## 502 – Exponentiation

This is the last homework assignment of the term: Assume ${\sf CH}.$ Evaluate the cardinal number $\aleph_3^{\aleph_0},$ the size of the set of all  functions $f:\omega\to\omega_3.$

2. ${\sf CH}$ suffices, but feel free to assume ${\sf GCH}$ if it helps.
3. With GCH, I think the problem is too easy, but I don’t know how I can use just CH. I have it bounded between $\omega_3$ and $2^{\omega_3}.$
• Here is a hint: If $f:\omega\to\omega_3$ then $\sup_n f(n)<\omega_3.$ So, we can write the set of functions from $\omega$ into $\omega_3$ as a union of $\omega_3$ many sets, the $\alpha$-th one being the set of functions from $\omega$ into $\alpha.$ Now, note that $|{}^\omega\alpha|=|{}^\omega|\alpha||.$