All Questions
1,142 questions
27
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3
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Using consistency to create new axioms in set theory
As everybody knows, the ZFC axioms may serve as a foundation for (almost)
all of contemporary mathematics, and it is also well-known that several results
are "indecidable" in ZFC, which means that ...
- 3,595
26
votes
2
answers
1k
views
When does the choice of the generic matter?
It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...
- 2,389
26
votes
4
answers
7k
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What would be some major consequences of the inconsistency of ZFC?
Update (21st April, 2019). Removed the reference / initial trigger behind my question (please see comment thread below for the reasons). Am retaining, of course, the actual question, noted both in the ...
- 28.6k
26
votes
3
answers
7k
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Presburger Arithmetic
Presburger arithmetic apparently proves its own consistency. Does anyone have a reference to an exposition of this? It's not clear to me how to encode the statement "Presburger arithmetic is ...
- 464
26
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4
answers
2k
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Does every set of reals contain a measure-zero set of the same cardinality? Does it contain a meager set of the same cardinality?
This question arises from an issue in my post on Ashutosh's excellent question on Restrictions of the null/meager ideal.
Question 1. Does every set of reals contain a measure-zero subset
of the same ...
- 236k
26
votes
9
answers
8k
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Why are proofs so valuable, although we do not know that our axiom system is consistent? [closed]
As a person who has been spending significant time to learn mathematics, I have to admit that I sometimes find the fact uncovered by Godel very upsetting: we never can know that our axiom system is ...
25
votes
7
answers
3k
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When can we prove constructively that a ring with unity has a maximal ideal?
Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian ...
- 118k
25
votes
2
answers
1k
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Detecting uncountable cardinals in $(\mathbb{R};+,\times,\mathbb{N})$
For a structure $\mathcal{X}=(X;...)$, say that a cardinal $\kappa$ is $\mathcal{X}$-detectable iff there is some sentence $\varphi$ in the language of $\mathcal{X}$ together with a fresh unary ...
- 20.5k
25
votes
3
answers
1k
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What spaces $X$ do have $\text{End}(X) \cong \text{End}(\mathbb{R})$?
This is a follow-up on the following question. Let $\text{End}(X)$ denote the endomorphism monoid of a topological space $X$ (that is, the collection of all continuous maps $f:X\to X$ with composition)...
- 48.9k
25
votes
1
answer
783
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Decidability of equality of elementary expressions
In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...
- 6,819
25
votes
2
answers
2k
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Axiom of choice: ultrafilter vs. Vitali set
It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set,
a set of ...
- 16.2k
24
votes
5
answers
2k
views
Lie groups vs Lie monoids
Does there exist a well developed theory of a class of objects which might rightfully be called Lie monoids? By this I mean with axioms similar to those of Lie groups, but with the axiomatic existence ...
- 2,099
24
votes
1
answer
1k
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Forcing and Family Contentions: Who wins the disputes?
The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible ...
24
votes
8
answers
6k
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Choice vs. countable choice
This question arose after reading the answers (and the comments to the answers) to Why worry about the axiom of choice?.
First things first. In my intuitive conception of the hierarchy of sets, the ...
- 1,848
24
votes
3
answers
3k
views
Non-abelian Grothendieck group
By general nonsense the forgetful functor from groups to monoids has a left adjoint. It maps a monoid $(X,\cdot,1)$ to the free group on $\{\underline{x} : x \in X\}$ modulo the relations $\underline{...
- 63.1k
24
votes
0
answers
3k
views
What's the smallest $\lambda$-calculus term not known to have a normal form?
For Turing Machines, the question of halting behavior of small TMs has been well studied in the context of the Busy Beaver function, which maps n to the longest output or running time of any halting n ...
- 1,734
24
votes
9
answers
2k
views
Self-containing structures
This question is partly inspired by this question: independently of the original context, I'm interested in the general claim* that an ill-founded set theory would represent certain mathematical ...
23
votes
4
answers
3k
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Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?
Last year, Andrej Bauer gave a talk showing that there is a topos in which the set of Dedekind reals is (sub)countable, and thus, you cannot prove that $\mathbb{R}$ is uncountable without LEM. He ...
