感谢@ASYNC #2.0.1的贡献
以下是求1+1=2的所有方法
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第 1–10 阶：算术与集合基础
第 1 阶：直观加法（小学算术）
1+1=21 + 1 = 2 1+1=2
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第 2 阶：集合基数（直观集合论）
设
A={a}A = \{a\}A={a}，B={b}B = \{b\}B={b}，且 A∩B=∅A \cap B = \varnothingA∩B=∅，则
∣A∣=1,∣B∣=1,∣A∪B∣=2|A| = 1,\quad |B| = 1,\quad |A \cup B| = 2 ∣A∣=1,∣B∣=1,∣A∪B∣=2
于是
1+1=21 + 1 = 2 1+1=2
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第 3 阶：皮亚诺公理（PEANO ARITHMETIC）
定义：
* 1=S(0)1 = S(0)1=S(0)
* 2=S(1)2 = S(1)2=S(1)
加法递归定义：
* a+0=aa + 0 = aa+0=a
* a+S(b)=S(a+b)a + S(b) = S(a + b)a+S(b)=S(a+b)
推导：
1+1=1+S(0)=S(1+0)=S(1)=21 + 1 = 1 + S(0) = S(1 + 0) = S(1) = 2 1+1=1+S(0)=S(1+0)=S(1)=2
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第 4 阶：一阶算术（形式系统）
在语言 L={0,S,+}\mathcal{L} = \{0, S, +\}L={0,S,+} 中，使用公理：
1. a+0=aa + 0 = aa+0=a
2. a+S(b)=S(a+b)a + S(b) = S(a + b)a+S(b)=S(a+b)
可构造有限步证明：
⊢1+1=2\vdash 1 + 1 = 2 ⊢1+1=2
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第 5 阶：ZFC 集合论（冯·诺依曼自然数）
定义：
* 0=∅0 = \varnothing0=∅
* 1={0}1 = \{0\}1={0}
* 2={0,1}2 = \{0, 1\}2={0,1}
加法递归定理保证：
1+1=21 + 1 = 2 1+1=2
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第 6 阶：整数构造（Z\MATHBB{Z}Z）
将自然数嵌入整数，定义等价类：
[(a,b)]+[(c,d)]=[(a+c,b+d)][(a,b)] + [(c,d)] = [(a+c, b+d)] [(a,b)]+[(c,d)]=[(a+c,b+d)]
可得：
1Z+1Z=2Z1_{\mathbb{Z}} + 1_{\mathbb{Z}} = 2_{\mathbb{Z}} 1Z +1Z =2Z
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第 7 阶：有理数构造（Q\MATHBB{Q}Q）
11+11=21\frac{1}{1} + \frac{1}{1} = \frac{2}{1} 11 +11 =12
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第 8 阶：实数构造（DEDEKIND 截）
设 1∗1^*1∗ 为所有小于 1 的有理数集合，则
1∗+1∗=2∗1^* + 1^* = 2^* 1∗+1∗=2∗
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第 9 阶：实数域公理（ORDERED FIELD）
R\mathbb{R}R 是有序域，满足加法结合律、交换律与单位元存在，因此：
1+1=21 + 1 = 2 1+1=2
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第 10 阶：极限定义（数列）
limn→∞(1+1n)=1\lim_{n \to \infty} \left(1 + \frac{1}{n}\right) = 1 n→∞lim (1+n1 )=1
limn→∞(1+1n+1)=2\lim_{n \to \infty} \left(1 + \frac{1}{n} + 1\right) = 2 n→∞lim (1+n1 +1)=2
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第 11–20 阶：分析与积分
第 11 阶：函数加法（初等函数）
设 f(x)=1f(x) = 1f(x)=1，则
f(x)+f(x)=2f(x) + f(x) = 2 f(x)+f(x)=2
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第 12 阶：黎曼和定义（RIEMANN SUM）
区间 [0,1][0,1][0,1] 分割 P={x0,x1,…,xn}P = \{x_0, x_1, \dots, x_n\}P={x0 ,x1 ,…,xn }，取样本点 ξi\xi_iξi ：
∑i=1n1⋅(xi−xi−1)=1\sum_{i=1}^n 1 \cdot (x_i - x_{i-1}) = 1 i=1∑n 1⋅(xi −xi−1 )=1
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第 13 阶：黎曼积分定义
∫011 dx=lim∥P∥→0∑i=1n1⋅Δxi=1\int_0^1 1 \, dx = \lim_{\|P\| \to 0} \sum_{i=1}^n 1 \cdot \Delta x_i = 1 ∫01 1dx=∥P∥→0lim i=1∑n 1⋅Δxi =1
