# Questions tagged [transfinite-induction]

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### Is strengthening Foundation in NBG sufficient to make it prove Con(ZFC)?

Can $\sf NBG$ class theory prove the foundation scheme: Foundation schema: if $\Phi(X)$ is a formula in which "$X$" occurs free and only free, and in which "$Y$" doesn't occur, ...
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### Is there a canonical mapping between countable transfinite ordinals and $\omega$? What about recursive ordinals?

Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built. I was wondering if there is a ...
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### Is there a stronger form of recursion?

I'm wondering if there are any recursion principles more general than the following, first given by Montague, Tarski and Scott (1956): Let $\mathbb{V}$ be the universe, and $\mathcal{R}$ be a well-...
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### careful exposition of transfinite recursion

Given a well ordered set W and a family of sets X(w) indexed by the elements w of W, transfinite recursion allows one often to define a function f on W that takes values f(w) in X(w). (I distinguish ...
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### Adding consistency statements to Peano arithmetic allows more instances of transfinite induction?

Consider the hierarchy given by $\cal S_0 =$ first-order Peano arithmetic, $\cal S_{\alpha+1}=\cal S_{\alpha} + Con(S_\alpha)$ (a consistency statement for $\cal S_\alpha$), and if $\alpha$ is a limit ...
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### Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...
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1 vote
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### Generalized connected components decomposition for Priestley spaces

Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal ...
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### How much of ZFC do I need to construct this cofinal, order-preserving class function?

EDIT: I'm bumping this, because while Joel ruled out some naive options, my question in bold below is not yet answered. Suppose I have a directed partially ordered set $(\Gamma,\leq)$ with a bottom ...
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### What is induction up to $\varepsilon_0$?
This is a question asked out of curiosity, and because I can't understand the Wikipedia page. I have often been told that PA cannot prove the validity of induction up to $\varepsilon_0$, which has ...