As I discuss in my answer here, it seems to me that Solomon Feferman shows, on pages 10-11 of his seminal 1964 paper "Systems of Predicative Analysis", that if you consider predicative second-order ...
Around page 300 of his book "Mathematical Thought and its Objects", Charles Parsons discusses the work of Edward Nelson, who believes that mathematical induction is impredicative, because it can be ...
A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...
I am trying to understand why induction up to exactly $\epsilon_0$ is necessary to prove the cut-elimination theorem for first-order Peano Arithmetic; or, as I understand, equivalently, why the length ...
I'm quite sure I don't understand very well the links between proof theoretical ordinals of theories, the axioms of transfinite induction and the objects a theory can prove to exist. For instance I'm ...
If we take Peano Arithmetic and restrict induction to formulas over various fragments of the arithmetic hierarchy, say to the $\Sigma^0_n$ formulas for various $n$ or some other interesting fragments, ...