# 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 is the weakest subsystem of Second-order Arithmetic (or its first-order part) that proves Szemeredi's Regularity Lemma?

The question is in the title. Szemeredi'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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### 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 ...
232 views

### 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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### 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 ...
232 views

### 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 , 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 ...
1 vote
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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 ...
632 views

### 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 ...
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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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### An axiom that shows that the real numbers are weakly countable?

Is there a model of Intuitionistic Higher-Order Logic in which the following axiom is true? Covering Axiom: Any true statement of the form $\forall x \in A, \exists y \in B, \phi(x,y)$ gives rise to ...
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### BISH: If a function is pointwise positive, is its infimum positive?

Let $f:[0,1] \to \mathbb R$ be a uniformly continuous function such that each value of $f(x)$ is greater than zero. Is its infimum greater than zero in BISH? I believe that it is indeed the case if ...
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### Why is weak Kőnig's lemma weaker than Kőnig's lemma?

Kőnig's lemma states that any finitely-branching tree with infinitely many nodes contains an infinite path. Weak Kőnig's lemma states the same thing about binary trees. It's known that these are not ...
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### Proof theory and subsystems of second-order arithmetic: in particular the reverse mathematics of Godel's system $T$

While doing some research on reverse mathematics, I came across the following document under the address, http://www.andrew.cmu.edu/user/avigad/Talks/survey1.pdf: Proof theory and Subsystems of ...
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### Is it possible to constructively prove that every quaternion has a square root?

Is it possible to constructively prove that every $q \in \mathbb H$ has some $r$ such that $r^2 = q$? The difficulty here is that $q$ might be a negative scalar, in which case there might be "too many"...
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### Proof-theoretic ordinals: inevitable consistency?

There are various different notions of the proof-theoretic ordinal of a theory; most of these are "notation-dependent" in that they're only nontrivial once we restrict attention to a class of "natural"...
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### Why restrict to $\Sigma_1^0$ formulas in $RCA_0$ induction?

I recently asked this question over on math.se, warmly welcomed by crickets. I hope it's appropriate here. I'm reading Stillwell's Reverse Mathematics, and the induction axiom was just introduced. ...
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### 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 ...
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### Reverse Mathematics of Euclid's theorem

Euclid's theorem that there are infinitely many prime numbers has multiple proofs, ranging from Euclid's original theorem that constructs a new prime from a finite list of such, to Euler's proof that ...
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### van der Waerden's theorem in Reverse Mathematics

What is known about weak systems of axiomata that allow one to prove van der Waerden's theorem? van der Waerden's theorem can be used to show that there are infinitely many primes (see below). Is ...
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### reverse mathematics of the Lebesgue measurability of analytic sets

Can the fact that all analytic sets are Lebesgue measurable be proven in $Z_2$, or in some weak subsystem such as $\Pi^1_1\text{-CA}_0$? Conversely, can certain set existence axioms be derived from ...
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### How can you formalize the metamathematics conventionally used to state Godel’s theorem?

Gödel’s incompleteness theorem states that for any sufficiently strong formal system $T$ there exists a statement $G$ such that if $T$ is consistent, then $G$ is true but not provable in $T$. But I’m ...
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### Does comprehension for formulas in the analytical hierarchy imply comprehension for all formulas in second-order arithmetic?

The proof that all formulas of second-order arithmetic are $\Pi^1_n$ for some $n$ (i.e. can be written with a bloc of second-order quantifiers followed by an arithmetical formula) uses the axiom of ...
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### Are all generalized Scott sets realized as generalized standard systems?

Below, I've focused on PA when lots of other theories would do. If replacing PA with a different theory leads to a more answerable question, feel free to do so. The standard system of a nonstandard ...
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### Am I counting quantifiers correctly?

I think this is right but I want to check. The theory $\mathsf{WKL}^*_0$ is conservative over EFA for $\Pi^0_2$ sentences. And the first order part of $\mathsf{WKL}^*_0$ is axiomatized by EFA plus ...
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### Are there amenable groups without explicit Folner sets?

This is essentially a follow-up to this previous discussion on how, in the absence of choice, the "invariant mean" and "Folner set" characterizations of amenability are no longer equivalent. Recently ...
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### If one adds an inductive subset to a model of $ACA_0$, do we always get a new model of $ACA_0$?

Suppose $(M, \mathcal X) \models ACA_0$. Recall that a subset $A \subseteq M$ is $inductive$ over $M$ if $M$ satisfies all instances of induction in the expanded language with a predicate for $A$. ...
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As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem? In particular, what is known about the arithmetic systems $PA + \... 4 votes 0 answers 158 views ### A forcing which can build weird models of$\neg\$ADS

There is a class of forcing notions I've been playing around with recently. They have a couple nice properties, and all have the same theme, but I've found them difficult to analyze beyond the basics. ...
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### What does it mean to suspect that two conjectures are logically equivalent?

Here's a familiar conversation: Me: Do you think Conjecture A and Conjecture B are equivalent? Friend: Yes, because I think they're both true. Me: [eye roll] You know what I mean... Does there ... 