All Questions
6,026 questions
6
votes
2
answers
320
views
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 ...
- 3,318
2
votes
2
answers
335
views
Are there models of ZF in which all uncountable sets are super/hyper/ultra-singular?
This question is a follow up to that posting.
Recall the definition of super/hyper/ultra-singular set given in the linked posting.
Is there a model of $\sf ZF$ in which every uncountable set is super-...
- 11.3k
5
votes
1
answer
422
views
What is the relationship between non-existence of those kinds of singular sets and AC?
Let's say a set $A$ is super-singular, if and only if, there exists a set $x$ such that $|A|=|\bigcup x|$ and $| A| > |x|$, and for each $y \in x$ we have: $ |y| \not > |x|$ .
A set $A$ is ...
- 11.3k
1
vote
0
answers
96
views
Determine equivalences in the generated collection of subgroups and quotients
Let $A$ be an abelian group, and $B_1, B_2, \dots, B_m$ be subgroups of $A$. Define the family of subgroups $\mathcal{D}_0 = \{ \{0\}, A, B_1, B_2, \dots, B_m \}$.
Let $\mathcal{C}_1$ be the ...
- 807
19
votes
2
answers
2k
views
Can there exist a definable "ultrafilter" on the ordinals?
Can there exist a model $\textit M$ of $\textit{ZF}$ (or $\textit{ZFC}$) that contains a definable nontrivial "ultrafilter" on $\sf Ord^{\textit M}$? By this I mean that there is some ...
- 576
7
votes
1
answer
221
views
Finitistic interpretation of Nelson's internal set theory
What does “standard” in internal set theory really mean?
Is it secretly a way of reconciling conventional mathematics with (ultra)finitism?
Until recently I thought “standard” was just a way of ...
- 15.9k
2
votes
0
answers
187
views
Semantic equivalence between mathematical proofs
Sometimes, we recognize two proofs of the same claim to be the "same" proof. In some cases, this sameness is obvious -- for example, the proofs that $\sqrt{2}$ and $\sqrt{3}$ are irrational ...
- 225
1
vote
0
answers
109
views
Name For Effective Cantor-Bendixsonish Derivitive
When dealing with a tree (substring closed subset of $\omega^{< \omega})$ a useful operation will frequently be to remove any nodes with finite ordinal rank (i.e., all nodes whose extensions on the ...
- 3,029
8
votes
1
answer
228
views
Examples of anti-classical theories in iFOL
An anti-classical axiom $\phi$ is one which is inconsistent with LEM
Are there any sources for good examples of anti-classical theories in intuitionstic first-order logic? There are many examples of ...
- 183
1
vote
1
answer
152
views
Looking for Fitch-style Natural Deduction system that allows for open formula
I find most Natural Deduction proof systems only allow for close formulas, which are not convenient for FOLs without a constant. Most Sequent Calculus systems instead allow for open formulas, but it ...
- 127
5
votes
1
answer
621
views
Non-atomic probability measures on N
One can intuitively imagine picking a random natural number and ask to what extent the intuition can be axiomatized.
Using the axiom of choice, there is a total finitely additive (monotonic) averaging ...
- 7,545
2
votes
0
answers
117
views
Can we have the set world obeying Quine's New Foundations with its well-founded realm obeying $\sf ZFC$?
Is this theory consistent?
Language: first order language of set theory,
Extra-logical axioms:
1. Extensionality: as in $\sf NF$.
2. Stratified Comprehension: as in $\sf NF$.
Define: a set is said ...
- 11.3k
2
votes
2
answers
172
views
Can the Category of that kind of small sets in $\sf NFU$ be Cartesian closed?
Working in Quine's $\sf NFU$, with urelements being at least as many as sets. Formally the latter is: $|Ur| \geq |Set|$.
Where $Ur$ is the set of all urelements and $Set$ is the set of all sets. We ...
- 11.3k
6
votes
1
answer
162
views
Can there exist a set of all transitive sets in a model of NF or NFU?
Is it consistent with $\sf NF$ or $\sf NFU$ to have a set of all transitive sets? Formally:
$\exists t \forall x (x \in t \leftrightarrow x \text { is transitive})$
Where "$x$ is transitive" ...
- 11.3k
4
votes
0
answers
214
views
Algebraic logic in the style of algebraic geometry
I am writing a thesis on algebraic logic, I wonder if there is any recent research on an idea mentioned in Yuri Manin's book on algebraic geometry and in another Russian textbook on differential ...
- 558
5
votes
1
answer
632
views
Consistency of ZFC with inaccessible cardinals but no measurable cardinals
Let $S$ be a set and k a infinite field. The injection $S \to k\mathrm{Alg}(k^S, k)$ (sending a point to the evaluation in it) is a bijection if and only if $S$ is a non-measurable cardinal (see for ...
- 6,962
5
votes
1
answer
266
views
What oracles make finding isomorphism (of finite structures) easy?
