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).
192 questions
3 votes
1 answer
165 views
Models of Second Order Arithmetic with non-standard length and all subsets of $\omega$?
Are there any non-standard models of RCA$_0$ such that every subset of $\omega$ appears as a restriction of a set in the second-order part to it's standard initial segment? In other words, does ...
- 3,987
4 votes
1 answer
242 views
Enumerating a $\Delta_1^1$-set in Reverse Mathematics
The system $\Delta_1^1$-CA$_0$ from Reverse Mathematics consists of the base theory RCA$_0$ and the comprehension axiom for $\Delta_1^1$-formulas, i.e. for any $\varphi_0, \varphi_1 \in \Sigma_1^1$ ...
- 5,004
3 votes
0 answers
223 views
How hard is it to just bound the torsion in the Mordell–Weil group of an elliptic curve over Q?
Given an elliptic curve over the rationals Mazur proved that the order of the torsion subgroup of $E(\mathbb{Q})$ is bounded by 16, and specified which groups can occur. If one only needs a bound of ...
- 36.8k
6 votes
1 answer
606 views
A form of reverse mathematics that works with hereditarily finite sets instead of numbers
Does anyone know of any texts where reverse mathematics is developed using hereditarily finite sets and subsets of $V_\omega$? Reverse mathematics is typically carried out in the framework of second-...
4 votes
0 answers
144 views
Can Witnessing theorems lead to code extraction from proofs? (bounded arithmetics)
In bounded arithmetic (or bounded reverse mathematics) we study formal systems so weak that the structures of their proofs correspond to some known complexity classes such as PTIME, LOGSPACE or AC0. ...
- 41
6 votes
1 answer
272 views
$\mathit{RCA}_0$ without the law of the excluded middle
$\newcommand\name{\mathit}$In Classical Reverse Mathematics, the most famous base theory is $\name{RCA}_0$. I want to work in the area of formal Constructive Reverse Mathematics. I wonder if "$\...
2 votes
0 answers
211 views
Fragments of set theory required to prove the independence of CH
What are the smallest fragments of set theory known to be sufficient to prove Cohen's independence theorems that if ZF is consistent then so is ZF plus the negation of the continuum hypothesis CH, or ...
- 5,493
3 votes
1 answer
115 views
Closed versus separably closed sets in Reverse Mathematics
In second-order Reverse Mathematics, a code for an open set $O$ of reals is a sequence of rationals $(a_n)_{n \in \mathbb{N}}$, $(b_n)_{n \in \mathbb{N}}$. We write $x\in O$ in case $(\exists n\in \...
- 5,004
2 votes
0 answers
223 views
Is there a compendium of the consistency strength between ZFC to Z2?
Backgrounds The part that goes beyond ZFC is complete in Cantor’s Attic. The portion below Second order arithmetic is complete ...
- 1,425
3 votes
0 answers
168 views
The third-order arithmetic subsystems that are stronger than second-order arithmetic
Backgrounds Two (non-peer-reviewed) papers on ordinal analysis of second-order arithmetic have appeared in the arxiv [1][2], which seems to imply that ordinal analysis of second-order arithmetic is ...
- 1,425
11 votes
1 answer
330 views
Does analytic WLLPO together with sequential LLPO imply analytic LLPO?
This question is about constructive mathematics, without any form of Choice except Unique Choice, such as in the internal logic of a topos with natural numbers object, or in IZF. The “reals” (and the ...
- 38k
9 votes
0 answers
336 views
Does the decomposability of $\mathbb{R} \setminus \mathbb{Q}$ imply the decomposability of $\mathbb{R} \setminus \{0\}$?
By $\mathbb{R}$ I mean Dedekind real numbers. By $X \setminus Y$ I mean $\{x \in X: \neg (x \in Y)\}$. Let's assume the following statements: ($\bf WLLPO$) For all binary sequence $(\alpha_n)$ with at ...
4 votes
0 answers
176 views
Are the representations of metric spaces expected to increase or decrease logical strength?