- 317
23
votes
10
answers
5k
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Completeness vs Compactness in logic
One standard approach to showing compactness of first-order logic is to show completeness, of which compactness is an easy corollary. I am told this approach is deprecated nowadays, as Compactness is ...
- 3,475
23
votes
2
answers
1k
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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 ...
23
votes
1
answer
3k
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What's the difference between ZFC+Grothendieck, ZFC+inaccessible cardinals and Tarski-Grothendieck set theory?
Say that "U" is the axiom that "For each set x, there exists a Grothendieck universe U such that x $\in$ U", where Grothendieck universes are defined in the usual way (or, if that'...
- 4,907
23
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4
answers
2k
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Can we recognize when a category is equivalent to the category of models of a first order theory?
Many of the most canonical early examples of categories
arise as the collection of models of a fixed first order
theory, with the related model-theoretic concept of
homomorphism. For example, the ...
- 236k
23
votes
1
answer
1k
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Is it consistent with ZF that $V\to V^{\ast \ast}$ is always surjective?
In a comment to a recent question, Jeremy Rickard asked whether it is consistent with ZF that the map $V \to V^{**}$ from a vector space to its double dual is always surjective. We know that "...
- 82.7k
23
votes
3
answers
2k
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An ultrafilter is a set of subsets containing exactly one element of each finite partition: reference request
There are probably dozens of ways of defining "ultrafilter". The definition I've seen most often involves first defining "filter", then declaring an ultrafilter to be a maximal filter.
But there's ...
- 27.7k
23
votes
4
answers
20k
views
Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?
In a conversation where it came up that the Pythagoreans probably found an enumeration of the rational numbers I erroneously remarked that Georg Cantor found a natural bijection from $\mathbb{N}$ to $\...
- 3,574
22
votes
4
answers
3k
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Impredicativity
I hope this question is not so elementary that it'll get me banned...
In mathematics we see a lot of impredicativity. Example of definitions involving impredicativity include: subgroup/ideal ...
- 231
22
votes
5
answers
2k
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Platonic Truth and 1st Order Predicate Logic
Consider the following simple example as motivation for my question. If it were the case that, say, the Riemann hypothesis turned out to be independent of ZFC, I have no doubt it would be accepted by ...
- 18.7k
22
votes
6
answers
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Is any interesting question about a group G decidable from a presentation of G?
We say that a group G is in the class Fq if there is a CW-complex which is a BG (that is, which has fundamental group G and contractible universal cover) and which has finite q-skeleton. Thus F0 ...
- 1,219
22
votes
4
answers
1k
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Is there a Leibnizian model with no definable elements, in a finite language?
A first-order structure $M$ is Leibnizian, if any two distinct
elements $a,b\in M$ satisfy different $1$-types; that is, if there
is some formula $\varphi$ such that $M\models\varphi(a)$ and
$M\models\...
- 236k
22
votes
1
answer
686
views
Is a model of set theory determined by the Cohen reals over it?
This question concerns the amount of information about a model $M$ that is contained in the collection of all reals Cohen over $M$.
Specifically, let $M$ and $N$ be countable transitive models of ZFC ...
- 2,389
22
votes
2
answers
2k
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Non standard Algebraic Topology
Let *$\mathbb R$ a field of non-standard real numbers (or any real closed field) equipped with its natural generalized metric $d(x,y)=|x-y|$. Equip *$\mathbb R^2$ and *$\mathbb R^3$ with the $\ell^1$-(...
- 6,034
22
votes
3
answers
2k
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Nice algebraic statements independent from ZF + V=L (constructibility)
Background and motivation
I've always been fascinated about algebraic statements independent from ZFC set theory. One such fascinating example comes from considering $\rm{Ext}^1_\mathbb{Z}(A,\mathbb{Z}...
user1437
22
votes
3
answers
3k
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Half Cantor-Bernstein without choice
I had a discussion with one of my teachers the other day, which boiled to the following question:
Assume ZF. Let $A,B$ be sets such that there exist $f\colon A\to B$ which is injective and $g\colon A\...
