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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14
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1answer
324 views

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-...
8
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1answer
221 views

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 ...
2
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1answer
153 views

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 $c_X$...
5
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1answer
218 views

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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2answers
210 views

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 ...
6
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0answers
208 views

$0^\#$ 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 ...
5
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2answers
492 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 ...
10
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1answer
369 views

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 ...
4
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174 views

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 ...
5
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1answer
362 views

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 ...
13
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2answers
1k views

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 ...
3
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0answers
172 views

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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2answers
2k views

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"...
4
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1answer
293 views

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"...
4
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1answer
157 views

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. ...
6
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2answers
426 views

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 ...
10
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428 views

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 ...
7
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1answer
380 views

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 ...
13
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1answer
416 views

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 ...
3
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3answers
459 views

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 ...
4
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0answers
120 views

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 ...
11
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2answers
303 views

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 ...
6
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1answer
258 views

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 ...
3
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0answers
158 views

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 ...
3
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1answer
87 views

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 $A$. ...
12
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1answer
1k views

What is known about the relationship between Fermat's last theorem and Peano Arithmetic?

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
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0answers
152 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. ...
54
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8answers
7k views

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 ...
2
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0answers
70 views

Is there a connection between the subsystems of second-order arithmetic and computational complexity?

The "big five subsystems of second-order arithmetic" in reverse arithmetic reveal the stratification of the structure of mathematics. What if any is the connection of these strata with complexity ...
5
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1answer
242 views

A game with boldface strength

This is a problem which has been bothering me for a while now; it doesn't seem inherently too hard, but I haven't been able to make any real headway, so I'm putting it out in the open since at this ...
2
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2answers
527 views

Is any Cauchy sequence for completion of rational semicomputable?

For the definition of a semicomputable real, see An Introduction to Kolmogorov Complexity and its Applications by Li and Vitanyi (1997). In fact, it is not true that every Cauchy sequence for ...
6
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1answer
217 views

Bounded-width Konig's lemma in reverse math

We define $\mathsf{BWKL}$ as follows: Every infinite binary tree of bounded width has an infinite path. This obviously follows from $\mathsf{WKL}$. Is this principle true in $\mathsf{RCA}_0$? If not, ...
7
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1answer
245 views

Axiomatizations of arithmetical parts of theories

For common theories that talk about something more general than first-order arithmetic (e.g. set theories and subsystems of second-order arithmetic), are there nice axiomatizations of their arithmetic ...
1
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1answer
210 views

How does $RCA_0$ achieve weak completeness?

Few days ago I asked about $WKL_0$ and the role of binary trees to provide for completeness for first order theories, and the question was nicely answered by Joel David Hamkins: Does $WKL_0$ plus CON(...
2
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1answer
104 views

Does $WKL_0$ plus CON(PA+X) give a binary tree model of PA+X?

In the context of reverse mathematics $WKL_0$ is considered equivalent to Gödel's completeness theorem over $RCA_0$. Does this mean that e.g. $WKL_0$ plus the consistency statement CON(PA+X) gives a ...
1
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1answer
220 views

Forcing in Second-Order Arithmetic

If I understand correctly, Stephen Simpson, in his book Subsystems of Second Order Arithmetic, deems second-order arithmetic as a two-sorted first-order theory. If this is correct, then it seems ...
88
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5answers
6k views

Is there a database for tracking the dependencies of mathematical theorems?

Given a proof for a result, one could denote the proof as a node on a graph, and then draw arrows to the node from axioms and previous results that the proof uses, and then draw arrows from the node ...
2
votes
1answer
250 views

Is there a model of ZF+ACC where transfer fails for the definable hyperreals?

In 2003 Kanovei and Shelah constructed a definable hyperreal field. The ultrapower used exploits a fairly large index set so that it is clear that the usual proof of Los and transfer does not go ...
2
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0answers
146 views

Circular reasoning in proof of bounded comprehension

Theorems II.3.7 and II.3.9 in Simpson's Subsystems of Second-Order Arithmetic appear to be circular. Specifically, theorem II.3.7 seems to make implicit use of theorem II.3.9. [Theorem II.3.9 is the ...
2
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1answer
203 views

Transfer with minimal choice

Let FUF postulate the existence of a Free UltraFilter on $\mathbb{N}$ and ACC the axiom of countable choice. Consider the superstructure on $\mathbb{R}$ and its inclusion in the bounded ultrapower. ...
12
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2answers
374 views

Can noncomputable sets be distinguishable in $RCA_0$?

Say that a set $X\subseteq\omega$ is distinguishable if there is some Turing machine $\Phi_e$ which, when given two sets exactly one of which is $X$, can determine which set is $X$. Formally, $X$ is ...
38
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2answers
5k views

Why is this new result such a big deal?

This popular article reports a recent result in reverse mathematics, showing that a certain theorem in Ramsey theory is provable from RCA$_0$, the base theory in SOSOA. Then there are a bunch of ...
5
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1answer
361 views

Constructive compactness for countable models?

The compactness theorem for countable (Tarski?) models is equivalent to the weak König's lemma by a result of H. Friedman and others as noted here, in the context of classical logic. The weak König's ...
2
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2answers
238 views

Compactness for countable models?

How and where is it proved that WKL$_0$ proves the compactness theorem for countable models? (This is a follow-up to a comment of F. Dorais.)
6
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1answer
320 views

Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?

I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...
4
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1answer
153 views

Proving moduli of uniform continuity in RCA_0

Simpson's Subsystems of Second Order Arithmetic (pp. 134ff.) uses RCA$_0$ to prove various theorems of analysis for all continuous functions with a suitable modulus of uniform continuity. And he ...
9
votes
1answer
347 views

Proof-theoretic ordinals after liberalizing induction to $RCA_0$

This is kind of a follow-up to this question. For a class $\Gamma$ of second-order formulas (here either $\Sigma_n^0$ or $\Sigma_n^1$), let $X\Gamma$ be a formal theory consisting of $RCA_0$ together ...
4
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2answers
176 views

What can be achieved by liberalizing induction for $RCA_0$?

$RCA_0$ has $\Delta_0$-comprehension and $\Sigma_1$ induction. Let $X\Sigma_{n}$ be $RCA_0$ plus $\Sigma_n$-induction and let $X\Sigma_{\omega}$-induction be $RCA_0$ plus the full induction schema. ...
6
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1answer
140 views

Does $WKL_0$ provide more comprehension than $RCA_0$?

$WKL_0$ extends $RCA_0$ with the statement that any infinite subset of the infinite binary tree has an infinite branch. Does $WKL_0$ Prove that there are sets which are not proven to exist by the $\...
5
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1answer
473 views

What are key $\Sigma^0_2$ or $\Pi^0_3$ theorems?

I am researching a logical system that is limited to $\Pi^0_2$ sentences and I am busy to prove that FOL + PA is a conservative extension of that system. Meaning that with $\Sigma^0_n$ sentences (that ...