Such a non-standard model cannot be isomorphic to the natural numbers with their successor function and zero

This means that the Löwenheim number[4] and the Hanf number[5] of the entire second-order logic are the same as those of the fragment \(\Pi^1_1\). Summing up, upon first inspection the levels \(\Sigma^1_n\) and \(\Pi^1_n\) of the hierarchy of second-order formulas grow strictly in expressive power as n increases, but a more careful analysis reveals that already the first level \(\Sigma^1_1\cup \Pi^1_1\) has the power of all the levels \(\Sigma^1_n, \Pi^1_n\) even if the power is somewhat im