 0 , . . . , f N ∈ A f_0,..., f_N \\in A f0,...,fN∈A 满足 a k = f 0 ・ a 0 + ・・・ + f N ・ a N a_k = f_0・a_0 +・・・+ f_N・a_N ak=f0・a0+・・・+fN・aN。
In other words, a countable ring is Hilbertian iff all sequences of its elements are finitely generated.
换言之，可数环是希尔伯特环当且仅当其所有元素序列都是有限生成的。
The version of Hilbert's basis theorem considered in this chapter is a theorem which states that for all countable rings A and natural numbers m \> 0, A \[ x 1 , . . . , x m \] A \[x_1,...,x_m\] A\[x1,...,xm\] is Hilbertian.
本章所研究的希尔伯特基定理版本为：对所有可数环 A A A 和自然数 m \> 0 m \> 0 m\>0，多项式环 A \[ x 1 , . . . , x m \] A \[x_1,...,x_m\] A\[x1,...,xm\] 是希尔伯特环。
The set of ordinals up to ω ω \\omega\^\\omega ωω, denoted O, is the set of (codes for) finite sequences of natural numbers along with a special code ∣ ω ω ∣ \|\\omega\^\\omega\| ∣ωω∣, which can be arbitrary as long as it is distinct from the other elements of O.
所有小于等于 ω ω \\omega\^\\omega ωω 的序数构成的集合记为 O O O，其元素为自然数有限序列（的编码）加上一个特殊编码 ∣ ω ω ∣ \|\\omega\^\\omega\| ∣ωω∣，该特殊编码可任意选取，只需与 O O O 中其他元素相异。
The code ∣ α ∣ = ( α 0 , . . . , α n ) \|\\alpha\| = (\\alpha_0,...,\\alpha_n) ∣α∣=(α0,...,αn) is intended to represent the ordinal α = α n ∗ ω n + α n − 1 ∗ ω n − 1 + ・・・ + α 0 \\alpha = \\alpha_n \*\\omega\^n + \\alpha_{n-1} \*\\omega {n-1} +・・・+ \\alpha_0 α=αn∗ωn+αn−1∗ωn−1+・・・+α0 with ∣ ω ω ∣ \|\\omega\^\\omega\| ∣ωω∣ of course representing ω ω \\omega\^\\omega ωω.
编码 ∣ α ∣ = ( α 0 , ... , α n ) \|\\alpha\| = (\\alpha_0, \\dots, \\alpha_n) ∣α∣=(α0,...,αn) 用于表示序数 α = α n ⋅ ω n + α n − 1 ⋅ ω n − 1 + ⋯ + α 0 \\alpha = \\alpha_n \\cdot\\omega\^n + \\alpha_{n-1} \\cdot\\omega\^{n-1} + \\dots + \\alpha_0 α=αn⋅ωn+αn−1⋅ωn−1+⋯+α0，而 ∣ ω ω ∣ \|\\omega\^\\omega\| ∣ωω∣ 显然用于表示序数 ω ω \\omega\^\\omega ωω。
The ordinals in O have the usual lexicographical ordering with ω ω \\omega\^\\omega ωω as the largest element.
集合 O O O 中的序数按常规字典序排序，其中 ω ω \\omega\^\\omega ωω 是最大元素。
The **natural sum** and **natural product** of ordinals are commutative binary operations on ordinals, defined by first ordering the summands or factors from largest to smallest and then taking the sum or product as it is usually defined for ordinals.
序数的**自然和** 与**自然积**是序数上的交换二元运算，定义方式为：先将被加数或因数按从大到小排序，再按序数的常规运算规则进行求和或求积。
For example, the natural sum of α = α m ∗ ω m + α m − 1 ∗ ω m − 1 + ・・・ + α 0 \\alpha = \\alpha_m \*\\omega\^m + \\alpha_{m-1} \*\\omega\^{m-1} +・・・+ \\alpha_0 α=αm∗ωm+αm−1∗ωm−1+・・・+α0 and β = β m ∗ ω m + β m − 1 ∗ ω m − 1 + ・・・ + β 0 \\beta = \\beta_m \*\\omega\^m + \\beta_{m-1} \*\\omega\^{m-1} +・・・+ \\beta_0 β=βm∗ωm+βm−1∗ωm−1+・・・+β0 where m \> n m \> n m\>n is simply α + β = β m ∗ ω m + β m − 1 ∗ ω m − 1 + ・・・ + β n + 1 ∗ ω n + 1 + ( α n + β n ) ∗ ω n + ・・・ + α 0 + β 0 \\alpha + \\beta = \\beta_m \*\\omega\^m + \\beta_{m-1} \*\\omega\^{m-1} +・・・+ \\beta_{n+1} \*\\omega\^{n+1} + (\\alpha_n + \\beta_n) \*\\omega\^n +・・・+ \\alpha_0 + \\beta_0 α+β=βm∗ωm+βm−1∗ωm−1+・・・+βn+1∗ωn+1+(αn+βn)∗ωn+・・・+α0+β0.
例如，设 α = α m ⋅ ω m + α m − 1 ⋅ ω m − 1 + ⋯ + α 0 \\alpha = \\alpha_m \\cdot\\omega\^m + \\alpha_{m-1} \\cdot\\omega\^{m-1} + \\dots + \\alpha_0 α=αm⋅ωm+αm−1⋅ωm−1+⋯+α0， β = β m ⋅ ω m + β m − 1 ⋅ ω m − 1 + ⋯ + β 0 \\beta = \\beta_m \\cdot\\omega\^m + \\beta_{m-1} \\cdot\\omega\^{m-1} + \\dots + \\beta_0 β=βm⋅ωm+βm−1⋅ωm−1+⋯+β0 且 m \> n m \> n m\>n，则二者的自然和为 α + β = β m ⋅ ω m + β m − 1 ⋅ ω m − 1 + ⋯ + β n + 1 ⋅ ω n + 1 + ( α n + β n ) ⋅ ω n + ⋯ + α 0 + β 0 \\alpha + \\beta = \\beta_m \\cdot\\omega\^m + \\beta_{m-1} \\cdot\\omega\^{m-1} + \\dots + \\beta_{n+1} \\cdot\\omega\^{n+1} + (\\alpha_n + \\beta_n) \\cdot\\omega\^n + \\dots + \\alpha_0 + \\beta_0 α+β=βm⋅ωm+βm−1⋅ωm−1+⋯+βn+1⋅ωn+1+(αn+βn)⋅ωn+⋯+α0+β0。
To say that an ordinal η ∈ O \\eta \\in O η∈O is **well-ordered** is to say that there does not exist a sequence { η k ∈ O } k ∈ N \\{\\eta_k \\in O\\}_{k\\in\\mathbb {N}} {ηk∈O}k∈N such that η 0 = η \\eta_0 = \\eta η0=η and η k + 1 \< η k \\eta_{k+1} \< \\eta_k ηk+1\<ηk for all k k k.
若序数 η ∈ O \\eta \\in O η∈O 满足：不存在序列 { η k ∈ O } k ∈ N \\{\\eta_k \\in O\\}_{k\\in\\mathbb {N}} {ηk∈O}k∈N 使得 η 0 = η \\eta_0 = \\eta η0=η 且对所有 k k k 有 η k + 1 \< η k \\eta_{k+1} \< \\eta_k ηk+1\<ηk，则称 η \\eta η 是**良序**的。
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* 逆数学导论（3）-CSDN博客