All Questions
1,135 questions
11
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3
answers
942
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What is the smallest variety of algebras containing all fields?
A field is a ring whose nonzero elements form a commutative group under multiplication. A field is also a commutative inverse semigroup with respect to multiplication. The unique multiplicative ...
- 2,464
10
votes
1
answer
2k
views
Generating family for the Lebesgue $\sigma$-algebra
Let $X$ be a set, and $\cal F$ a family of subsets of $X$, let $\Sigma(\cal F)$ denote the smallest $\sigma$-algebra containing $\cal F$. We can also define $\Sigma(\cal F)$ internally using a ...
- 39.8k
11
votes
1
answer
2k
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What is needed to prove the consistency of Tarski's Euclidean geometry?
This question might be too elementary for MO, in which case I would gladly move it to math.stackexchange.com
Consider Tarski's axiomatization of Euclidean Geometry. It is stated in the wikipedia ...
- 26k
11
votes
1
answer
711
views
Can we separate the almost-disjointness sunflower numbers?
This question concerns a new cardinal characteristic of the
continuum that arose out of issues in my answer to the question,
Sunflowers in maximal almost disjoint
families.
A family $\cal A$ of ...
- 236k
11
votes
1
answer
614
views
Does every nonempty definable finite set have a definable member?
I asked this on MSE yesterday ( https://math.stackexchange.com/q/197873/39378 ) but no one has answered it yet. I hope it's not too soon to post it here.
Here are a few ways to formalize the ...
- 5,471
10
votes
0
answers
514
views
Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < \delta$)...
- 2,231
11
votes
3
answers
1k
views
The concept "conjugate class" in monoids.
Is there any concept in monoids that is similar to the concept "conjugate class" in groups? For example, are there any such similar concept in symmetric inverse monoids? Thank you very much.
- 6,211
11
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1
answer
427
views
Is there a finitely axiomatizable class of structures whose equality-free theory is not finitely axiomatizable?
This was originally an MSE question, but I was told to ask it on MathOverflow. Does there exist a class $C$ of $L$-structures for a finite signature $L$, which is finitely axiomatizable in first-order ...
- 2,023
11
votes
2
answers
808
views
What is the depth of the "provability hierarchy"?
I am not a logician or set theorist, so hopefully this makes sense. Let $T$ be a theory which is expressive enough to make statements like "Statement $A$ has a proof in $T$"; for example, $...
- 23k
11
votes
1
answer
924
views
Causality, if any, in mathematics itself
Mathematicians often express comments like "X is true because Y and Z are true". One's sense of mathematical causation is also a major part of mathematical intuition.
But causality per se is ...
- 2,858
11
votes
1
answer
1k
views
The (un)decidability of Robinson-Arithmetic-without-Multiplication?
I asked this over at math.stackexchange, and though a number of people were interested enough to vote up the question, I didn't get an answer -- which makes me wonder whether it isn't quite so trivial/...
- 1,599
10
votes
2
answers
455
views
Is equivalence of functions built from nested exponentiations a decidable problem?
Let $\mathcal{E}$ be the minimal set of symbolic expressions (without any predefined meaning) such that
The symbol $x$ is in $\mathcal{E}$, and
If expressions $P,Q\in\mathcal{E}$, then the ...
- 1,699
11
votes
1
answer
730
views
Set-theoretical multiverses and their representation as functors? Why *the* multiverse?
In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...
- 213
11
votes
3
answers
794
views
When are two forcing posets "the same"?
Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a ...
- 877
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3
answers
2k
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Use of Conjectures to Prove a Theorem
Name a theorem T that has a proof based upon the truth of a conjecture C, and also has another proof based upon the falsehood of the same conjecture C, but for longtime has no known direct proof that ...
- 113
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1
answer
564
views
Is the inclusion version of Kunen inconsistency theorem true?
The relations $\in$ and $\subsetneq$ seem so similar in some sense. For example they are equal on ordinal numbers. So there is a natural question about their possible similar behaviors on the ...
user36136
11
votes
1
answer
441
views
Concerning Silver's result
Jack Silver proved that if $x$ is a real so that every $x$-admissible ordinal is a cardinal in $L$, then $0^{\sharp}$ exists.
