# Constructing Baire functions of some class which are not of the previous class

How to construct a real Baire function of class $\alpha$ which is not in the class $\alpha-1$, where $\alpha\in \mathbb{N}$? I believe the solution to this problem is equivalent to constructing a $\Sigma_{\alpha+1}^0$ set which is not in $\Sigma_{\alpha}^0$ (using characteristic functions), any such construction in $\mathbb{R}$?

• Kechris's book "Classical Descriptive Set Theory" is a good reference for this sort of thing – Trevor Wilson Mar 2 '13 at 17:23
• One of the standard arguments for producing sets in $\Sigma_{\alpha+1}^0 \setminus \Sigma_\alpha^0$ is using universal sets and diagonalization. See Theorems 22.3 and 22.4 in Kechris's book. – Martin Mar 3 '13 at 8:54
• A recent post on MathOverflow on the same topic: Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$. – Martin Sleziak May 31 '18 at 10:33