All Questions
Tagged with set-theory foundations
157 questions
3
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0
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Formal foundations done properly [closed]
I would want to do mathematics properly, so that the proofs of results can be trusted on instead of them being just suggestions on which results could perhaps apply. This means formulating the math in ...
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3
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446
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How do we formally construct the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$?
A set $\mathscr{U}$ is a universe if the following conditions are met:
For any $x \in \mathscr{U}$ we have $x \subseteq \mathscr{U}$
For any $x,y \in \mathscr{U}$ we have $\{x,y\} \in \mathscr{U}$,
...
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12
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1
answer
603
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Translating Grothendieck axiom UB into ZFC
In SGA 4, Grothendieck introduced a set-theoretic device called a Grothendieck universe. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but ...
- 6,700
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1
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Enhancing Grothendieck's universes and Grothendieck's axiom: Feferman's universe
A Grothendieck's universe is such a set $U$ so that
$\forall x \in U, x \subseteq U$,
$\forall x,y \in U, \{x,y\} \in U$,
$\forall x \in U, \mathcal{P}(x) \in U$,
given a family $(X_i)_{i \in I}$ ...
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5
votes
3
answers
488
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Counting without one-to-one correspondence? [closed]
Ash and Gross in their wonderful book Fearless Symmetry found it worth mentioning (and thus suggesting) another way of counting for which "we do not even need to know how to count" (in the sense of ...
- 4,786
12
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3
answers
649
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Has the Ramified Theory of Types been applied to NBG?
Questions of predicativity are well-studied in the context of arithmetic. We have a base theory, first-order Peano arithmetic. Some people, like Edward Nelson (in chapter 1 of his book) and Charles ...
- 2,167
15
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2
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Does foundation/regularity have any categorical/structural consequences, in ZF?
(Prompted by reflection on this old answer, and its suggestion of the “harmlessness” of the axiom of regularity.)
In ZFC, one may justify the axiom of foundation (AF, aka the axiom of regularity) as ...
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4
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0
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424
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What are the requirements of a foundational theory?
There are multiple languages to describe all of mathematics, and there are some equivalences between them, some more successful then others.
My question is can we describe some requirements (in some ...
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1
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317
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Is it natural to hold that Ur-elements, small & big sets and proper classes exists? [closed]
The topic of this post was shifted to
https://philosophy.stackexchange.com/questions/49504/is-it-natural-to-hold-that-big-sets-and-proper-classes-exist
Since it was deemed to be a philosophical ...
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2
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0
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92
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Constructing the Von Neuman Hierarchy at ω+ω in a structural set theory
I'm working in SEAR which is a relatively new structural set theory, and I am trying to prove the existence of big sets.
SEAR has the collection axiom which is, loosely speaking, that for every ...
8
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1
answer
1k
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Category theory without axiom of choice
I'm looking for references on the development of (some of) Category theory without the axiom of choice. One possible axiom system (that, to me, seems the natural setting) is ZF + there are arbitrarily ...
- 66.8k
50
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4
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Do set-theorists use informal set theory as their meta-theory when talking about models of ZFC?
Here, Noah Schweber writes the following:
Most mathematics is not done in ZFC. Most mathematics, in fact, isn't done axiomatically at all: rather, we simply use propositions which seem "intuitively ...
- 403
8
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1
answer
1k
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Does equality between sets contradict the philosophy behind structural set theory?
Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. This means that for every two objects/sets $a,b$ one can ask ...
- 2,754
2
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0
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150
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Is there equality between sets in structural set theory?
In material set theory, the axiom of extensionality defines equality between sets: two sets are equal iff they have the same elements. In structural set theory, one cannot formulate this.
But however,...
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1
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How are material set theory and structural set theory related from the point of view of category theory?
In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman ...
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19
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2
answers
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Which kind of foundation are mathematicians using when proving metatheorems?
Zermelo-Fraenkel set theory (with choice) is commonly accepted as the standard foundation of mathematics. It is a material set theory. For every two objects/sets $a,b$ one can ask whether $a=b$ or not....