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第 14 阶：积分线性性定理（RIEMANN）
对任意可积函数 f,gf, gf,g：
∫ab(f+g) dx=∫abf dx+∫abg dx\int_a^b (f+g) \, dx = \int_a^b f \, dx + \int_a^b g \, dx ∫ab (f+g)dx=∫ab fdx+∫ab gdx
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第 15 阶：用积分表示 1+11 + 11+1
1+1=∫011 dx+∫011 dx1 + 1 = \int_0^1 1 \, dx + \int_0^1 1 \, dx 1+1=∫01 1dx+∫01 1dx
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第 16 阶：积分合并
∫011 dx+∫011 dx=∫01(1+1) dx\int_0^1 1 \, dx + \int_0^1 1 \, dx = \int_0^1 (1 + 1) \, dx ∫01 1dx+∫01 1dx=∫01 (1+1)dx
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第 17 阶：积分计算
∫01(1+1) dx=∫012 dx=2\int_0^1 (1 + 1) \, dx = \int_0^1 2 \, dx = 2 ∫01 (1+1)dx=∫01 2dx=2
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第 18 阶：LEBESGUE 积分（测度论）
在测度空间 ([0,1],B,μ)([0,1], \mathcal{B}, \mu)([0,1],B,μ) 上：
∫[0,1]1 dμ=1\int_{[0,1]} 1 \, d\mu = 1 ∫[0,1] 1dμ=1
线性性仍成立，故：
1+1=21 + 1 = 2 1+1=2
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第 19 阶：L1L^1L1 空间（泛函分析）
L1([0,1])L^1([0,1])L1([0,1]) 是线性空间，范数：
∥f∥1=∫01∣f(x)∣ dx\|f\|_1 = \int_0^1 |f(x)| \, dx ∥f∥1 =∫01 ∣f(x)∣dx
对 f(x)=1f(x) = 1f(x)=1：
∥1+1∥1=2\|1 + 1\|_1 = 2 ∥1+1∥1 =2
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第 20 阶：分布 / 广义函数
常数函数 111 是分布，作用在函数 φ\varphiφ 上：
⟨1,φ⟩=∫011⋅φ(x) dx\langle 1, \varphi \rangle = \int_0^1 1 \cdot \varphi(x) \, dx ⟨1,φ⟩=∫01 1⋅φ(x)dx
线性性：
⟨1+1,φ⟩=2⟨1,φ⟩\langle 1 + 1, \varphi \rangle = 2 \langle 1, \varphi \rangle ⟨1+1,φ⟩=2⟨1,φ⟩
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第 21–30 阶：测度与概率
第 21 阶：Σ-代数
F 是 Ω 的子集构成的 σ-代数\mathcal{F} \text{ 是 } \Omega \text{ 的子集构成的 σ-代数} F 是 Ω 的子集构成的 σ-代数
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第 22 阶：测度定义
μ(⋃i=1∞Ai)=∑i=1∞μ(Ai)\mu\left(\bigcup_{i=1}^\infty A_i\right) = \sum_{i=1}^\infty \mu(A_i) μ(i=1⋃∞ Ai )=i=1∑∞ μ(Ai )
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第 23 阶：LEBESGUE 测度
λ([0,1])=1\lambda([0,1]) = 1 λ([0,1])=1
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第 24 阶：可测函数
f−1(B)∈F,∀B∈Bf^{-1}(B) \in \mathcal{F},\quad \forall B \in \mathcal{B} f−1(B)∈F,∀B∈B
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第 25 阶：简单函数逼近
s(x)=∑i=1naiχAi(x)s(x) = \sum_{i=1}^n a_i \chi_{A_i}(x) s(x)=i=1∑n ai χAi (x)
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第 26 阶：LEBESGUE 积分定义
∫f dμ=sup{∫s dμ:s≤f}\int f \, d\mu = \sup\left\{ \int s \, d\mu : s \le f \right\} ∫fdμ=sup{∫sdμ:s≤f}
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第 27 阶：积分单调收敛定理
若 0≤fn↑f0 \le f_n \uparrow f0≤fn ↑f，则
∫fn dμ↑∫f dμ\int f_n \, d\mu \uparrow \int f \, d\mu ∫fn dμ↑∫fdμ
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第 28 阶：FATOU 引理
