Questions tagged [foundations]
Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.
330 questions
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Replacement axiom and the von Neumann hierarchy
Within ZFC, the von Neumann hierarchy consists of sets $V_\alpha$ indexed by ordinals, subject to the following conditions:
$V_0=\varnothing$.
$V_{\alpha+1}=\mathcal P(V_\alpha)$.
$V_\lambda=\bigcup_{...
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Is the centroid property equivalent to the middle line property of the triangle?
By a Tarski plane I understand a set $X$ endowed with a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ satisfying all Tarski axioms except ...
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Why do we care about small sets?
I have been a user of category theory for a long time. I recently started studying a rigorous treatment of categories within ZFC+U. Then I become suspecting the effect of the smallness of sets.
We ...
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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 ...
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Is it possible to set up multiple automorphisms over a structureless object inside single-sort defined category?
I was trying to understand the behaviour of the primitive equality (=) in the axiomatization of category, which takes morphisms as primitives and objects as derivatives in bijection to identity ...
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Comparing the areas of polygons via equidecomposability in the hyperbolic plane
It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles.
Question. Is an ...
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A name for a mathematical structure of geometric type
I am looking for (maybe existing) name for a mathematical structure $(X,\leqslant)$ consisting of a set $X$ and a transitive relation ${\leqslant}\subseteq X^2\times X^2$ such that $xx\leqslant yz\...
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Does the concept of a $\infty$-category have a natural definition in the $\infty$-world?
I start with a thesis: the natural notion of equality is additional data (paths/morphisms), not a binary relation (the fact that they exist). So, in particular, with such a constructivization (...
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For which Sheaf topoi is Brouwer's fan theorem true?
Brouwer's fan theorem is the standard result that the Cantor space is compact, or equivalently that the Cantor space viewed as a locale is spatial. Since it is a compactness result for a countable ...
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Are the categories of sets, abelian groups, and commutative rings unique?
Are the categories of sets, abelian groups, and commutative rings unique? Independence results like the independence of the generalized continuum hypothesis, the Whitehead problem, and the global ...
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Does the Segment-Circle Axiom imply the Circle-Circle Axiom in a non-Euclidean Tarski plane?
By a Tarski plane I understand a mathematical structure $(X,B,\equiv)$ consisting of set $X$, a betweenness relation $B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\times X^2$ ...
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6
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1
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The algebraic structure of a line in a (Tarski) plane
By a Tarski plane (resp. plane) I understand a mathematical structure $(X,B,\equiv)$ consisting of a set $X$, a ternary betweenness relation $B\subseteq X^3$ and the 4-ary congruence relation ${\equiv}...
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A textbook on foundations of geometry in spirit of Tarski
I am interested in a textbook for studying (and teaching) foundations of geometry in the spirit of Tarski. I know that there is a rather old German book [W. Schwabhäuser, W. Szmielew, A. Tarski, ...
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How strong is separation + reflection of unbounded quantifiers?
Consider a set theory with the following axioms:
separation: $\exists y \forall x (x \in y \leftrightarrow \phi \land x \in a)$ where $y$ is not free in $\phi$
reflection: $\phi \to \exists u \phi^u$
...
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Is ZFC set theory a satisfactory foundation for mathematics?
The conventional wisdom seems to be that it is, but there are problematic mismatches. Some are well-known: the use of first-order logic, and the many different implementations of the axioms, neither ...
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Higher inductive types in higher observational type theory
Mike Shulman gave the following set of talks on higher observational type theory earlier this year (part 1, part 2, part 3). However, while he talked about how the identity types are defined and ...
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What is the consistency strength of this addition on simple type-set theory?
Language: multi-sorted first order logic with equality and membership, where for each natural $n$ there is a set $x^n$ of sort $n$. Equality "$=$" only occurs between variables of the same ...
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Is $\sf \Gamma_0$ the proof theoretic ordinal of this kind of predicative class theory?
Adopting the approach of Mono-sorted $\sf NBG$, define sets as elements of classes, then axiomatize:
Extensionality, Predicative Class comprehension, emptyset, in the usual manner along mono-sorted $\...
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Axiomatic construction of trigonometric functions
I am able to construct functions $\sin,\cos\colon \mathbb R \to \mathbb R$ satisfying the following properties:
$\sin^2 x + \cos^2 x = 1$,
$\sin(x+y)=\sin x \cos y + \sin y\cos x$, $\cos(x+y)=\cos x \...
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Principle of unique choice in homotopy type theory
In the MathOverflow thread Mathematics without the principle of unique choice, Mike Shulman defines the principle of unique choice to be
if $R$ is a relation between two sets $A$, $B$, and for every $...
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What is the proof theoretic ordinal of this kind of predicative type-set theory?
The following is a kind of Predicative Type Set Theory.
The question is about what is exactly the proof theoretic ordinal of this theory? Is it lower than the one expected for predicative theories, i....
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Joyal arithmetic universes and the Box operator
Last month @godelian, alias Christian Espíndola, in a FOM post has mentioned Joyal's proof of Godel's second incompleteness via the so-called Arithmetic Universes, introduced by Joyal around 1973, ...
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Can this type theory interpret second order arithmetic?
Language: multi-sorted first order logic with equality and membership, where for each natural $t$ there is a set $x^t$ of sort $t$. Equality "$=$" only occurs between variables of the same ...
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Is definability in $V$ in $\sf Ack+MK$ expressible in its language?
Recall Ackermann set theory. If we extend Ackermann's set theory by adding all axioms of $\sf MK$ to it. We shall denote the universe of all elements by $W$, while $V$ is the primitive constant symbol ...
