Questions tagged [constructive-mathematics]
The term "constructive mathematics" refers to the discipline in mathematics in which one proves the existence of mathematical objects only by presenting a construction that provides such an object. Indirect proofs involving proof by contradiction and law of excluded middle are considered nonconstructive. Constructivism is the philosophical stance that the only "true" mathematics is constructive mathematics.
625 questions
1 vote
1 answer
41 views
Function continuous nowhere whose domain and range are $[0,1]$
Find a function continuous nowhere, whose domain and range are both $[0,1]$. My intuition was to start with $f(x)=x$ and exchange to $f(a)=b$, $f(b)=a$ for sufficiently many pairs of $(a,b)$. So I ...
- 4,882
2 votes
1 answer
66 views
Monotonicity of odd powers of reals in strictly constructive/choice-free setting
Working with the Dedekind real numbers, in a fully constructive, choice-free context: can we show that if $f(x) = x^{2n+1}$ is monotonic? That is, if $x \le y$, then $x^{2n+1} \le y^{2n+1}$? I have ...
- 195
4 votes
1 answer
131 views
Injectivity implies left inverse in Bishop's constructive mathematics
In Bishop and Bridges' "Constructive analysis" on page 17 there is a note with explanation that If $f\colon A\to B$, $g\colon B\to A$, and $g(f(a))=a$ for all $a$ in $A$, then the function $...
7 votes
1 answer
380 views
In constructive math, does "$\mathbb R$ is the disjoint union of two non-empty sets" imply $\mathbb R = (-\infty,0] \cup (0,+\infty)$?
Consider IZF (Intuitionistic Zermelo-Fraenkel set theory) with the added axiom that there exist two non-emtpy subsets $A, B$ of $\mathbb R$ such that $\mathbb R$ is the disjoint union of $A$ and $B$. ...
- 2,206
3 votes
0 answers
99 views
Bounded abstract inductive definitions seem hard to provide
In the paper, "On Tarski's fixed point theorem" by Giovanni Curi, the notion of abstract inductive definition is explored (as a generalization of Aczel's inductive definitions) in the ...
- 31
5 votes
1 answer
151 views
constructive representation of real numbers
For simplicity, let’s assume the axiom of countable choice, that is, any countable product of merely inhabited sets is also merely inhabited. Then we can define Cauchy reals as modulated Cauchy ...
- 650
0 votes
0 answers
140 views
Classical theorem which cannot be modified to be proven constructively
To which extent constructive mathematics can recover classical mathematics? Is there classical theorem about computable objects and operations which can not be proven constructively? I know that many ...
- 650
9 votes
1 answer
281 views
Constructive subtleties about the Sierpinski Space
Classically, $\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\IS}{\mathbb{S}}\newcommand{\P}{\mathscr{P}}\newcommand{\upset}{\!\uparrow}$ the Sierpinski space is given by the set $\mathbb{S} = \{0, 1\}$ ...
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