The coding of metric spaces (via countable dense subsets) in the language of second-order arithmetic is well-known from recursion theory and reverse mathematics. During a recent discussion of said ...
- 5,004
8 votes
2 answers
450 views
Status of "the unit interval is connected" in second-order Reverse Mathematics
Simpson's SOSOA and some papers cited therein mention connectedness. Which subsystem proves that the unit interval is not the union of two disjoint and non-empty (codes for) open sets? It feels like ...
- 5,004
2 votes
0 answers
557 views
What is the weakest theory that settles all interesting first-order arithmetic sentences?
Woodin's program of refuting CH, as summarized in 1, continues the following assertions (roughly as in Propositions 7, 13, and 20 of that paper): In any model of $\text{ZFC}$, the theory of $(H(\omega)...
- 333
4 votes
0 answers
172 views
Complexity of comparison between mice below a strong cardinal
How does the complexity of comparison between mice increase as a function of their large cardinal strength, especially for mice below a strong cardinal? For example, what is the first point at which ...
- 8,648
5 votes
1 answer
308 views
Does the axiom of countable choice constructively imply an omniscience principle?
$\newcommand\name{\mathrm}\newcommand\BISH{\name{BISH}}\newcommand\ACC{\name{ACC}}\newcommand\LPO{\name{LPO}}\newcommand\WLPO{\name{WLPO}}\newcommand\LLPO{\name{LLPO}}\newcommand\WLLPO{\name{WLLPO}}\...
14 votes
2 answers
670 views
The power of Harrison's order
Due to Joseph Harrison we know that there exists a recursive linear order such that every recursive ordinal is isomorphic to an initial segment of this order. What is the "logical power" of ...
- 143
7 votes
1 answer
342 views
Mathematical strength of the statement "Heyting Arithmetic admits Markov's rule"
Consider the following theorem about Heyting arithmetic (HA): For every arithmetical formula $\phi$ whose only free variable is $n$, if $\text{HA} \vdash \forall n. \phi \lor \lnot \phi$ and $\text{...
- 7,128
15 votes
1 answer
911 views
Are key theorems finitistically reducible?
Simpson writes on page 378 of his Subsystems of Second Order Arithmetic: "For example, all of the following key theorems of infinitistic mathematics are provable in WKL$_0$ and therefore, by ...
- 18k
7 votes
2 answers
480 views
Numerical choice and reverse mathematics
Consider the following fragment of numerical choice in the language of second-order arithmetic: for any arithmetical $\varphi$, we have: $$ (\forall n\in \mathbb{N})(\exists m\in \mathbb{N})(\forall X\...
- 5,004
9 votes
0 answers
501 views
Consequences of recent claims of Ordinal Analysis of $Z_2$
Recently Toshiyasu Arai submitted "An ordinal analysis of $\Pi_{N}$-Collection" and Henry Towsner submitted "Proofs that Modify Proofs", both of which claim ordinal analysis of ...
- 161
9 votes
1 answer
447 views
Does the decomposability of $\mathbb{R}$ imply analytic LLPO?
By "BISH" I mean constructive mathematics without axiom of countable choice. By $\mathbb{R}^f$ I mean real numbers as fundamental sequences of rational numbers and by $\mathbb{R}^d$ I mean ...
15 votes
0 answers
293 views
Natural examples of Borel surjections without right inverse
As discussed in this question, in general a Borel surjection $f:\mathbb{R}\rightarrow\mathbb{R}$ may not have a Borel right inverse, namely a $g$ such that $f\circ g=id$, although there is always a ...
- 903
4 votes
0 answers
202 views
Can one formalize the prevalence of the Big Five systems of reverse math?
Simpson's systems of second order arithmetic turn out to be five in number; to simplify notation let's denote them A, B, C, D, E. What seems to be an empirical observation is that most theorems in ...