∫lim infn→∞fn dμ≤lim infn→∞∫fn dμ\int \liminf_{n\to\infty} f_n \, d\mu \le \liminf_{n\to\infty} \int f_n \, d\mu ∫n→∞liminf fn dμ≤n→∞liminf ∫fn dμ
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第 29 阶：控制收敛定理
若存在 ggg 使得 ∣fn∣≤g|f_n| \le g∣fn ∣≤g，则
limn→∞∫fn dμ=∫limn→∞fn dμ\lim_{n\to\infty} \int f_n \, d\mu = \int \lim_{n\to\infty} f_n \, d\mu n→∞lim ∫fn dμ=∫n→∞lim fn dμ
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第 30 阶：概率空间
(Ω,F,P),P(Ω)=1(\Omega, \mathcal{F}, P),\quad P(\Omega) = 1 (Ω,F,P),P(Ω)=1
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第 31–40 阶：泛函与算子
第 31 阶：赋范空间
∥x+y∥≤∥x∥+∥y∥\|x + y\| \le \|x\| + \|y\| ∥x+y∥≤∥x∥+∥y∥
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第 32 阶：巴拿赫空间
每一个 Cauchy 序列在 X 中收敛\text{每一个 Cauchy 序列在 } X \text{ 中收敛} 每一个 Cauchy 序列在 X 中收敛
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第 33 阶：线性算子
T(ax+by)=aT(x)+bT(y)T(ax + by) = aT(x) + bT(y) T(ax+by)=aT(x)+bT(y)
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第 34 阶：对偶空间
X∗={f:X→R∣f 有界线性}X^* = \{ f: X \to \mathbb{R} \mid f \text{ 有界线性} \} X∗={f:X→R∣f 有界线性}
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第 35 阶：里斯表示定理（HILBERT）
f(x)=⟨x,y⟩,∃y∈Hf(x) = \langle x, y \rangle,\quad \exists y \in H f(x)=⟨x,y⟩,∃y∈H
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第 36 阶：索伯列夫空间
∥u∥Wk,p=(∑∣α∣≤k∥Dαu∥pp)1/p\|u\|_{W^{k,p}} = \left( \sum_{|\alpha| \le k} \|D^\alpha u\|_p^p \right)^{1/p} ∥u∥Wk,p = ∣α∣≤k∑ ∥Dαu∥pp 1/p
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第 37 阶：弱收敛
xn⇀x ⟺ f(xn)→f(x), ∀f∈X∗x_n \rightharpoonup x \iff f(x_n) \to f(x),\ \forall f \in X^* xn ⇀x⟺f(xn )→f(x), ∀f∈X∗
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第 38 阶：弱*收敛
xn⇀∗x ⟺ xn(x)→x(x), ∀x∈Xx_n \overset{*}{\rightharpoonup} x \iff x_n(x) \to x(x),\ \forall x \in X xn ⇀∗x⟺xn (x)→x(x), ∀x∈X
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第 39 阶：哈恩–巴拿赫定理
线性泛函可保范延拓\text{线性泛函可保范延拓} 线性泛函可保范延拓
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第 40 阶：一致有界原理
supn∥Tnx∥<∞, ∀x ⟹ supn∥Tn∥<∞\sup_n \|T_n x\| < \infty,\ \forall x \implies \sup_n \|T_n\| < \infty nsup ∥Tn x∥<∞, ∀x⟹nsup ∥Tn ∥<∞
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第 41–50 阶：流形与几何
第 41 阶：拓扑流形
局部同胚于 Rn\text{局部同胚于 } \mathbb{R}^n 局部同胚于 Rn
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第 42 阶：光滑结构
φα∘φβ−1∈C∞\varphi_\alpha \circ \varphi_\beta^{-1} \in C^\infty φα ∘φβ−1 ∈C∞
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第 43 阶：切空间
TpM={在 p 处的导子}T_pM = \{ \text{在 } p \text{ 处的导子} \} Tp M={在 p 处的导子}
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第 44 阶：余切丛
T∗M=⋃p∈MTp∗MT^*M = \bigcup_{p \in M} T_p^*M T∗M=p∈M⋃ Tp∗ M
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第 45 阶：微分形式
