# Questions tagged [transfinite-recursion]

Questions dealing with set-theoretic functions defined by transfinite recursion.

90 questions
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### A Proof of Transfinite Recursion Theorem

This proof takes me a huge amount of time to formulate, so I hope that someone can help me verify it. There're possibly subtle mistakes that I'm unable to recognize. Thank you for your dedicated help! ...
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### Strong induction with recursive definition function

Look at following recursive function definition for function $F :\mathbb{N}​\times\mathbb{N}​ \to \mathbb{N}$​: $$\begin{split} F(x,0) & = 0\\ F(x,n) & = x + F(x, n-1) \end{split}$$ Prove ...
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### Is Rudy Rucker's Theta (θ) the same as Θ and also the same as the omega-fixed point (the first aleph-fixed point) and the first beth-fixed point?

Apologies for the long title! So I've been reading Rucker's Infinity and the Mind and came across this: $$\theta = \aleph_{\aleph_{\aleph_{\aleph_{\ddots}}}}$$ Previously, theta was defined by ...
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### Prove equivalence of Axiom of choice and Zorn's lemma

I have the following proof and there are some steps I do not understand. Can anyone explain to me what is going on? Claim: (AC) $\implies$ Zorn's lemma Proof: Assume that $S$ is a partial ordered ...
### Proving that Monotone Convergence implies Least Upper Bound in $\mathbb{R}$.
I tried proving that Every bounded increasing sequence converges in $\mathbb{R}$. implies that $\mathbb{R}$ has the least upper bound. Here, $\mathbb{R}$ is taken as an ordered field which ...
Given the level $N$ at which a node $X$ is located in a binary tree, to search for node $X$ according to level-order traversal, we can use the knowledge of level $N$ where $X$ is located to narrow our ...