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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How strong is exponentiation with only open induction? (Or: "how low can we go?")
Do the strongest theories currently known to be unconstrained by Tennenbaum's theorem ($IOpen$ and some modest extensions) remain so when augmented with a definition of exponentiation and axiom $\...
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Logical strength of the pigeon-hole principle for measure spaces
In his book on measure theory, Tao discuss the pigeon-hole principle for measure spaces, which expresses that the union of measure zero sets is again measure zero.
I am interested in the logical ...
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Set-theoretical reverse mathematics of the reals
While reading through a nice old question/answer about the behavior of measures on the reals in $ZFC$ that popped back up today, I began to wonder how much of $ZFC$ is required for various things we ...
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What is the strongest form of the Axiom of Choice available in $\mathsf{Z}_{2}$?
$\mathsf{Z}_{2}$ denotes second-order arithmetic. Some forms of AC are expressible in $\mathsf{Z}_{2}$; for example the $\mathsf{\Sigma}_{1}^{1}$ axiom of choice is part of the theory $\mathsf{ATR}_{0}...
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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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"At most one" versus "at most finitely many"
As shown in Simpson's excellent Subsystems of Second Order Arithmetic, the ‘big five’ system ATR$_0$ from second-order reverse mathematics is equivalent to the following principle:
For arithmetical $\...
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Equivalences between statements of (seemingly) different order
In Steve Simpson's excellent monograph SOSOA, we find Theorem X.4.4 which contains an equivalence (over RCA$_0^*$) between the following statements:
The induction axiom for $\Sigma_1^0$-formulas (...
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Doing reverse mathematics by regarding modal logic as weak first-order logic
Reverse mathematics seeks to find subsystems of second-order logic that are equivalent to certain mathematics theorems, say over $\mathsf{RCA}_0$.
Modal logic can be regarded as a weak version of ...
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Is Monsky's theorem provable in $\mathsf{RCA}_0$?
Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. Monsky's proof attracted attention in part because it unexpectedly made use of the ...
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How does one prove the consistency of $\mathrm{PA}$ in $\mathrm{Z_2}$?
It is "well-known" (e.g. stated here without proof and sketched here) that $\mathrm{Z_2}$ proves $\mathrm{Con(PA)}$ using the "usual" model-theoretic proof, that is one can build a ...
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What is the weakest subsystem of Second-order Arithmetic (or its first-order part) that proves Szemerédi's Regularity Lemma?
The question is in the title. Szemerédi's Regularity Lemma is the following (according to the Wikipedia entry):
For every $\epsilon \gt 0$ and positive integer $m$ there exists an integer $M$ such ...
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Weaker uniformisation theorems
An interesting topic in Reverse Mathematics is uniformisation theorems (see VI.2 and VII.6 in Simpson's SOSOA). Now, these theorems all express the following: for a suitable formula $\varphi$, there ...
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Comprehension axiom that helps in the opposite direction
Usually, having more comprehension axiom means the more you can prove. We wonder if the converse can be the case.
Is there a natural problem $\mathsf{P}$ so that $\mathsf{P}+\neg(\Gamma-\mathsf{...
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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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Reverse Mathematics strength of fixed radius covering theorem
I am curious about the reverse math status of the below statement. Note that we work in second-order RM, i.e. 'closed set' is interpreted as in Simpson's excellent SOSOA.
For any closed $E\subset [0,...
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Different definitions of 'countable set'
There are a number of different definitions of 'countable set', all equivalent given a strong enough (classical) system. The obvious ones (injection to $\mathbb{N}$, bijection to $\mathbb{N}$, ...
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Does ACA prove categoricity of the reals?
$\def\f#1{\text{#1}}$Does $\f{ACA}$ prove that any two internally complete ordered fields are isomorphic?
Here internal completeness is expressed roughly as "every sequence of reals with an upper ...
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Coding third-order objects via second-order ones
As is well-known, the language of second-order arithmetic only has variables for natural numbers and sets of natural numbers. Higher-order objects, like functions on $\mathbb{R}$, have to be ...
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Relationship between provable in $RCA_0$ and effectively true
Question: What is the relationship between provability in $RCA_0$ and effectively true?
In other words: Given a problem, if a statement asserting the existence of a solution of the problem is provable ...
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Is anything known about $\Delta_n$ bounding?
For a class $\Gamma \in \{ \Sigma_n, \Pi_n, \Delta_n \}$ in the arithmetical hierarchy, we can consider the induction, bounding, and least number principles for $\Gamma$:
$\mathsf{I}\Gamma$ is $\big[ ...
