Kreisel and Boolos demonstrated that second-order consequence is not axiomatizable, meaning 'derivation from axioms' in second-order logic is inherently incomplete relative to standard semantics.

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Boolos(named as the key philosopher making an argument about logic): George Boolos was a 20th-century American philosopher and logician who developed new ways of thinking about logic and mathematics, particularly through something called 'plural logic.'
Incomplete(How Aristotle saw infinity—always in the process of continuing rather than fully done): Not finished or not whole; missing something or still in the process of becoming.
Kreisel(as a historical figure in logic and philosophy of mathematics): Georg Kreisel (1923–2015), a mathematical logician who studied how mathematical reasoning works and whether we can be certain about mathematical truths.
Second-order logic(as used in mathematical logic): A formal system that goes beyond basic logic by allowing you to quantify over (talk about) properties and relations themselves, not just individual objects.

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