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Solutions to HW 10 are also available.

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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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First argue that it suffices to prove the result for of the form . Then proceed by induction: Assume is a permutation of that does not contain a monotone -AP. Consider the sequence .

**Problem 2**

Choose and a prime number . Consider the set

**Problem 3**

Find a way how you can identify with , so that a line in can be identified with an -dimensional subspace of . Then apply the Hales-Jewett Theorem.

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Scott Aaronson’s “Who can name the bigger number” is a really nice introduction to fast-growing functions.

Finally, here is the promised photo from Niagara Falls

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