# Proof: the element $0 \in \mathbb{Z}$ is unique.

If I symbolize the argument in First-Order logic, I think this would be the argument (the conclusion is an expansion of $$\exists!$$ definition):

$$\exists z \forall x(x+z=x) \vdash \exists y(\forall x(x+y=x) \land \forall z(\forall x(x+z=x) \to y=z)$$

$$\def\fitch#1#2{\quad\begin{array}{|l}#1\\\hline#2\end{array}} \def\Ae#1{\qquad\mathbf{\forall E} \: #1 \\} \def\Ai#1{\qquad\mathbf{\forall I} \: #1 \\} \def\Ee#1{\qquad\mathbf{\exists E} \: #1 \\} \def\Ei#1{\qquad\mathbf{\exists I} \: #1 \\} \def\R#1{\qquad\mathbf{R} \: #1 \\} \def\ci#1{\qquad\mathbf{\land I} \: #1 \\} \def\ce#1{\qquad\mathbf{\land E} \: #1 \\} \def\ii#1{\qquad\mathbf{\to I} \: #1 \\} \def\ie#1{\qquad\mathbf{\to E} \: #1 \\} \def\be#1{\qquad\mathbf{\leftrightarrow E} \: #1 \\} \def\bi#1{\qquad\mathbf{\leftrightarrow I} \: #1 \\} \def\qi#1{\qquad\mathbf{=I}\\} \def\qe#1{\qquad\mathbf{=E} \: #1 \\} \def\ne#1{\qquad\mathbf{\neg E} \: #1 \\} \def\ni#1{\qquad\mathbf{\neg I} \: #1 \\} \def\IP#1{\qquad\mathbf{IP} \: #1 \\} \def\x#1{\qquad\mathbf{X} \: #1 \\} \def\DNE#1{\qquad\mathbf{DNE} \: #1 \\}$$ $$\fitch{1.\, \exists z \forall x(x+z=x)}{ \fitch{2.\, a+0'=a}{ \fitch{3.\, \neg(0 = 0')}{ \fitch{4.\, \forall x(x+0=x)}{ 5.\, 0'+0=0' \Ae{3} 6.\, 0=0 \qi{} 7.\, 0+0'=0 \qe{5,2} 8.\, 0'=0 \qe{6,4} 9.\, \bot \ne{} }\\ 10.\, \bot \Ee{} }\\ 0=0' } }$$

I have as an axiom: $$\exists z \forall x(x+z=x)$$, used here as a premise.

The problem I have is that I cannot instantiate z to $$0$$ (step 4), as it occurs in a previous undischarged assumptions (step 3).

Are there any routes to complete the proof and effectively use Existential Elimination ?

EDIT: based on comments, I rewrote the proof. I am a step forward, but still cannot close $$\mathbf{\exists E}$$ proof, since both $$0$$ and $$0'$$ appear in undischarged assumptions.

EDIT 2: $$\fitch{1.\, \exists z \forall x(x+z=x)\\2.\, \forall x\forall y(x+y=y+x)}{ \fitch{2.\, \forall x(x+0'=x)}{ 3.\, 0+0'=0 \Ae{1} \fitch{4.\, ¬(0=0')}{ \fitch{5.\, \forall x(x+0=x)}{ 6.\, 0'+0=0' \Ae{4} 7.\, 0+0'=0'+0 \Ae{2} 8.\, 0+0'=0' \qe{6,5} 9.\, 0=0' \qe{3,7} 10.\, \bot \ne{4,8} }\\ 11.\, \bot \Ee{1,5-10} }\\ 12.\, 0=0' \IP{4-11} }\\ 13.\, \forall x(x+0'=x) \to 0=0' \ii{2-12} 14.\, \forall z(\forall x(x+z=x) \to 0=z) \Ai{13} 15.\, \exists y(\forall x(x+y=x) \land \forall z(\forall x(x+z=x) \to y=z) \Ei{14} }$$

• $\exists !y [P(y)]$ is $\exists y[P(y) \land \forall z[P(z) \to z = y]$. In your case $P(y)$ is $\forall x[ x + y = x ]$, so your expansion is incorrect. Apr 24, 2020 at 14:30
• @Magdiragdag deleted. You were correct. Apr 24, 2020 at 15:02
• Your conclusion should be $\exists y \forall x(x+y=x \land \forall z(\color{red}{\forall x} \ x+z=x \to y=z))$ which is equivalent to $\exists y (\forall x \ x+y=x \land \forall z(\forall x \ x+z=x \to y=z))$, which as a symbolization I like better Apr 24, 2020 at 15:22
• Thank you very much, @Magdiragdag, Bram28 and David. Could you look at my updated proof ? Apr 24, 2020 at 17:03
• You still haven't fixed the statement you have to prove. And in line 3, instead of instantiating with an arbitrary element $a$, instantiate with $0$. Then, in line 5, don't instantiate with $a$, but with $0'$. You then need commutativity of $+$ to finish. Apr 24, 2020 at 17:08

$$\forall x(x+0'=x) \land \forall z(\forall x(x+z=x) \to 0'=z)$$
... and so all that needs to go inside the subproof. Here is a proof in Fitch. The $$\forall \ I$$ rule works a little differently than yours: it is looking for a subproof where you introduce a new constant, i.e. the analogue to 'Let $$b$$ be an arbitrary object':