9. Using our logical axioms and structural rules as discussed in class (but not the axioms of ZFC), give a formal proof that equality is symmetric: i.e., prove ⊢∀x∀y(x=y→y=x) in our formal system.

Pf:

- $\phi\vdash\phi$ by axiom
- $\left(\Sigma\vdash\left(\phi\rightarrow\psi\right)\right)\wedge\left(\phi\vdash\left(\Sigma\rightarrow\psi\right)\right)$ by $\rightarrow$I
- $\Sigma\vdash\phi$ by $\rightarrow$E
- $\phi\vdash\Sigma$ by weakening

10 a. Using our logical axioms and structural rules as discussed in class (but not the axioms of ZFC), give a formal proof of ⊢∃x(x=x). You could say that our logic assumes that at least one thing exists.

b. Can you prove that there exist at least two things? (If nothing else, give an unofficial wff expressing that there exist at least two things.)

Pf of (a):

- $\vdash\left(x=x\right)$ by =I
- $\left[y|x\right]\left(y=y\right)$ by $\exists I$
- thus $\exists x\left(x=x\right)$

Pf of (b):

Unofficial WFF version: $\exists x\exists y \left(x\neq y\right)$

Given (a), must do the same for another element $z\neq x$

- $\vdash\left(z=z\right)$ by =I
- $\left[y|z\right]\left(y=y\right)$ by $\exists I$
- $\text{since} x\neq z, \text{then there exist two separate things}$

Comments - Andrew Furash

You do something very general in 9 which is not true. You should have just stuck to the notation given in the prompt.

10A is great, but 10B is incorrect. There could be a universe of only one thing.