All Questions
Tagged with set-theory foundations
157 questions
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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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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 $\...
CommunityBot
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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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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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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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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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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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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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Why would the category of sets be intuitionistic?
This question is probably really naive. And, I hope the title doesn't come off as too combative. I think that topoi of $\mathbf{Set}$-valued sheaves provide an excellent motivation for higher-order ...
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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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Category theory and set theory: just a different language, or different foundation of mathematics?
This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics.
I am asking for a reference. In order to make the reference request as ...
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Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?
I've asked a related question about nine months ago here, however, apparently, I lacked expertise to ask the precise question I want to ask here, as I wish to revisit the matter of universes. I hope ...
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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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12
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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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3
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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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When size matters in category theory for the working mathematician
I think a related question might be this (Set-Theoretic Issues/Categories).
There are many ways in which you can avoid set theoretical paradoxes in dealing with category theory (see for instance ...
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1
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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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2
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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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6
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2
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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 $...
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1
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1
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Complete and consistent first-order theories that contain interesting phenomena
Gödel has shown that a consistent recursively axiomatizable first-order theory that can interpret Robinson arithmetic is incomplete.
I think there is some sentimental value in working with a theory ...
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4
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Where is the end of universe?
In some sense the empty set ($\emptyset$) and the global set of all sets ($G$) are the ends of the universe of mathematical objects. The world which $ZFC$ describes has an end from the bottom and is ...
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22
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Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?
Let the proof theoretic ordinal $\alpha$ of a theory $T$ be the least recursive ordinal such that $T$ does not prove that $\alpha$ is well-founded. This ordinal is intended to quantify in some sense ...
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Erroneous proof of recursion theorem examples
In his book Elements of Set Theory, Herbert Enderton defines (p. 70) a Peano system as a triple $(N, S, e)$ where $N$ is a set, $S$ is an $N$-valued function defined on $N$ and $e$ is a member of $N$ ...
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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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7
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(Non?)-linearity of the consistency strength ordering in ZF
Much of the research taking place in set theory, is related to the classification of sentences according to their consistency strength relative to ZF. In order to be more specific, we say that for all ...
157
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5
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What makes dependent type theory more suitable than set theory for proof assistants?
In his talk, The Future of Mathematics, Dr. Kevin Buzzard states that Lean is the only existing proof assistant suitable for formalizing all of math. In the Q&A part of the talk (at 1:00:00) he ...
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What is difference between working with small and large category of spaces?
The following construction have always bugged me. This is p328, Remark 5.1.6.1 in Lurie's Higher Topos Theory. Lurie begins with the following:
Construction: Let $C$ be a simplicial set. $S$ denote ...
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1
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Set Theoretic Geology II: The structure of the directed partial order of grounds
In my previous question Set-theoretic geology: controlled erosion?
and the great answer by Jonas Reitz, I have learned a few things, starting from the awareness that I understand the fine-grain ...
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4
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775
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Self-contained formalization of random variables?
I have not been able to find any formalization of random variables that supports construction of new random variables dependent on previously constructed ones. In what I have found, a random variable $...
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5
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Class theory of ZF-minus-Powerset as classical predicative system?
I've been thinking about some mathematics in weaker foundational systems a little bit, largely from a structural viewpoint, and with particular attention to classes.
Some categories I've been keeping ...
CommunityBot
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4
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On a weak choice principle
[PLEASE SEE EDITS AT BOTTOM OF QUESTION]
Consider the following set-theoretic axiom:
For each set $X$ there exists a set-indexed collection $\{C_i \to X\}_{i\in I_X}$ of surjections such that for ...
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2
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Could Kronecker accept a proof of Goodstein's theorem?
A famous result of Goodstein asserts that the Goodstein sequence of integers terminates.
For a precise statement and a short proof, see https://en.wikipedia.org/wiki/Goodstein%27s_theorem.
A well ...
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6
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1
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Historical origin of the empty set
The question is in the title:
Who first claimed the existence / necessity of the empty set ? When did this happen ?
