# Questions tagged [reverse-math]

The general enterprise of calibrating the strength of classical mathematical theorems in terms of the axioms, typically of set existence, needed to prove them; originated in its modern form in the 1970s by H. Friedman and S. G. Simpson (see R.A. Shore, "Reverse Mathematics: The Playground of Logic", 2010).

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### What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?

There have been a couple of recent questions, here and here, regarding the role of the axiom of choice in real-analytic results with applicability to general relativity. This lead me to look at some ...
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### Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC+$\lnot$CH?

We work in ZFC. Let $C(X)$ be the ring of continuous functions $f:X\to\mathbb{R}$, and $M$ a maximal ideal. We call $C(X)/M$ a hyperreal field if it's not isomorphic to $\mathbb{R}$. A field $E$ is ...
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### Coding fourth-order objects in second-order Reverse Mathematics

Reverse Mathematics (RM for short) generally takes place in the language of second-order arithmetic. Thus, higher-order objects need to be "coded" or "represented" indirectly. ...
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### How much determinacy do you need for second order arithmetic to be as strong as ZFC?

From Wikipedia (I couldn't find the original source): $\text{ZFC} + \{\text{there are$n$Woodin cardinals:$n$is a natural number}\}$ is conservative over $\text{Z}_2$ with projective determinacy. ...
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### Entailment in one-point extensions of standard-enough models

This is one of two questions about the power of "one-point extensions" in reverse mathematics. This one focuses on what separations can be achieved as one-point extensions of as-closed-as-...
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### What is the meaning and proof of Harvey Friedman’s ultrafinite incompleteness sentence?

On page 7 of his paper “Adventures in Incompleteness”, Harvey Friedman states the following: IN ANY LONG ENOUGH SEQUENCE $x_1,...,x_n$ FROM $\{1,2,3\}$, SOME $(x_i,...,x_{2i})$ IS A SUBSEQUENCE OF ...
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### Is the IVT internally true in Johnstone's topological topos?

By IVT, I mean that for any continuous function $f:[0,1]\to\mathbb R$ for which $f(0)\leq 0 \leq f(1)$, there is a $t \in [0,1]$ for which $f(t)=0$. I don't mean any "constructive" ...
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### Does $\text{ACA}_0$ + True Arithmetic prove the well-foundedness of every recursive ordinal?

As discussed in Noah Schweber's answer to What is the proof-theoretic ordinal of true arithmetic?, it is somewhat ambiguous what “the proof-theoretic ordinal of True Arithmetic” might mean. In one ...
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### Is the Intermediate Value Theorem strictly stronger than LLPO?

(The context is Intuitionistic ZF set theory, or HoTT, or the internal logic of a topos with a Natural Number Object. The real numbers here mean the Dedekind reals.) By LLPO, I mean the statement that ...
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### What is the strength of “if $c≥0$ then $[0,c] = c·[0,1]$” in constructive math (w.r.t., LPO, WLPO, LLPO, etc.)?

Context: This question is about constructive mathematics, such as in the internal logic of a topos with natural numbers object, or in IZF. (I wish to avoid the axiom of countable choice if possible, ...
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### When more is less in logic

I am looking for examples of theorems where adding a 'trivial' extra condition makes the theorem provable in weaker systems. By 'trivial' I mean that the extra condition is trivial in strong enough ...
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### Reverse mathematics of Banach-Mazur games

Given $\mathcal{A}\subseteq\omega^\omega$, the Banach-Mazur game with payoff set $\mathcal{A}$ consists of players $1$ and $2$ alternately playing nonempty finite strings of naturals with player $1$ ...
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### Enumerating unions of arithmetical sets

In Simpsons's excellent Subsystems of Second-order Arithmetic, we find V.4.10 which tells us the following: The following is provable in ATR$_0$. Let $(A_n)_{n\in \mathbb{N}}$ be a sequence of ...
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### Two questions regarding the reverse mathematics of Siegel's lemma

In a short proof of the Roth theorem regarding the rational approximation of algebraic reals I found online (which made use of Siegel's Lemma), it was stated that "Siegel's lemma is a corollary ...
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### Independence of $\Pi^1_1$-induction from ATR$_0$

Is it known that $\Pi^1_1$-induction is independent of ATR$_0$? Simpson's book shows this for $\Pi^1_1$ transfinite induction ($\Pi^1_1$-TI), but I'm only interested in inducting on $\omega$. I can ...
The monadic theory of the real line is the set of all sentences in the monadic second-order language of order which are true in $\mathbb{R}$. In this 1982 paper, Gurevich and Shelah show that true ...