Boolos showed in 'The Consistency of Frege's Foundations of Arithmetic' that arithmetic can be recovered from Hume's Principle without Basic Law V, confirming Basic Law V is the locus of contradiction.

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Arithmetic: Arithmetic is the branch of mathematics dealing with basic number operations—addition, subtraction, multiplication, and division. It's the foundation of math that you use in everyday life, like calculating change at a store, splitting a bill, or figuring out measurements. Essentially, it's the practical math skill everyone learns early in school to work with numbers in simple, straightforward ways.
Basic Law V(Foundational axiom of Frege's logicist program): Frege's axiom committing to the existence of a set corresponding to every predicate; also called the unrestricted Comprehension Axiom
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.'
Frege's Foundations of Arithmetic(as the historical work being discussed): A foundational work by philosopher Gottlob Frege attempting to show that all of arithmetic (basic math) can be derived from pure logic alone.

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