- 39.8k
21
votes
2
answers
1k
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Antirandom reals
This is a crossposting of https://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it ...
- 20.5k
21
votes
9
answers
5k
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Was the early calculus inconsistent?
This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.
George Berkeley wrote in 1734 with reference to the early calculus that ...
- 16.6k
21
votes
4
answers
2k
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Is there a non-Hopfian lacunary hyperbolic group?
The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then ...
- 25k
20
votes
1
answer
4k
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Where can I find Gonthier's Coq code proving the four color theorem?
In a 2008 article in the Notices, Georges Gonthier announced a computer-checked proof of the four color theorem using Coq:
Gonthier, Georges. Formal proof—the four-color theorem.
Notices Amer. ...
- 29.8k
20
votes
5
answers
2k
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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 ...
- 21.3k
20
votes
3
answers
2k
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Can FPA really prove its consistency?
I will ask the question first and then explain.
QUESTION: FPA can prove its own consistency in the Godelian sense. But can it really prove its consistency?
FPA is a multi-sorted first-order theory,...
- 1,974
20
votes
2
answers
2k
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Non-definability of graph 3-colorability in first-order logic
What is a proof that graph 3-colorability is not definable in first order logic? Where did it first appear?
- 309
20
votes
2
answers
1k
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An order type $\tau$ equal to its power $\tau^n, n>2$
(This is a re-post of my old unanswered question from Math.SE)
For purposes of this question, let's concern ourselves only with linear (but not necessarily well-founded) order types.
Recall that:
$...
- 6,819
20
votes
4
answers
2k
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Nuances Regarding Naturality
It's frequently said, informally, that a natural isomorphism is one that doesn't depend on arbitrary choices.
But the phrase "arbitrary choices" lends itself to different interpretations. Consider ...
- 23k
20
votes
4
answers
3k
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A New Continuum Hypothesis (Revised Version)
Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...
user42090
20
votes
1
answer
1k
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Is the one-point compactification of $\mathbb{N}$ computably countable?
The one-point compactification $\mathbb{N}_\infty$ of $\mathbb{N}$ is obtained from the discrete space $\mathbb{N}$ by adjoining a limit point $\infty$. It may be identified with the subspace of ...
- 48.8k
20
votes
1
answer
731
views
Model-completeness of real exponential fields
Let $\mathbb R$ denote the ordered field of the reals (in a language with $+$, $\cdot$, and possibly $<$, $0$, $1$, or $-$; these are all existentially definable in terms of $+$ and $\cdot$ alone).
...
- 47.5k
20
votes
1
answer
2k
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Joyal's construction of the spectrum of a commutative ring
I am trying to understand bits and pieces of Lawvere's article Continuously Variable Sets; Algebraic Geometry = Geometric Logic. I'm not doing very well.
I know this is a lot to ask, but basically, I ...
- 10.5k
20
votes
1
answer
1k
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Axiom of Choice versus V=L in opposition to large cardinals
Consider the following two observations:
The axiom $V=L$ is incompatible with large cardinal axioms that are somehow "too large", like measurable cardinals.
The axiom of Choice is incompatible with ...
- 32.5k
20
votes
1
answer
2k
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Axiom of choice and bases of vector spaces over a fixed field
Let $k$ be a field. In 1984 Andreas Blass proved that the axiom "for every extension $K|k$, every vector space over $K$ has a basis" implies the axiom of choice. He also raised the question
Does ...
- 16.2k
19
votes
1
answer
1k
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Goodstein's theorem without transfinite induction
Goodstein's theorem is an example of a theorem that is not provable from first order arithmetic. All proofs of the theorem seem to deploy transfinite induction and I've wondered if one could prove the ...
user10891
19
votes
3
answers
3k
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Can you show rank E(Q) = 1 exactly for infinitely many elliptic curves E over Q without using BSD?
Let $K$ be a number field and let $\mathcal O_K$ be the ring of integers. Following this paper of Cornelissen, Pheidas, and Zahidi, a key ingredient needed to show that Hilbert's tenth problem has a ...
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