ω=∑ai1⋯ikdxi1∧⋯∧dxik\omega = \sum a_{i_1\cdots i_k} dx^{i_1} \wedge \cdots \wedge dx^{i_k} ω=∑ai1 ⋯ik dxi1 ∧⋯∧dxik
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第 46 阶：外微分
d(dω)=0d(d\omega) = 0 d(dω)=0
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第 47 阶：斯托克斯定理
∫Mdω=∫∂Mω\int_M d\omega = \int_{\partial M} \omega ∫M dω=∫∂M ω
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第 48 阶：黎曼度量
gp(u,v)=⟨u,v⟩pg_p(u,v) = \langle u, v \rangle_p gp (u,v)=⟨u,v⟩p
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第 49 阶：测地线方程
∇γ˙γ˙=0\nabla_{\dot{\gamma}} \dot{\gamma} = 0 ∇γ˙ γ˙ =0
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第 50 阶：高斯–博内定理
∫MK dA=2πχ(M)\int_M K \, dA = 2\pi \chi(M) ∫M KdA=2πχ(M)
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第 51–60 阶：复分析与调和
第 51 阶：全纯函数
∂f∂zˉ=0\frac{\partial f}{\partial \bar{z}} = 0 ∂zˉ∂f =0
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第 52 阶：柯西–黎曼方程
∂u∂x=∂v∂y,∂u∂y=−∂v∂x\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y},\quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x} ∂x∂u =∂y∂v ,∂y∂u =−∂x∂v
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第 53 阶：柯西积分公式
f(z0)=12πi∮γf(z)z−z0 dzf(z_0) = \frac{1}{2\pi i} \oint_\gamma \frac{f(z)}{z - z_0} \, dz f(z0 )=2πi1 ∮γ z−z0 f(z) dz
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第 54 阶：留数定理
∮γf(z) dz=2πi∑Res(f,zk)\oint_\gamma f(z)\,dz = 2\pi i \sum \operatorname{Res}(f, z_k) ∮γ f(z)dz=2πi∑Res(f,zk )
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第 55 阶：调和函数
Δu=uxx+uyy=0\Delta u = u_{xx} + u_{yy} = 0 Δu=uxx +uyy =0
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第 56 阶：泊松积分公式
u(r,θ)=12π∫02πPr(θ−ϕ)u(ϕ) dϕu(r,\theta) = \frac{1}{2\pi} \int_0^{2\pi} P_r(\theta - \phi) u(\phi)\,d\phi u(r,θ)=2π1 ∫02π Pr (θ−ϕ)u(ϕ)dϕ
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第 57 阶：共形映射
f 全纯且 f′≠0f \text{ 全纯且 } f' \neq 0 f 全纯且 f′=0
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第 58 阶：黎曼映射定理
单连通区域≅D\text{单连通区域} \cong \mathbb{D} 单连通区域≅D
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第 59 阶：椭圆函数
f(z+ω1)=f(z),f(z+ω2)=f(z)f(z + \omega_1) = f(z),\quad f(z + \omega_2) = f(z) f(z+ω1 )=f(z),f(z+ω2 )=f(z)
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第 60 阶：模形式
f(az+bcz+d)=(cz+d)kf(z)f\left( \frac{az + b}{cz + d} \right) = (cz + d)^k f(z) f(cz+daz+b )=(cz+d)kf(z)
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第 61–70 阶：概率与随机过程
第 61 阶：随机变量
X:Ω→R,X−1(B)∈FX: \Omega \to \mathbb{R},\quad X^{-1}(B) \in \mathcal{F} X:Ω→R,X−1(B)∈F
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第 62 阶：数学期望
E[X]=∫ΩX dPE[X] = \int_\Omega X \, dP E[X]=∫Ω XdP
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第 63 阶：大数定律
