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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How strong is separation + reflection without transitivity?
Consider a theory $T$ with a binary relation $\in$ and the following axiom schemas:
$\exists u \forall x (x \in u \leftrightarrow x \in a \land \phi)$ where $u$ is not free in $\phi$. This is the ...
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Can the axiom of choice be expressed in 4 quantifiers?
This 2007 paper presents a 5-quantifier $(\in, =)$-expression that is ZF-equivalent to the axiom of choice, but leaves open the 4-quantifier case:
Thus the gap is reduced to the undecided case of a 4 ...
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Is there a $\Pi_2$ sentence $A$ such that $\text{ZFC}^- + A$ proves powerset?
This is a follow-up to this question.
Let $\text{ZFC}^-$ be ZFC without powerset and with collection rather than replacement, as described here.
Is there a $\Pi_2$ (or perhaps $\Sigma_2$) sentence $A$ ...
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Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here).
However, it ...
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CH in non-set theoretic foundations
I asked this question on stack exchange and got little attention, barring a nice example I intend to look into. The original post can be found here: https://math.stackexchange.com/q/4941233/1053681
I ...
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Consistency strength of HoTT
What is the consistency strength of Homotopy type theory (HoTT) relative to various set theories (e.g., are there any known set theories that it can interpret)? Does this question even make sense?
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How to define Dedekind reals and Eudoxus reals such that they are equivalent to unmodulated Cauchy reals
In constructive mathematics without choice, we have three different versions of the real numbers (each embedding into the next).
Regular Cauchy reals (functions $f : \mathbb N \to \mathbb Q$ such ...
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Foundations and contradictions of Scholze's work: the category of presentable infinity categories contains itself
Preface: I am not an expert in the work of Scholze, or anything for that matter.
Question
Has Scholze stated what axioms he is using to develop his theory of motives and analytic geometry. In the ...
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Can these short set-building expressions of the finite set world extend to the infinite set world?
A formula of the form $\forall \vec{p}\, \exists x \, \forall y\, (y \in x \leftrightarrow \phi(y,\vec{p}))$
is to be named a "set-building" formula.
Now, when $\vec{p}$ includes a predicate ...
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Type vs. Set Theory: Expressive Ability
In the modern mathematical arena, the two primary contenders for the ‘correct’ foundation of mathematics are set and type theory.
Set theory, very roughly, captures the intuition that we frequently ...
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How big can function spaces get without extensionality?
In what follows we work in the usual formulation of Martin-Löf Type Theory including Axiom K [1]. Boldface numbers $\mathbf{n}$ denote the usual finite type with $n$ elements.
Motivation
Postulating ...
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Negating fundamental axioms
It is commonplace to consider standard axiomatic systems (e.g. $ZF$) with one of the 'less essential' axioms negated, like infinity, 'less essential' here having some ambiguous definition related to ...
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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 ...
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Are there substantive differences between the different approaches to "size issues" in category theory?
In category theory, there are different ways to approach the "size issues" that crop up when we try to formalise the subject in axiomatic set theory. As far as I can tell, there are two main ...
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Recording of 2009 lecture on Harvey Friedman's work
On December 13--20 2009 at Bristol, there was a meeting devoted to thorough dissection of Harvey Friedman's work on the foundations of mathematics and his statements claimed to be equivalent to ...
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Which is richer Set or Graph Theory?
This theory about structures, defined as abstractions over isomorphic graphs, can interpret Set Theory in a rather creepy manner. Though the theory is largely technical, yet it is not far from being ...
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Suitability of formal type theory for mathematical thinking (vs. traditional set theory)
Type theory has advantages over set theory for the (computer) formalisation of mathematics, but has anybody who does mathematics with pen and paper found proof assistants or automated theorem provers, ...
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What governs our "perception?" about the platonic realm of sets?
Here, I want to delve into what do we exactly feel about what constitutes a platonic existence of a set? Or what makes us think or actually a kind of feel or sense the existence of a set in the ...
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The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
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Why not $\sf ZFC+[V=HOD]$?
Why not $\sf ZFC+[V=HOD]$ as the standard set theory?
It implies the existence of a definable global choice and well-order, and it is compatible with all large cardinal axioms extending $\sf ZFC$, so ...
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Does inductive definitions must be supported by the set theoretical definition of natural numbers?
In page 4 of Gödel's book The Consistency Of The Axiom Of Choice and Of The Generalized Continuum Hypothesis With The Axioms Of Set Theory, Gödel defined the $n$-tuple as
$\langle x \rangle = x$;
$\...
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Is Bounding Reflection consistent?
Working in the first order language of set theory.
Let $\varphi^{*B}$ be the formula obtained from $\varphi$ by merely bounding all open quantifiers in $\varphi$ by the symbol "$B$".
Here a ...
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What is the proof of consistency of anterior reflection?
Let Anterior Reflection be the following principle: $$\forall \vec{v}~ \exists X: \operatorname {transitive} (X) \land \, (\varphi \to \varphi^{X"}) $$
where $\varphi$ is a formula in $\sf FOL(=,\in)$ ...
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Can this form of reflection be consistent?
Is this form of reflection consistent?
First I'll begin by clarifying the notation I'm using here:
By a quantifier being relativized or bounded it means that the first occurrence of the quantified ...
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Is this form of replacement suitable for ZF - Powerset + well-ordering principle?
