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      Questions tagged [constructive-mathematics]

      6
      votes
      2answers
      332 views

      Constructivist defininition of linear subspaces of $\mathbb{Q}^n$?

      Let me preface this by saying I'm not someone who has every studied mathematical logic or philosophy of math, so I may be mangling terminology here (and the title is a little tongue in cheek). I (and ...
      11
      votes
      0answers
      128 views

      Ordinal-valued sheaves as internal ordinals

      Let $X$ be a topological space (feel free to add some separation axioms like “completely regular” if they help in answering the questions). Let $\alpha$ be an ordinal, identified as usual with $\{\...
      8
      votes
      1answer
      198 views

      Kleene realizability in Peano arithmetic

      For completeness of MathOverflow and for clarity of the question, I will first recall a few things, including the the definition of Kleene realizability: experts can jump directly to the question ...
      7
      votes
      5answers
      1k views

      Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle?

      I became interested in mathematics after studying physics because I wanted to better understand the mathematical foundations of various physical theories I had studied such as quantum mechanics, ...
      7
      votes
      1answer
      519 views

      Explaining the consistency of PRA and ZF from predicative foundations

      Recently I got interested in predicative foundations, mostly because of Laura Crosilla's work and because Agda employs a predicative type theory. From the point of view of a predicative foundation to ...
      13
      votes
      1answer
      273 views

      Is there a constructive proof that in four dimensions, the PL and the smooth category are equivalent?

      Summary Famously, the categories of 4-dimensional smooth manifolds and 4-dimensional piecewise linear manifolds are equivalent. Is there a constructive proof for this theorem or does it depend on the ...
      9
      votes
      0answers
      222 views

      Reflection principle for intuitionistic Zermelo–Fraenkel?

      The well-known reflection principle for classical Zermelo–Fraenkel states: For any formula $\varphi(x_1,\ldots,x_n)$ of the language of ZFC with free variables $x_1,\ldots,x_n$, ZFC proves $$ \...
      3
      votes
      1answer
      434 views

      Going beyond the strength of Peano arithmetic “without sets”

      First-order arithmetic is fairly weak, as measured for example by its consistency strength. When a stronger theory is desired, it is common to work with (fragments of) second-order arithmetic or set ...
      9
      votes
      2answers
      591 views

      What did the Intuitionists want to do with applied mathematics?

      Oversimplification: Newton & Leibnitz &c build the calculus and other methods that solve a vast number of practical problems. Weierstrass, Dedekind, Cantor &c build a foundation under it ...
      10
      votes
      5answers
      429 views

      Locales as spaces of ideal/imaginary points

      I posted this question on MSE a few days ago, but got no response (despite a bounty). I hope it will get more answers here, but I'm afraid it might not be appropriate as I'm not sure it's actually ...
      3
      votes
      1answer
      140 views

      Partitions of unity in constructive mathematics

      Can someone point me to any substitutes for the partition of unity in Bishop's constructive mathematics? In particular, under what circumstances can we construct a partition of unity subordinate to ...
      8
      votes
      1answer
      296 views

      Is every set smaller than a regular cardinal, constructively?

      Constructively, my only interest in regular cardinals is in terms of the "$\Sigma$-universes" they generate. By a $\Sigma$-universe, I mean a collection of triples $(X,Y,f: X \to Y)$ closed under base ...
      17
      votes
      4answers
      1k views

      Constructively, is the unit of the “free abelian group” monad on sets injective?

      Classically, we can explicitly construct the free Abelian group $\newcommand{\Z}{\mathbb{Z}}\Z[X]$ on a set $X$ as the set of finitely-supported functions $X \to \Z$, and so easily see that the unit ...
      19
      votes
      4answers
      910 views

      Mathematics without the principle of unique choice

      The principle of unique choice (PUC), also called the principle of function comprehension, says that if $R$ is a relation between two sets $A,B$, and for every $x\in A$ there exists a unique $y\in B$ ...
      16
      votes
      1answer
      760 views

      New articles by Errett Bishop on constructive type theory?

      Recently two formerly unknown articles by Errett Bishop (1928-1983) were posted online by Martín Escardó. One is entitled "A general language", deals with constructive type theory, and ...

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