# Hints Homework 10

**Problem 2**

As we saw in class, you can define the relation using the formula . Furthermore, given any number , we can represent it as , where we apply -times.

Now show that .

**Problem 3**

For any *specific* finite number , you can find a formula that says: “There exist *exactly* distinct elements”.

**Problem 4**

One direction is trivial. For the other direction, use the completeness theorem. Argue that a theory is consistent if and only if it has a model. Then use the fact that proofs are finite, in particular can use at most finitely many formulas.

**Problem 5**

Extend the language of arithmetic by a new constant symbol . Add sentences of the form , , , … to . Apply the compactness theorem.

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