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Consistency strength of an attempt at higher order set theory
Work in a theory with (deep breath) a countable number of primitives denoted with capital letters from the end of the alphabet with numerical subscripts $\{X_n,Y_n,Z_n,\dots\}_{n<\omega}$ ...
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Can Frege set extensions of second order unary predicates serve as a foundation for mathematics?
To second order logic, add a primitive partial one place function symbol "$\epsilon$" from unary predicate symbols [upper cases] to object symbols [lower cases].
Define: $ x=y \equiv_{df} \...
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A formal definition of a useful theorem?
Sorry if this feels a bit squishy, but I'm wondering if there is any published work trying to give a fully formal definition of the notion of a useful theorem. I mean, in mathematics we all know that ...
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An axiomatic approach to the multiverse of sets
Work in a theory where the primitives are classes $X,Y,Z,\dots$, and class membership $X\in Y$, and add an individual constant $\mathcal{M}$ called 'the multiverse'. Classes $V$ which are members of ...
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Is there a purely constructive presentation of the HK integral?
Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
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Consistency of Generalised Continuum Hypothesis and univalence in HoTT
In homotopy type theory, propositional excluded middle and the axiom of choice sets are both consistent with univalence, both of which yields type theoretic models for classical mathematics. However, ...
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'Maximising interpretative power entails maximising consistency strength'?
I'm hoping there is a clear mathematical answer to this question (hence asking it here) rather than anything more exegetical (in which case it's presumably not appropriate for this site).
In his paper ...
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Coinduction for all?
Every undergraduate in mathematics learns about proofs by mathematical induction. Moreover, every undergraduate taking a course in theoretical computer science or logic learns about inductive ...
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Homotopical realizability
After a long story of dancing around the effective topos $ \mathcal{Eff}$, I finally resolved to get to the bottom of it. To this effect, working as it were backward, I ended up revisiting Kleene's ...
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The free complete lattice on three generators, beyond ZF
This was originally asked at MSE; although it is still under bounty it seems unlikely to be answered there.
$\mathsf{ZF}$ proves that there is no free complete lattice on three generators since any ...
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Harvey Friedman's minimalist axioms for set theory
[This is a question on the FOM mailing list.]
In 1997, Harvey Friedman introduced the following theory: Let $\in$ be a binary predicate and $U$ be a constant. Add the following axioms:
Subworld ...
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Does foundationless Ackermann set theory prove replacement?
From Ackermann's set theory equals ZF (1970) by William N. Reinhardt:
Let A be the theory determined by the following axioms:
Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x = y$
...
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Can we interpret ZFC in GEM?
I have long held similar views as put forth here. As professor Hamkins points out though, $(M,\subseteq)$ is insufficient to model most of mathematics. However, this is unsatisfactory to me, as this ...
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0-valued and 1-valued logics?
In addition to classic two-valued logic, there are many many-valued logics, including Łukasiewicz's and Kleene's three-valued logics, Gödel's many-valued logic $G_k$, and infinite-valued fuzzy logic ...
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Does the small object argument need replacement?
Does one need the axiom of replacement in the small object argument and in the transfinite construction of free algebras?
My motivation for the question is that I heard that the axiom of replacement ...
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Context of set theory in which one doesn't have to worry about size issues
In this beautiful talk by Colin McLarty, McLarty quotes Grothendieck:
It would be nice to have a context where one doesn't add any real axioms to set theory, and yet one can work with categories ...
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What do you call the generalisation of the direct image?
This question was posted on Math Stack Exchange, but did not attract an answer. Here is the question:
Informal Description
Let me start with an example. Let $X$ be the set $\{a, b, c, d, e\}$ and $E$ ...
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Applications of ZFA-Set Theory
The set theory with atoms (ZFA), is a modified version of set theory, and is characterized by the fact that it admits objects other than sets, atoms. Atoms are objects which do not have any elements.
...
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Can this graph theory serve as a foundational theory of mathematics?
Working in mono-sorted first order logic, add primitives of equality and its axioms, set membership $\in$, a partial ternary relation $\to$ denoting is the direction from to, and at last a total unary ...
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Resource request on "$\in$-homomorphisms" in Set Theory
Very loosely put, this is the intuitive idea behind an $\in$-homomorphism:
Let $\mathcal{U}$ and $\mathcal{W}$ be universes of sets. A function $f \colon \mathcal{U} \to \mathcal{W}$ is said to be an $...
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What structure do all kinds of theories, models, interpretations, proofs and all that form?
This is a question about a structure that can be used to investigate all kind of structures that can be investigated. Many years ago with Joseph Gubeladze we discussed something similar but I only ...
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Practical Benefits of HTT/univalent foundations for assisted proofs
I'm trying to understand what the claimed practical benefits of HTT/univalent foundations are for doing computer assisted proofs and while I've seen a lot of claims of benefits they all seem to be ...
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Can the opposite of an elementary topos be an elementary topos?
This question is not really about elementary topoi, it is much more about a category $(\mathcal{E}, \Omega)$ admitting a subobject classifier, or about a category with power objects, you can choose ...
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Is material set theory conservative over structural set theory?
Suppose a statement $\phi$ that doesn't use the global $\in$-relation or the global $=$-relation in an essential way is provable in some material set theory, say bounded Zermelo with choice. (So that ...
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Top-down mathematics, or "Where it all begins"
Sorry if this is off-topic.
It was my attempt to take a top-down approach to mathematics.
Being an inexperienced undergraduate (so please take my writing here lightly), I've been presented with ZFC as ...
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When the definition of a set starts to matter in category theory
In most introductory courses to category theory, the precise definition of a set is more-or-less ignored. The idea being that all basic results in the subject hold for any reasonable definition of a ...
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