I wonder whether various weaker or stronger versions of Silver's result ...
- 4,201
11
votes
1
answer
541
views
Is every set being cardinal definable consistent with ZF + negation of Choice?
Recall the definition of cardinal definable, where every set being cardinal definable is proved consistent relative to ZF + V=HOD. To re-iterate it:
$Define: X \text { is cardinal definable} \iff \\\...
- 11.3k
11
votes
3
answers
534
views
Who proved "sets in every generic are already in the ground model?"
Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if ...
- 20.5k
11
votes
1
answer
625
views
Cut-free proofs in ZFC
If a statement $P$ has a ZFC proof of length $n$, must it also have a cut-free ZFC proof of length polynomial in $n$?
By a cut-free ZFC proof, I mean a proof in sequent calculus without cut rule of ...
- 7,545
11
votes
2
answers
2k
views
Can linear logic be used to resolve unexpected hanging/surprise examination paradox?
In the Unexpected Hanging Paradox, the prisoner tries to narrow down their date of execution using seemingly sound logical reasoning. They instead arrive at a contradiction. When the paradox is ...
- 4,943
10
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0
answers
216
views
Can we find minimal-diameter metrics without computability?
A beautiful argument by Nabutovsky and Weinberger (see http://math.uchicago.edu/~shmuel/fractal.ps) shows that, if $M$ is any smooth compact manifold of dimension $\ge 5$, then the diameter functional ...
- 20.5k
10
votes
1
answer
580
views
Can you have many independent reals?
Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:
$\Bbb P_\alpha$ is c.c.c.
$\Bbb P_\alpha$ adds a real which ...
- 39.8k
10
votes
1
answer
786
views
Can an ultrapower be undone by forcing?
I am not 100% certain this question is appropriate for MO; I may just be missing something obvious. Also, I vaguely recall a similar question being asked here a while ago, but I can't find it; if it ...
- 20.5k
10
votes
1
answer
599
views
Is Vopenka's Principle + "ORD has the tree property" consistent?
Vopenka's principle implies the existence of weakly compact cardinals (a proper class of them, I believe). My question is whether Vopenka's principle is consistent with the assertion that the universe ...
- 64k
10
votes
2
answers
716
views
On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
- 613
10
votes
5
answers
1k
views
On the notion of partial semigroup
A partial binary operation on a set $X$ is just a (partial) function $\varphi: X \times X \rightharpoonup X$ (I'm using \rightharpoonup for partial maps), and a partial magma is a pair $\mathbb M = (M,...
- 10.5k
10
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1
answer
2k
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Finite order arithmetic and ETCS
I'm looking for a reference to the statement that Lawvere's Elementary Theory of the Category of Sets (ETCS) is equal in proof-theoretic strength to finite order arithmetic. The person who informed ...
- 27.7k
10
votes
2
answers
363
views
Source on smooth equivalence relations under continuous reducibility?
This question was asked and bountied at MSE, but received no answer.
In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
- 20.5k
10
votes
1
answer
262
views
Does every linear cover contain a minimal cover?
This is a follow-up question to an older question.
Let $X\neq \emptyset$ be a set. We say that ${\cal C}\subseteq {\cal P}(X)$ is a cover if $\bigcup {\cal C} = X$, and we call ${\cal C}$ linear if $|...
- 48.9k
9
votes
0
answers
440
views
A new maximality principle and its consequences
Let us consider the following maximality principle:
$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$
and all trees of height and size $\kappa$ are specialized.
It ...
- 32.2k
10
votes
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: ...
- 7,853
10
votes
1
answer
510
views
Examples of proofs using induction or recursion on a big recursive ordinal
There are many proofs use induction or recursion on $\omega$, or on an arbitary (may be uncountable) ordinal. Are there some good examples of proofs which use a big but computable ordinal?
The ...
- 475
10
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1
answer
480
views
Is every set smaller than a regular cardinal, constructively?
Constructively, my only interest in regular cardinals is in terms of the “$\Sigma$-universes” they generate. By a $\Sigma$-universe, I mean a collection of triples $(X,Y,f: X \to Y)$ closed under base ...
