Homework Problems 9 & 10 J.O. - starting to learn some LaTeX at least!

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:

1. $\phi\vdash\phi$ by axiom
2. $\left(\Sigma\vdash\left(\phi\rightarrow\psi\right)\right)\wedge\left(\phi\vdash\left(\Sigma\rightarrow\psi\right)\right)$ by $\rightarrow$I
3. $\Sigma\vdash\phi$ by $\rightarrow$E
4. $\phi\vdash\Sigma$ 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):

1. $\vdash\left(x=x\right)$ by =I
2. $\left[y|x\right]\left(y=y\right)$ by $\exists I$
3. 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 $z\neq x$

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