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55
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10
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How should a "working mathematician" think about sets? (ZFC, category theory, urelements)
Note that "a working mathematician" is probably not the best choice of words, it's supposed to mean "someone who needs the theory for applications rather than for its own sake". Think about it as a ...
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1
vote
0
answers
223
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What should one know about abstract sets and structural foundations?
Recently I came by accident across the book sets for mathematics by Lawvere. It says:
First we deplete the object of nearly all content. We could think of an idealized
computer memory bank that ...
0
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1
answer
678
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Why do we try to encode every mathematical object as a set? [closed]
Probably everyone of us has seen set-theoretic encodings of mathematical objects which we wouldn't naturally consider to be sets. May it be the "definition" of a function from $A$ to $B$ as a relation ...
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4
votes
3
answers
915
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Compactness of existential second order logic and definability of certain quantifiers
It is known (as a slogan) that the "existential fragment of second-order logic (ESO) is compact".
My first question is:
(1) Is ESO compact for:
(a) uncountable languages
(b) languages with ...
2
votes
0
answers
264
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About the limitation by size
This could be a big post, so I'll try to summarize my thoughts and divide them into several questions.
When working in category theory, I used to choose the following definition. A category $C$ is ...
- 1,017
10
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4
answers
1k
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Direct axiomatization of ordinal and cardinal numbers
Again, this question is related (**) to a previous one:
in standard books on basic set theory, after stating the axioms of ZFC, ordinal numbers are introduced early on. Afterwards cardinals appear: ...
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9
votes
1
answer
856
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Taller models of ZFC
This question is somewhat related to a previous one, where I asked for new forms of infinite beyond the cardinal hierarchy.
Using forcing techniques, at least the ones I know of, one starts from a ...
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7
votes
1
answer
429
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How many closed measure zero sets are needed to cover the real line, really?
This is a refinement of an earlier question.
This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
For the reader's convenience, I reproduce below the ...
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9
votes
1
answer
603
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How many closed measure zero sets are needed to cover the real line?
This question assumes familiarity with combinatorial cardinal characteristics of the continuum.
Let $\mathcal{E}$ be the $\sigma$-ideal generated by closed measure zero subsets of the real line. It ...
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19
votes
0
answers
703
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The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?
For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
- 3,104
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1
answer
262
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An axiomatic system with a set of constants that form a complete ordered field [closed]
I am developing a ZFC axiomatic system where together with the empty set, there is a singular (and huge) set of constants that are themselves sets and form a complete ordered field (cof) these ...
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9
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3
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2k
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What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?
Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
- 437
6
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2
answers
1k
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Why can't mathematics be formalised in terms of classes rather than sets? [closed]
I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...
- 42.8k
5
votes
1
answer
191
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Class theory with support for self-application of class functions?
To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be:
$\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$
$\underline{n+1}(f) = \underline{n}...
- 236k
14
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1
answer
2k
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Martin's "Philosophical Issues about the Hierarchy of Sets"
Some months ago (October 2010), in the context of the Workshop on Set Theory and the Philosophy of Mathematics, Professor Donald A. Martin gave a talk entitled "Philosophical issues about the ...
- 3,793
2
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1
answer
609
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Would a non-constructible set become constructible if we had oracles of arbitrarily high cardinality for the halting problems of ordinal computers?
I still have trouble to grasp the concept of a non-constructible set, my intuition is that we could "avoid" the non-constructibility of many of them if we assume we have "ordinal computers" extended ...
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12
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2
answers
748
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Ways to define "definability"
The notion of a definable set is not expressible in the language of set theory: there is no formula $\delta(x)$ that is equivalent with there being a formula $\phi(y)$ such that $x = \lbrace y : \phi(...
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4
votes
2
answers
845
views
Functor category's objects fail to be a class?
Given two categories, we can form the functor category whose objects are functors. Functors by definition consist of two mappings from in general classes to classes, which makes it fail to be a set. ...