- 64k
10
votes
4
answers
978
views
On surjections, idempotence and axiom of choice
The following assertion is trivial in ZFC, or even in much weaker theories. Is it also true in ZF?
(I couldn't find it in the Consequences site so far.)
If $A$ is an infinite set such that $A$ can ...
- 39.8k
10
votes
1
answer
889
views
What is the theory of the random poset?
$\DeclareMathOperator\Th{Th}$The random poset is the Fraisse limit of the class of finite posets, just like the random graph is the Fraisse limit of the class of finite graphs? That is, the random ...
- 4,597
10
votes
2
answers
2k
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Second-order term in first-order logic?
Could a function in FOL take functions as arguments? FOL only limits on the order of the individuals being quantified, but if an expression does not involve quantifying over second-order or higher ...
- 103
10
votes
1
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462
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Reverse mathematics of meromorphic functions on Riemann surfaces
Various sources touch briefly on the reverse mathematics of measure theory and complex analysis. But I have found none on the uniformization theorem for Riemann surfaces or the existence of non-...
- 11.1k
10
votes
2
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1k
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What's the exact consistency strength of this axiom system for classes and sets?
Notation: Let $\phi$ be any formula in $\mathsf{FOL}({=},{\in}, W)$; let $\varphi$ be any formula in $\mathsf{FOL}({=},{\in})$ having $x$ free, and whose parameters are among $x_1,\dotsc,x_n$.
Note: “$...
- 11.3k
10
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1
answer
986
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Applications of Morley's Categoricity Theorem
I just attended a lecture by Rami Grossberg and he mentioned that he is not aware of any applications of Morley's Categoricity Theorem. This is exactly my question.
Question: Do you know of any ...
- 2,882
10
votes
0
answers
377
views
Model for "$\kappa$ limit cardinal iff $2^\kappa$ limit cardinal"
Is there a model of ${\sf (ZFC)}$ such that in the model we have that $\kappa$ is a limit cardinal if and only if $2^\kappa$ is a limit cardinal?
- 48.9k
10
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1
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257
views
Can we always "sharpen" interpretations?
For the purposes of this question, a $T$-interpretation with arity $n$ will be a tuple $\Phi=(\delta,\eta,F)$ where
$\delta$ and $\eta$ are individual formulas of arity $n$ and $2n$ respectively,
$T$...
- 20.5k
10
votes
2
answers
510
views
Definability of the ring of integer in algebraic extensions of $\mathbb Q$
J. Robinson has proved that exists a formula $\psi(x)$ in the language of rings which,applied to the rational numbers, defines the the ring integers (making the theory of $\mathbb{Q}$ undecidable, due ...
- 101
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1
answer
440
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Reference for a generalization of Γ-spaces to monoidal model categories
Γ-spaces were introduced by Segal in 1969 as models for what can be now described
as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
- 37.8k
10
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1
answer
2k
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Who invented Monoid?
I was trying to find (and failed) the original author of either
the concept of Monoid (set with binary associative operation and identity)
the name (which sounds french ? and also Dioid (for what ...
- 203
10
votes
1
answer
3k
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Axiom of choice and non-measurable set
We know that existence of a Lebesgue non-measurable set follows from the Axiom Of Choice. Is the converse true? That is, does the existence of a Lebesgue non-measurable set imply the Axiom Of Choice?...
- 101
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1
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514
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Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$
Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$.
...
- 42k
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3
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1k
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Is set-induction relatively consistent?
One way to state the axiom of foundation is that the $\in$ relation on any transitive set is well-founded in the following sense:
A relation $(X,\prec)$ is well-founded if for any subset $S\subseteq ...
- 66.8k
10
votes
3
answers
700
views
Is there a theory in a finite language that is computably axiomatizable but not by a finite number of axiom schemas?
I was told to ask this question on mathoverflow. I asked on math stack exchange whether there is a computably axiomatizable theory that can't be axiomatized by a finite number of axiom schemas. I got ...
- 2,023
10
votes
2
answers
805
views
Does every consistent extension of ZF have a model in the minimal transitive model of ZFC?
Suppose there exists a transitive model of $\sf ZFC$. Is it the case that every consistent theory that extends $\sf ZF$ must have a model that is an element of the minimal transitive model of $\sf ZFC$...
- 11.3k