For which classes of functions this inverse function formula gives a closed form expression?

Lets consider this method of finding inverse function: $$f^{-1}(x) = \sum_{k=0}^\infty A_k(x) \frac{(x-f(x))^k}{k!}$$ where coefficients $A_k(x)$ recursively defined as $$\begin{cases} A_0(x)=x \\ A_{n+1}(x)=\frac{A_n'(x)}{f'(x)}\end{cases}$$ It is evident that for some classes of functions starting from some point $A_k(x)$ becomes zero and thus the inverse function can be expressed in closed form. For example, the expression has … Read more

Induction for quantum group

I am confused about a claim in the article Representation of quantum algebras by Henning Haahr Andersen, Patrick Polo and Wen Kexin. I probably misunderstood a definition, but I found two claims about the induction functor that seems contradictory to me. Let Uq be Lusztig version of the usual Drinfeld-Jimbo quantum group associated to a … Read more

Mathematical induction vis-a-vis primes

One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone sequence 2=p1<p2<⋯<pn<⋯. But, knowing the prime pn does not tell us the exact location of the next. My inquiry here is this: … Read more

Why restrict to Σ01\Sigma_1^0 formulas in RCA0RCA_0 induction?

I recently asked this question over on, warmly welcomed by crickets. I hope it’s appropriate here. I’m reading Stillwell’s Reverse Mathematics, and the induction axiom was just introduced. For a Σ01 formula ϕ, [ϕ(0)∧∀n(ϕ(n)⇒ϕ(n+1))]⇒∀nϕ(n) I’m wondering why we restrict to Σ01. All that he offers about the decision is in a footnote Σ01 induction … Read more

Inductive Definitions in Category Theory

I’m trying to pin down a notion of inductive definability in category-theoretic terms. The sorts of inductively defined sets (and classes) I’m most interested in are those that admit of induction and recursion. So far, I’ve noticed that (weakly?) initial algebras of an endofunctor do a pretty good job of this. We can capture natural … Read more

Limit of alternated row and column normalizations

Let E0 be a matrix with non-negative entries. Given En, we apply the following two operations in sequence to produce En+1. A. Divide every entry by the sum of all entries in its column (to make the matrix column-stochastic). B. Divide every entry by the sum of all entries in its row (to make the … Read more

Is ∈\in-induction provable in first order Zermelo set theory?

Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which ∈-induction fail? I asked this question on Mathematics Stack Exchange here 4 days ago, to receive no answer yet, I thought this question already has a well known answer that is somehow … Read more

Cases where multiple induction steps are provably required

I am looking for references for theorems of the form: 1) Any proof of theorem X requires n applications of induction axioms and especially 2) Any proof of theorem X requires n nested applications of induction axioms. I’ve seen similar statements for applications of axioms other than induction axioms (and I’d be happy to receive … Read more

Independent/Easy fraction of sentences over PA

Let S(n) be the set of all sentences over PA of length at most n (counting the quantifier symbols, boolean connectives, arithmetic operations and constants, and counting each variable as length 1). Let I(n) = \{ φ : φ \in S(n) \land \text{$φ$ is independent of PA} \}. Let E(n) = \{ φ : φ … Read more