# Tagged Questions

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### Induction on Primitive Recursive Function

The set $F_{n}$ of primitive recursive function symbols of arity $n$ can be defined inductively as: \begin{align} & Z, \text{Succ} \in F_{1} & \\ &\pi_{j}^{n} \in F_{n} \quad \text{for ...
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### Proofs with Induction Imply Proofs Without Induction?

Assume we can prove $\forall x P(x)$ in first order Peano Arithmetic (PA) using induction and modus ponens. Does this mean we can prove $\forall x P(x)$ from the other axioms of PA without using ...
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### Why do sequences exist? What does “constructing a sequence” mean formally?

Everybody knows arguments like: "We can construct such a sequence inductively. Let $a_0$ be chosen as [..]. Then we can choose $a_{k+1}$ out of the set $A_{k+1}$ (which was shown to be non-empty)." ...
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### is it possible to prove the method of mathematical induction itself?

Since the method of mathematical induction follows some sort of 'algorithm', would the method itself be provable? namely, give that the method of mathematical induction is as follows: if S is a ...
Is there any examples of statements P(n) such that "for all $n>1$, P(n)" is provable, but P(n)=>P(n+1) is not provable? (without using some mild deformation of "for all $n>1$, ...
Following on from my question about $\omega$-models, I'm interested in the interaction between subsystems of second order arithmetic with restricted induction such as $\mathsf{RCA}_0$ and those which ...