The constructive-mathematics tag has no wiki summary.

**6**

votes

**3**answers

401 views

### In the category of sets epimorphisms are surjective - Constructive Proof?

The statement that surjective maps are epimorphisms in the category of sets can be shown in a constructive way.
What about the inverse?
Is it possible to show that every epimorphism in the category ...

**4**

votes

**2**answers

201 views

### What is the status of the extreme value theorem in forms of constructive mathematics, such as Smooth Infinitesimal Analysis?

In certain intuitionistic frameworks the extreme value theorem cannot be proved. Depending on the exact framework, counterexamples can be constructed as well; see for example pp. 294-295 in
...

**12**

votes

**3**answers

445 views

### Formal/rigorous treatment of (im)predicativity/predicativism

There are several places on the web where one may find quite intuitively understandable accounts of (im)predicativity; here on MO I found two questions with very good detailed answers (Predicative ...

**11**

votes

**1**answer

332 views

### Is it possible for a theorem to be constructive only in a non-constructive metatheory?

There are several theorems in category-theoretic logic which say something like, "any proposition in X logic that is provable in topos logic assuming (the law of excluded middle and) the axiom of ...

**6**

votes

**2**answers

479 views

### Are all functions in Bishop's constructive mathematics continuous?

When I started to write this question I was really confused, but now I think I am starting to get it. Nonetheless I'd like some confirmation that my understanding is correct.
I have read, heard ...

**16**

votes

**3**answers

762 views

### Function extensionality: does it make a difference? why would one keep it out of the axioms?

Yesterday I was shocked to discover that function extensionality (the statement that if two functions $f$ and $g$ on the same domain satisfy $f\left(x\right) = g\left(x\right)$ for all $x$ in the ...

**33**

votes

**1**answer

1k views

### Wanted: a “Coq for the working mathematician”

Sorry for a possibly off-topic question -- there are four StackExchange subs each of which could be construed as the proper place for this question, and I've just picked the one I'm most familiar ...

**1**

vote

**2**answers

333 views

### Bishop's paradox of the countability of sequences

In 'Foundations of Constructive Analysis', in the notes at the end of the first chapter, Bishop poses an apparent paradox as an exercise for the reader:
Since every sequence of rational numbers ...

**4**

votes

**1**answer

364 views

### Why is adopting Russell's Axiom of Reducibility as strong as eliminating the Ramified Hierarchy?

In order to respond to concerns of impredicativity, Bertrand Russell developed a system of ramified second-order logic, which is like regular second-order logic except the comprehension schema is ...

**7**

votes

**0**answers

104 views

### Constructively correct notion of unique factorization domain

Recall the well-known proof that a unique factorization domain is a GCD domain:
Let $x, y \in R \setminus \{ 0 \}$. Factor $x$ and $y$ into pairwise non-associated irreducible elements: ...

**1**

vote

**0**answers

158 views

### Has the Ramified Theory of Types been applied to Predicative Set Theories?

Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...

**4**

votes

**2**answers

366 views

### Did Gödel prove that the Ramified Theory of Types collapses at $\omega_1$?

Second-Order Arithmetic is considered impredicative, because the comprehension scheme allows formulas with bound second-order variables that range over all sets of natural numbers, including the set ...

**4**

votes

**3**answers

405 views

### Infinite Partitions of the Primes and Sums of Reciprocals (Revised)

I have revised my original post. The questions I asked there were not well-put or even thought through. I don't want to delete, however, since some of the comments may be of interest to other MO ...

**2**

votes

**0**answers

364 views

### Is there a notion of “predicative given the real numbers”?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...

**4**

votes

**2**answers

272 views

### Does the Feferman-Schutte analysis give a precise characterization of Predicative Second-Order Arithmetic?

A definition is called impredicative if it involves quantification over a domain that contains the thing being defined. For instance, if you define hereditary property to be a property which applies ...

**22**

votes

**4**answers

1k views

### Illustrating Edward Nelson's Worldview with Nonstandard Models of Arithmetic

Mathematician Edward Nelson is known for his extreme views on the foundations of mathematics, variously described as "ultrafintism" or "strict finitism" (Nelson's preferred term), which came into the ...

**7**

votes

**1**answer

211 views

### Barr's theorem and constructivity?

Barr's covering theorem assert that any Gorthendieck topos can be covered by a Grothendieck topos (even a locale) satisfying the axiom of choice (and hence also the law of excluded middle). Its ...

**8**

votes

**3**answers

300 views

### reference request : constructive measure theory

As the title said, I would like to know if constructive measure theory has been developed somewhere ?
I am more precisely interested in the (constructive) theory of completely continuous valuation on ...

**9**

votes

**1**answer

165 views

### How hard is Heyting satisfiability, i.e. the constructive version of SAT? In particular, is 2-HSAT NL-complete or is it harder?

First of all, is it clear what I mean by $k$-HSAT?
I'm assuming that for $k>2$, $k$-HSAT is NP-complete, but the details of the reductions between $k$-HSAT and $k$-SAT aren't obvious to me.
I'm ...