Xˉn→a.s.E[X]\bar{X}_n \xrightarrow{a.s.} E[X] Xˉn a.s. E[X]
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第 64 阶：中心极限定理
n(Xˉn−μ)→dN(0,σ2)\sqrt{n}(\bar{X}_n - \mu) \xrightarrow{d} N(0, \sigma^2) n (Xˉn −μ)d N(0,σ2)
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第 65 阶：布朗运动
Bt∼N(0,t),独立增量B_t \sim N(0,t),\quad \text{独立增量} Bt ∼N(0,t),独立增量
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第 66 阶：伊藤积分
It=∫0tHs dBsI_t = \int_0^t H_s \, dB_s It =∫0t Hs dBs
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第 67 阶：伊藤引理
df(Bt)=f′(Bt)dBt+12f′′(Bt)dtdf(B_t) = f'(B_t) dB_t + \frac12 f''(B_t) dt df(Bt )=f′(Bt )dBt +21 f′′(Bt )dt
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第 68 阶：随机微分方程
dXt=μdt+σdBtdX_t = \mu dt + \sigma dB_t dXt =μdt+σdBt
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第 69 阶：费曼–卡茨公式
u(t,x)=E[f(XTx)]u(t,x) = E[f(X_T^x)] u(t,x)=E[f(XTx )]
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第 70 阶：马尔可夫过程
P(Xt+s∈A∣Ft)=P(Xt+s∈A∣Xt)P(X_{t+s} \in A \mid \mathcal{F}_t) = P(X_{t+s} \in A \mid X_t) P(Xt+s ∈A∣Ft )=P(Xt+s ∈A∣Xt )
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第 71–80 阶：偏微分方程
第 71 阶：拉普拉斯方程
Δu=0\Delta u = 0 Δu=0
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第 72 阶：热方程
ut=Δuu_t = \Delta u ut =Δu
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第 73 阶：波动方程
utt=c2Δuu_{tt} = c^2 \Delta u utt =c2Δu
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第 74 阶：弱解
∫Ω∇u⋅∇v dx=∫Ωfv dx\int_\Omega \nabla u \cdot \nabla v \, dx = \int_\Omega f v \, dx ∫Ω ∇u⋅∇vdx=∫Ω fvdx
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第 75 阶：变分法
minu∫Ω∣∇u∣2dx\min_u \int_\Omega |\nabla u|^2 dx umin ∫Ω ∣∇u∣2dx
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第 76 阶：欧拉–拉格朗日方程
∂L∂u−ddt∂L∂ut=0\frac{\partial L}{\partial u} - \frac{d}{dt} \frac{\partial L}{\partial u_t} = 0 ∂u∂L −dtd ∂ut ∂L =0
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第 77 阶：索伯列夫嵌入
Wk,p(Ω)↪Cm(Ω‾)W^{k,p}(\Omega) \hookrightarrow C^m(\overline{\Omega}) Wk,p(Ω)↪Cm(Ω)
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第 78 阶：正则性理论
弱解 ⟹ 光滑解\text{弱解} \implies \text{光滑解} 弱解⟹光滑解
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第 79 阶：最大值原理
maxΩ‾u=max∂Ωu\max_{\overline{\Omega}} u = \max_{\partial \Omega} u Ωmax u=∂Ωmax u
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第 80 阶：边值问题
−Δu=f,u∣∂Ω=0-\Delta u = f,\quad u|_{\partial\Omega} = 0 −Δu=f,u∣∂Ω =0
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第 81–90 阶：代数拓扑
第 81 阶：基本群
π1(X,x0)=[环路]/同伦\pi_1(X, x_0) = [\text{环路}] / \text{同伦} π1 (X,x0 )=[环路]/同伦
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第 82 阶：覆盖空间
p:X~→X,p 局部同胚p: \tilde{X} \to X,\quad p \text{ 局部同胚} p:X~→X,p 局部同胚