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Can $\mathsf{RCA}_0$ prove that every nonempty c.e. set $A \subseteq \mathbb{N}$ has a least element?
In other words, can $\mathsf{RCA}_0$ prove that for every function $f\colon \mathbb{N} \to \mathbb{N}$, there is $b \in \mathbb{N}$ such that
$$ \exists k \in \mathbb{N},\ f(k) = b\quad \land\quad \...
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Why doesn't $\mathsf{B}\Sigma_2$ hold in $\mathsf{RCA}_0$?
For a formula $\varphi(i,u)$ of arithmetic, the bounding principle for $\varphi$ is the statement
$$\forall m \, \Big( \big( \forall i<m\ \exists u\ \varphi(i,u) \big) \to \big( \exists v\ \forall ...
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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 ...
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What is the Turing degree of the monadic theory of the real line?
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 ...
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Paris-Harrington principles parametrized by functions $f:\mathbb N \to \mathbb N$
Recall that the Paris-Harrington Principle, $\mathsf{PH}$, is the statement that for each $e, r, k < \omega$ there is an $N < \omega$ so that given any coloring $c:[N]^e \to r$ there is an $H \...
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What is the strength of the second-order statement 'an uncountable closed set in $\mathbb{R}$ has a limit point'?
Perhaps surprisingly, we work in the language of second-order arithmetic. I was wondering if the strength of the following statement LP was known:
An uncountable closed set in $\mathbb{R}$ has a ...
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From Vitali to Heine-Borel in reverse mathematics
The Vitali and Heine-Borel covering theorems are house-hold names of analysis, and rightly well-studied in reverse mathematics. As shown in Simpson's excellent monograph [1], for countable coverings ...
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Connection between second-order arithmetic and Hilbert-Bernays' Grundlagen
What is the exact (historical) connection between second-order arithmetic and Hilbert-Bernays' Grundlagen der Mathematik?
Some background: the literature on Reverse Mathematics contains a number of ...
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What subsystem of third order arithmetic proves the real numbers are Dedekind complete?
Reverse mathematics is mainly about subsystems of second-order arithmetic, but in recent years it’s expanded to cover subsystems of third-order arithmetic as well. Now the fact that the real numbers ...
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What subsystem of second-order arithmetic is needed for the recursion theorem?
In its simplest version, the recursion theorem states that for any $m\in\mathbb{N}$ and any function $g:\mathbb{N}\rightarrow\mathbb{N}$, there exists a function $f:\mathbb{N}\rightarrow\mathbb{N}$ ...
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Reverse mathematics of Cousin's lemma
This paper by Normann and Sanders apparently caused a stir in the reverse mathematics community when it came out a couple years ago. It says that Cousin's lemma, which is an extension of the Heine-...
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How to understand the interface of the consistency strength hierarchy, reverse mathematics, and proof-theoretic ordinal analysis?
I am aware of three major "hierarchies" of mathematical theories, but I don't know how to relate these hierarchies to one another. Here are the hierarchies I have in mind:
Consistency strength. My ...
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Detecting comprehension topologically
This question basically follows this earlier question of mine but shifting from standard systems of nonstandard models of $PA$ to $\omega$-models of $RCA_0$. For $X$ a Turing ideal we get the map $c_X$...
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Every complex number has a square root via LLPO without weak countable choice
Is it possible to prove that every complex number has a square root using analytic LLPO, but avoiding Weak Countable Choice or Excluded Middle? Unique Choice is allowed.
(Analytic LLPO is the ...
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The "higher topology" of countable Scott sets
Fix some computable bijection $b$ between $\omega$ and $2^{<\omega}$. For $r\in 2^\omega$, let $$[r]=\{f\in 2^\omega: \forall\sigma\prec f(b^{-1}(\sigma)\in r)\}$$ be the closed subset of Cantor ...
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$0^\#$ in weak theories vs large cardinals in $L$
To better understand the transition from large cardinal axioms consistent with the constructible universe $L$ to large cardinal axioms transcending $L$, I am looking for natural equiconsistencies ...
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Uncountability of the real numbers from LLPO without countable choice
Does there exist a proof of the uncountability of the real numbers that uses analytic LLPO (the statement that any real number $x$ satisfies either $x \leq 0$ or $x \geq 0$) but avoids Excluded Middle ...
- 4,732
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Examples of proofs using induction or recursion on a big recursive ordinal
There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal?
The ...
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