2nd order logic and mathematics?

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Most mathematics primarily utilizes first-order logic, although many theories can be expressed in second-order logic. Studying the foundations of mathematics typically begins with 0th and first-order logic before addressing second-order logic. While second-order logic can encapsulate complex theories, it is often viewed as embedded within first-order logic. The Zermelo-Fraenkel (ZF) set theory, formulated in first-order logic, contains infinitely many axioms, suggesting that its second-order version may not be significantly more complex. Understanding the interplay between these logical levels is essential for grasping the foundations of mathematics.
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Does most of mathematics use 2nd order logic? If so would studying the foundations of mathematics involve mostly using 2nd order logic?
 
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Studying the foundations of logic usually starts with 0th level and moves to first, then either touches on second and returns to first or 'graduates' to second.

There's value in simplicity, so while many theories can be expressed in second-order logic I generally see them as being embedded in first-order logic. But there's no escaping real complexity; ZF in first-order logic has infinitely many axioms, so perhaps one would argue that its second-order formulation is 'no worse'.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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