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第 83 阶：塞弗特–范坎彭定理
π1(X∪Y)=π1(X)∗π1(X∩Y)π1(Y)\pi_1(X \cup Y) = \pi_1(X) *_{\pi_1(X\cap Y)} \pi_1(Y) π1 (X∪Y)=π1 (X)∗π1 (X∩Y) π1 (Y)
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第 84 阶：单纯同调
Hn(X)=Hn(C∙(X))H_n(X) = H_n(C_\bullet(X)) Hn (X)=Hn (C∙ (X))
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第 85 阶：奇异同调
Hnsing(X)=Hn(S∙(X))H_n^{\text{sing}}(X) = H_n(S_\bullet(X)) Hnsing (X)=Hn (S∙ (X))
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第 86 阶：长正合序列
⋯→Hn(A)→Hn(X)→Hn(X,A)→Hn−1(A)→⋯\cdots \to H_n(A) \to H_n(X) \to H_n(X,A) \to H_{n-1}(A) \to \cdots ⋯→Hn (A)→Hn (X)→Hn (X,A)→Hn−1 (A)→⋯
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第 87 阶：切除定理
Hn(X∖U)≅Hn(X),U 小H_n(X \setminus U) \cong H_n(X),\quad U \text{ 小} Hn (X∖U)≅Hn (X),U 小
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第 88 阶：庞加莱对偶
Hk(M)≅Hn−k(M)H^k(M) \cong H_{n-k}(M) Hk(M)≅Hn−k (M)
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第 89 阶：德拉姆上同调
HdRk(M)={闭 k-形式}{恰当形式}H^k_{\text{dR}}(M) = \frac{\{\text{闭 }k\text{-形式}\}}{\{\text{恰当形式}\}} HdRk (M)={恰当形式}{闭 k-形式}
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第 90 阶：霍奇分解
Hk=Hk⊕dΩk−1⊕d∗Ωk+1H^k = \mathcal{H}^k \oplus d\Omega^{k-1} \oplus d^*\Omega^{k+1} Hk=Hk⊕dΩk−1⊕d∗Ωk+1
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第 91–100 阶：现代物理与终极统一
第 91 阶：哈密顿力学
q˙=∂H∂p,p˙=−∂H∂q\dot{q} = \frac{\partial H}{\partial p},\quad \dot{p} = -\frac{\partial H}{\partial q} q˙ =∂p∂H ,p˙ =−∂q∂H
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第 92 阶：拉格朗日力学
L=T−V,S=∫L dtL = T - V,\quad S = \int L \, dt L=T−V,S=∫Ldt
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第 93 阶：最小作用量原理
δS=0\delta S = 0 δS=0
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第 94 阶：薛定谔方程
iℏ∂tψ=H^ψi\hbar \partial_t \psi = \hat{H} \psi iℏ∂t ψ=H^ψ
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第 95 阶：路径积分
⟨qf∣e−iHt/ℏ∣qi⟩=∫Dq eiS[q]/ℏ\langle q_f | e^{-iHt/\hbar} | q_i \rangle = \int \mathcal{D}q\, e^{iS[q]/\hbar} ⟨qf ∣e−iHt/ℏ∣qi ⟩=∫DqeiS[q]/ℏ
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第 96 阶：量子场论
ϕ^(x) 满足对易关系\hat{\phi}(x) \text{ 满足对易关系} ϕ^ (x) 满足对易关系
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第 97 阶：重整化群
β(g)=μdgdμ\beta(g) = \mu \frac{dg}{d\mu} β(g)=μdμdg
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第 98 阶：杨–米尔斯方程
DμFμν=0D_\mu F^{\mu\nu} = 0 Dμ Fμν=0
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第 99 阶：爱因斯坦场方程
Gμν=8πGTμνG_{\mu\nu} = 8\pi G T_{\mu\nu} Gμν =8πGTμν
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第 100 阶：终极统一视角
在所有自洽的物理理论中，经典极限下仍有1+1=2\text{在所有自洽的物理理论中，经典极限下仍有} \quad \boxed{1 + 1 = 2} 在所有自洽的物理理论中，经典极限下仍有1+1=2