The following scheme can be understood as a form of replacement. Axiomatizing $\sf ZF$ with it instead of the usual replacement schema renders it immune to removal of extensionality; see here.
In an ...
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What's the earliest result (outside of logic) that cannot be proven constructively?
Although mathematicians usually do not work in constructive mathematics per se, their results often are constructively valid (even if the original proof isn't).
An obvious counter-example is the law ...
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Is univalence equivalent to every type function being a functor over equivalence?
Introduce a rule in type theory that if $\Gamma \vdash f : \text{Type} \to \text{Type}$ and $\Gamma \vdash e : A \simeq B$ then $\Gamma \vdash f[e] : f(A) \simeq f(B)$.
It may seem like such a rule is ...
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Are there times when replacement is "more natural" than collection?
There are a couple examples I'm aware of where choosing to axiomatize $\mathsf{ZF(C)}$ using collection instead of replacement results in a much nicer (or at least less surprising) picture:
Let $\...
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Bounded alternatives to powerset that interpret ZFC
In set theory, many properties/relations of interest can be expressed as $\Delta_0$ formulas (formulas with only bounded quantifiers):
\begin{align}
\text{empty}(a) &\equiv \forall x \in a . \...
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How much Dependent Choice is provable in $Z_2$? And what about Projective Determinacy?
So, second order arithmetic, $Z_2$, is capable of proving quite a few things. One thing which would be of use is dependent choice for $\mathbb{R}$.
Basically, dependent choice on $\mathbb{R}$ says ...
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Existence of skeletons in ZFC
Influenced by this question from a fellow lagomorph, I would like to get to the bottom of existence of a skeleton of a category. I want to stay in ZFC, so I do not assume the global axiom of choice. ...
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Second order theory of a real-closed field
It is well known that the first-order theory of any real-closed field is complete, and consequently not capable of interpreting the majority of modern mathematics.
Is this still true for the second-...
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Lists as a foundation of mathematics
I am wondering if there is a foundation of mathematics where not sets or "set-like objects" (such as objects of a suitable topos as in ETCS) are the primitive notion, but rather lists. These ...
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Is Vopěnka's principle inherited by Grothendieck topoi?
I call the Vopěnka's principle:
Every subfunctor of an accessible functor is accessible
but other formulations (which may lose equivalence in weak contexts?) are also interesting to me.
If this is ...
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In HoTT with LEM, are sets and pointed sets the same thing?
The operations of adding and removing a point (where removing is a consideration of a subset of elements x such that $(x = *) \to 0$) implements the equivalence of these 1-types, as far as I can see. ...
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How did Szmielew prove that Pasch's axiom is a consequence of the circle axiom?
It is alleged that Szmielew proved that Pasch's axiom is a consequence of the circle axiom. The source is said to be
The Pasch axiom as a consequence of the circle axiom, Bull.Acad.Polon.Sci.Sér.Sci....
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Propositional calculus, first order theories, models, completeness
In the usual context of model theory one studies first order theories: the Gödel completeness theorem asserts that $\varphi$ is a theorem of a theory $T$ (i.e. $\varphi$ is provable from the axioms of ...
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How much of the axiom of choice do you need in mathematics?
Say we have DC-λ where λ is some inaccessible cardinal. Is that enough to develop all of ordinary mathematics? If not, is there a strengthening that is but that nevertheless does not assume full ...
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Real reverse mathematics
Standard mathematical developments, be they set theoretic, type theoretic, synthetic, etc. all follow the same basic pattern:
Lay out a language, assume some stuff in this language, then prove that ...
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What computable pseudo-ordinals are there with initial segment $\omega_1^{CK}(1+\eta+1)$?
The notion of a “computable pseudo-ordinal”, i.e. a computable linear order with no hyperarithmetical descending chains, is an old one going back to Stephen Kleene. Joe Harrison wrote the definitive ...
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Higher-order oracle computation of reals and axiom of constructibility
Certain real numbers can be approximated arbitrarily well by computable functions. If we introduce halting oracles, then more real numbers can be "computed", like Chaitin's constant or the ...
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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 $\...
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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 ...
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Freiling's question
(I've asked this question at Math StackExchange here but haven't gotten any response, so I decided to take a shot here as well.)
In his paper "Axioms of Symmetry: Throwing Darts at the Real ...
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Infinite decreasing sequence for class relation without minimal elements
Let us assume $<$ is some class relation without minimal elements, meaning $\forall a, \exists b, b< a$. This means that for every $n\in\omega$, one can build a decreasing function $f$ with ...
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Statements in differential geometry independent from ZFC
It is well known that some problems in functional analysis and in general topology are independent from ZFC: to name a few, Kaplansky's conjecture, the existence of outer automorphisms of the Calkin ...
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On the definition of small categories in SGA4
We assume ZFC+U.
A category is an ordered pair $(\operatorname{Ob} \mathcal{C},\operatorname{Mor} \mathcal{C},\operatorname{dom},\operatorname{codom},e,∘)$ of sets (not classes) and maps satifying ...
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Harvey Friedman: The expanding mind
In reference 1, Friedman writes:
I discuss my efforts concerning 3 crucial issues in the foundations of mathematics that are deeply connected with the great work of Kurt Gödel.
[...]
B. Are there ...
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Does every Tarski plane embed into a 3-dimensional Tarski space?
By a Tarski space 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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Can the axiom of choice be proved with ZF+Tarski axiom?
Can choice be proved with ZF+Tarski axiom?
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