Unreachability of ${\mathbf{\Gamma }}_{2n+1,\mathbf{m}}$

Abstract

We find bounds for the maximal length of a sequence of distinct ${\mathbf{\Gamma }}_{2n+1,\mathbf{m}}$-sets under $AD$ and show there is no sequence of distinct ${\mathbf{\Gamma }}_{2n+1}$-sets of length ${\delta }_{2n+3}^1$. As a special case, there is no sequence of distinct ${\mathbf{\Gamma }}_{1,\mathbf{m}}$-sets of length ${\aleph }_{m+2}$. These are the optimal results for the pointclasses ${\mathbf{\Gamma }}_{2n+1}$ and ${\mathbf{\Gamma }}_{1,\mathbf{m}}$.