**7**

votes

**1**answer

279 views

### Constructing Metrics for specific Topological Spaces, and Refinements of the Cantor-Space in particular

I have a Problem in general, given some some Topological Space $(X, \tau)$ from which I know it is metrisable, how can I find a metric (that is at best in some sence constructive and easy, at the very ...

**5**

votes

**6**answers

231 views

### Strength of Bishop style constructive mathematics vs $\mathsf{RCA}_0$

This question came out of this other MO question of mine. My question is
Is there a formal comparison between $\mathsf{RCA}_0$ and $\mathsf{BISH}$ (Bishop style constructive mathematics as used ...

**5**

votes

**0**answers

154 views

### Feasible Type Theories

I am looking for references about efficient type theories,
efficiency in the sense of computational complexity,
and type theory in the sense of Martin-Lof's type theories.
Has there been any studies ...

**5**

votes

**0**answers

180 views

### constructive Serre classes

A Serre class (of abelian groups) is a class of abelian groups closed under subgroups, quotients, and extensions. For instance, finitely generated groups and finite groups are both Serre classes.
...

**3**

votes

**2**answers

263 views

### Understanding Troelstra's Uniformity Principle in Constructive Mathematics

I have seen Troelstra's Uniformity Principle stated as:
$\forall x \exists n R(x,n) \rightarrow \exists n \forall x R(x,n)$
where $x$ ranges over $\mathbb{P(N)}$ and $n$ ranges over $\mathbb{N}$.
...

**1**

vote

**1**answer

362 views

### Cardinal Arithmetic, foundations and constructive math

This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one ...

**5**

votes

**1**answer

484 views

### Intuitionistic consistency of surjection from naturals to reals

Is it consistent intuitionistically (in the sense of topos theory) for there to be a surjection from the natural numbers to the (Dedekind, let us say) real numbers? [I've managed to convince myself ...

**4**

votes

**1**answer

513 views

### intensional equality in type theory

I want to know why we add an intensional equality in type theory to definitional equality ?
What is the aim with this intensional equality ?
thanks

**6**

votes

**3**answers

748 views

### What is the status of irrational numbers within finitism/ultrafinitism?

According to constructivism a mathematical object to prove that it exists". There are several formulas to calculate pi, such as:
so I take it ...

**4**

votes

**1**answer

712 views

### effective/constructive/algorithmic probability theory

What sort of "alternative" probability theories are out there in which the methods of proof are inherently constructive?
I know of a number of theorems that say that if you take an infinite sequence ...

**5**

votes

**1**answer

272 views

### Higher computability : Constructive ordinal and $\Delta^1_1$ predicates

Everything I know on this subject comes from Sacks book : "Higher recursion theory"
Let $\mathcal{O^Y}$ be the set of codes for ordinals constructive in $Y$.
We should have the result that $A ...

**2**

votes

**0**answers

206 views

### Non-Computational classical subterms

Assume we have a proof term of the form $(a^{A\rightarrow^c B\rightarrow^{nc} C}b^Ac^B)^C$, where $c$ is classical (that is, contains free instances of duplex negatio affirmat). The extracted term ...

**6**

votes

**0**answers

942 views

### Is there a finite-dimensional vector space whose dimension cannot be found? [closed]

Is there a finite-dimensional vector space whose dimension cannot be found? Assume, we have somehow constructed a vector space whose dimension is finite, but yet unknown. Is there always an algorithm ...

**2**

votes

**0**answers

283 views

### Artin Jacobson-semisimple rings are semisimple. Constructively, too?

Notation. When I say "ring", I mean "ring with unity" (not necessarily commutative).
Definition. A ring $R$ is said to be left-Artinian if for every sequence $I_0\supseteq I_1\supseteq I_2\supseteq ...

**3**

votes

**3**answers

3k views

### About the proof of the proposition “there exists irrational numbers a, b such that a^b is rational”

What does the classical proof of the proposition "there exists irrational numbers a, b such that $a^b$ is rational" want to reveal? I know it has something to do with the difference between classical ...

**26**

votes

**4**answers

3k views

### How to make Ext and Tor constructive?

EDIT: This post was substantially modified with the help of the comments and answers. Thank you!
Judging by their definitions, the $\mathrm{Ext}$ and $\mathrm{Tor}$ functors are among the most ...

**18**

votes

**3**answers

1k views

### What facts in commutative algebra fail miserably for simplicial commutative rings, even up to homotopy?

Simplicial commutative rings are very easy to describe. They're just commutative monoids in the monoidal category of simplicial abelian groups. However, I just realized that a priori, it's not clear ...

**48**

votes

**8**answers

5k views

### Is there any formal foundation to ultrafinitism?

Ultrafinitism is (I believe) a philosophy of mathematics that is not only constructive, but does not admit the existence of arbitrarily large natural numbers. According to wikipedia, it has been ...

**17**

votes

**4**answers

2k views

### How constructive is Doyle-Conway's 'Division by three'?

In the (whimsically written) article Division by three, Doyle and Conway describe a proof, (apparently) not using Choice, that an isomorphism $A \times 3 \simeq B\times 3$ of sets (where $3$ is a ...