- 54.6k
4
votes
2
answers
452
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On wild behavior of $\omega_{1}$ in the absence of some essential axioms of $ZFC$
The regularity of $\omega_{1}$ is one of the most well known facts of set theory. But it seems that in order to prove this simple fact we need the "full power" of mathematics! For example by an ...
- 236k
1
vote
0
answers
257
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Is there a non-trivial consistency preserving transformation?
In set theory "equiconsistency" (and not "consistency") of the theories is the main part of researches. So we usually try to construct a new model using a given one. In the ...
- 32.1k
18
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4
answers
2k
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Does the existence of the von Neumann hierarchy in models of Zermelo set theory with foundation imply that every set has ordinal rank?
Let $T$ be the theory consisting of Zermelo's original set theoretic axioms (extensionality, empty set, pairing, union, powerset, infinity, separation, choice) together with foundation. Put more ...
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21
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6
answers
3k
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Where in ordinary math do we need unbounded separation and replacement?
[I have updated the question after initial comments in the hope of clarifying it.]
I do quite a bit of reasoning, typically about topology and metric spaces, in "non-standard" foundations, such as ...
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-3
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3
answers
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Can different extensions of ZF have contradictory consequences for first-order arithmetic?
My question is basically, does there exist a statement X independent of ZF such that ZF + X implies a statement P of first-order arithmetic, but ZF + not X implies not P?
Now X cannot be the axiom ...
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6
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1
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Surreal numbers and large cardinals
This is a question in two parts about the interaction of surreal numbers and large cardinals, in both cases just a request for references on the subject.
Part 1 is about foundations. Much of the ...
- 73.2k
1
vote
1
answer
261
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Finite level super classes over ZFC
My question is: "Is it possible to have a sound and rigorous legitimation of the following construction ?". This construction is:
0/ Let ZFC be the usuel set theory, and let us add to the language ...
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10
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1
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Set-theoretical multiverse and foundations
I just had a look to the article The set theoretical multiverse by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, ...
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0
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1
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Good set theory in which to study ordinal-indexed sequences?
I'd like to "model" the absolute complement of a set $X$ as the ordinal-indexed sequence $\alpha \mapsto V_\alpha \setminus X$ where $V_\alpha$ is the $\alpha$ stage of the cumulative hierarchy. My ...
- 3,793
3
votes
1
answer
422
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Should functions be assumed to behave like the identity function when evaluated outside their domain?
Suppose we have a set $f$ of ordered pairs (so not a triple $(X,Y,f)$ but just the $f$) and suppose that $f$ has the appropriate property such that we can view $f$ as a function. Formally, we wish to ...
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11
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A function that is defined everywhere but has unknown values [closed]
For pedagogical purposes I am looking for a function $\mathbb{N}\to\mathbb{N}$ that is defined everywhere but has most of its values unknown. Although such a function cannot be simple by definition, ...
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5
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2
answers
1k
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Large cardinals without the ambient set theory?
In an attempt to understand a bit better large cardinals, I have been thinking along the following lines, which could be summarized under the slogan
Talk about cardinals without the
(ambient) ...
- 21.3k
1
vote
1
answer
535
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Tarski-Grothendieck set theory, the axiom of pairing and the axiom of specification
I am building upon MO question 102846 concerning the Tarski-Grothendieck set theory (TG).
I have two questions;
1/ I think that it is possible that the axiom of pairing (axiom 4 of the TG theory ...
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9
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1
answer
798
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Ultimate Maximality Principle
I wonder if it's possible to formulate an "ultimate" maximality principle (UMP) and prove its consistency. I envision UMP to express the idea that no matter how we enlarge the universe of set theory V ...
- 236k
2
votes
1
answer
275
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comprehension and ideal elements
A not uncommon thought in philosophy is that we should distinguish (in philosophy, anyway) between "sparse" ("real", "serious") and "abundant" ("ideal", "superficial") properties/classes and relations....
- 236k
6
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1
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Some questions about Ackermann set theory
In a comment on this site Andreas Blass stated:
"To fit this situation into my philosophical point of view, I'd say that what Ackermann's theory
calls proper classes are really certain sets. That ...
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