- 18k
8 votes
2 answers
673 views
Completing half of Hilbert's program: Foundations that are conservative over Peano Arithmetic
The goal of the Hilbert program was to find a complete and consistent formalization of mathematics. Gödel's first incompleteness theorem establishes that completeness is impossible with first-order ...
- 7,128
3 votes
1 answer
228 views
Reverse mathematics on lightface $\Pi^1_1$-uniformization for unary relation
It is known that the following form of $\Pi^1_1$-uniformization is equivalent to $\Pi^1_1$-Comprehension over $\mathsf{ATR}_0$ (cf. VI.2.6 of Simpson's book) : (Kondo's uniformization theorem) For ...
- 3,326
3 votes
1 answer
180 views
Kleene normal form theorem for r.e. relations proven in arithmetical theories
After reading the relevant chapters of Classical Recursion Theory (freely available from here), I have the following questions concerning Theorem II.1.10 (Normal form theorem) and Theorem IV.1.9 (...
- 31
3 votes
0 answers
267 views
Reverse-mathematical strength of Banach-Tarski
What is the reverse mathematical strength of the Banach-Tarski paradox? The usual proof of Banach-Tarski should carry out in $\mathrm{ZF}+\mathrm{AC}_\kappa$, where $\kappa$ is the supremum of the ...
- 2,848
7 votes
1 answer
392 views
Proving finiteness in Reverse Mathematics
In (second-order) Reverse Mathematics, a (code for an) open set $U\subset \mathbb{R}$ is given by two sequences of rationals $(a_n)_{n \in \mathbb{N}}, (b_n)_{n \in \mathbb{N}}$. The idea is that $U$ ...
- 5,004
12 votes
2 answers
501 views
Trading Choice for Comprehension (or Replacement)
This question is basically a request for clarification about a remark made by Sam Sanders in a comment to another question: IIUC what he's saying, there are statements that can be proved either with a ...
- 38k
7 votes
1 answer
726 views
Proof of global Peano existence theorem in ZF?
By global Peano existence theorem I mean the existence of a maximal interval of solution of a first order ODE $x'=f(x,t)$ with continuous $f$. The proofs of the global Peano Theorem found in the ...
- 18k
10 votes
1 answer
368 views
How much of second-order arithmetic do you need for $\mathbf{\Sigma}^1_1$-determinacy to give you countable transitive models of $\mathsf{ZFC}$?
This is in some sense a follow-up to this question. The answer there says that over $\mathsf{Z}_2$ (second-order arithmetic), (boldface) $\mathbf{\Sigma}^1_1$-determinacy is enough to entail the ...
- 19k
8 votes
1 answer
431 views
Strength of Borel determinacy
In this blog post by Gowers on Borel determinacy, Andres Caicedo says the following in a comment (slightly rephrased). Let $\mathsf{ZFC^-}$ be $\mathsf{ZFC}$ without power set and $\mathsf{ZC^-}$ be $...
- 1,109
20 votes
2 answers
2k views
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 ...
- 19k
13 votes
1 answer
822 views
Is "All hyperreal fields $C(\mathbb{R})/M$ are isomorphic" independent of ZFC?
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 field-isomorphic to $\mathbb{R}$. For example,...
- 2,287
4 votes
0 answers
136 views
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. ...
- 5,004
14 votes
1 answer
697 views
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. ...
- 7,128
5 votes
0 answers
114 views
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-...
- 19.4k
8 votes
0 answers
208 views
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 $\...
- 3,809
3 votes
0 answers
162 views
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 ...
- 5,004
15 votes
1 answer
743 views
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 ...
- 10.4k
7 votes
0 answers
353 views
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}...
- 2,206
3 votes
0 answers
344 views
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 ...
- 4,583
7 votes
0 answers
215 views
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" ...
- 4,993
6 votes
1 answer
446 views
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 ...
- 4,583
12 votes
2 answers
701 views
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 ...
- 4,993
7 votes
2 answers
625 views
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, ...
- 38k
14 votes
2 answers
1k views
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 ...
- 5,004
7 votes
0 answers
139 views
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 $1$ ...
- 19.4k