Of course I know that the notation $\emptyset$ goes back to André Weil, and that ...
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8
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2
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weakening naive comprehension to avoid the paradoxes
Weakening the axiom of naive comprehension has not been a popular way of escaping from the set-theoretic paradoxes because no consistent weakenings seem to be particularly well motivated or even to ...
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2
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Shortest axiom of infinity for foundationless set theory
Let $T$ be the theory with a binary symbol $\in$, an unary symbol $S$, and the following axioms:
Axiom of extension:
\begin{equation}
\forall x \forall y (\forall z (z \in x \leftrightarrow z \in ...
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3
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How strong is this set theory?
In the spirit of this related question, consider a set theory with the following axioms:
Axiom of extension:
$$ \forall x \forall y (\forall z (z \in x \leftrightarrow z \in y) \rightarrow x = y) $$
...
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1
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Set of definable real numbers?
Is there a set theory at least as strong as $KP\omega$ which has as a theorem that there is a set $\mathbb{D}$ of precisely the definable real numbers?
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What does second order set theory give us that is new?
There is a natural analogy between the theories PA and ZFC. See the linked question by Gro-Tsen here.
Peano arithmetic (PA) is a first order approximation to the natural numbers. As is well known, ...
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Weak power set - what strength may it have? [closed]
In The Consistency of Classical Set Theory Relative to a Set Theory with Intuitionistic Logic in THE JOURNAL OF SYMBOLIC LOGIC Volume 38, Number 2, June 1973 page 316 Harvey Friedman's axiom 8* $Weak \...
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A countable set theory providing choice?
Instead of Zermelo set theory $Z$ take $Y$ = $Z$ minus the power set axiom plus
Enumerability: $\forall x(x\neq \emptyset \to\exists f[f:\mathbb{N}\overset{onto}{\frown}x ])$
$\imath$ is the ...
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Is ETCS well-founded?
I can't find a statement about the axiom of regularity anywhere in treatments of ETCS. Perhaps this is due to the unfortunate clash of terminology with 'foundations'.
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Anti-foundational set theory with a universal set
There are alternative set theories that allow for a universal set, e.g. NF(U), positive set theory and and topological set theory. There are also alternative set theories like ZFA that allow for the ...
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Formalizing ontological optimism
Inform speaking ontological optimisms means that everything that possibly exists in the abstract reality actually exists. From this principle we (again informally) get the Axiom of infinity, the Power ...
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4
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Is well-pointedness the reason that the internal/external distinction seems not to apply to $\mathbf{Set}$?
When reasoning about the category of sets, we usually don't have to worry about the internal/external distinction. For example, if $f : X \rightarrow Y$ is a morphism of sets, then $f$ is either ...
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Why not replace reflection by bounded reflection in Muller's approach?
Bounded Reflection: If $\phi$ is a formula in the language of set theory [i.e.; $\small \sf FOL(=,\in)$], in which all and only symbols $``x,x_1,..,x_n"$ occur free, and $\phi^V$ is the formula ...
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Cardinals in $ZFC+\neg CH$
Before asking my question I like to admit that i am not an expert in the foundation of mathematics and I am interested in this issue more from a philosophical perspective. So my question may be ...
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Which branches of mathematics can be done just in terms of morphisms and composition?
Consider the first-order language $L_{\omega\omega}$ of the signature $L:=\{\mathrm{dom}, \mathrm{cod}, \mathrm{comp}\}$, where $\mathrm{dom}$ and $\mathrm{cod}$ are unary function symbols and $\...
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ZF(C) and category theory
Is there an axiomatisation of some kind of category theory and a definition of sets in this framework such that the axioms of ZF resp. ZFC are theorems?
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Does this axiomatic system satisfy requirements for founding mathematics?
In this article, the author, F.A.Muller, suggests criteria for a founding theory of mathematics (pp:14-16). The author proposes $ARC$ Class Theory to embody these requirements. The motivation is ...
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