Below, all structures are finite, in a finite language, with underlying set an initial segment of the natural numbers. This has been edited to fix errors pointed out by Emil Jerabek in his answer ...
- 20.9k
2
votes
0
answers
102
views
Direct construction of an arithmetically high degree below $0^{(\omega)}$
The existence of a high arithmetic degree (meaning the degrees induced by the notion of relative arithmetic definability) below $0^\omega$ can be established by using Harrington/Simpson's ...
- 3,029
3
votes
0
answers
90
views
Existence of symmetric total measures
Is it consistent that there is a total finitely additive measure $μ$ on $ℝ$ extending the Lebesgue measure such that for every Borel Lebesgue-measure-preserving bijection $f$ of $ℝ$, $∀α∈Ord \, ∀s∈Ord^...
- 7,545
-4
votes
1
answer
140
views
About the definitions of well-foundedness in this extension of NFU that interprets ZFC?
Lets see how the world of sets could look like from the perspective of $\sf NFU$. So, here we work within the first order language of set theory, with the following extra-logical axioms:
1. Quine atom:...
- 11.3k
7
votes
0
answers
260
views
A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice ...
- 7,545
2
votes
0
answers
232
views
Is the poset in the following construction stationary $\aleph_{\alpha + 2}$-linked?
Definition: A poset P is $\mathbf{stationary}$ $\kappa^{+}$-$\mathbf{linked}$ if for every sequence of conditions $(p_{\gamma} | \gamma < \kappa^{+})$, there is a regerssive function $f: \kappa^{+} ...
4
votes
1
answer
239
views
True or false? Every left or right cancellative, duo semigroup is cancellative
A semigroup $S$ is duo if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$; for instance, every commutative semigroup is duo, and so is every group. On the other hand, ...
- 10.5k
4
votes
0
answers
206
views
Fine structure without choice
In set theory, are there approaches to fine structure that give fine-structural models that do not satisfy the axiom of choice?
We can build fine-structural models above a given set (such as $\mathbb ...
- 7,545
11
votes
0
answers
430
views
Is $(\mathbb{R}, +)$ still injective as long as $(\mathbb{Q},+)$ is?
It is known that the existence of nontrivial injective abelian groups is independent of choice in ZF (or, rather, ZFA). In particular, $\mathbb{Q}$ is not provably injective, much less $\mathbb{R}$, ...
- 433
6
votes
2
answers
523
views
Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?
In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?.
In the topos of simplicial sets, the subobject ...
- 18.4k
13
votes
1
answer
291
views
Descriptive complexity of analytic continuation
Consider the set of complex power series
$$
f(z)=\sum_{n=0}^\infty a_nz^n
$$ that have radius of convergence $1$ and can be analytically continued to the neighborhood of some point on the unit circle. ...
- 623
-5
votes
1
answer
233
views
First research papers in mathematical logic [closed]
Hello I'm a software engineer who just wants to start research in mathematics soon. I'm interested in the foundations and hence I'm picking mathematical logic. As I have never touched undergraduate-...
8
votes
1
answer
322
views
Does every cancellative duo semigroup embed into a group?
Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:
Q. Does every cancellative duo semigroup embed into a group?
A (multiplicatively ...
- 10.5k
6
votes
3
answers
551
views
Conjecture about commutative semigroups
Conjecture: given any commutative semigroup $S$ of order $n \ge 4$, there exist $a, b \in S$ with $a \ne b$, an integer $m \ge \lfloor (n-1)/2 \rfloor$, and two $m$-element subsets $X = \{x_1, \ldots, ...
- 1,000
7
votes
2
answers
398
views
Numerical choice and reverse mathematics
Consider the following fragment of numerical choice in the language of second-order arithmetic:
for any arithmetical $\varphi$, we have:
$$
(\forall n\in \mathbb{N})(\exists m\in \mathbb{N})(\forall X\...
- 4,359
6
votes
0
answers
180
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the ...
- 7,545
8
votes
2
answers
596
views
If a semigroup embeds into a group, then is it a subdirect product of groups?
The title has it all:
Q. If a semigroup $S$ embeds into a group, then is $S$ (isomorphic to) a subdirect product of groups?
If yes, then $S$ is a subdirect product of subdirectly irreducible groups,...
- 10.5k
3
votes
0
answers
153
views
What is known about the word problem on free algebraic models?
Consider the fragment of first-order logic with equality and universal quantification as the only logical symbols; we call this logic the logic of universal algebra. I am interested in languages $\...
7
votes
2
answers
488
views
Is every cancellative semigroup a subdirect product of subdirectly irreducible cancellative semigroups?
By a classical result of Birkhoff (that is, Theorem 2 in [G. Birkhoff, Subdirect unions in universal algebra, Bull. AMS, 1944]) and the trivial fact that the class of semigroups is closed under the ...
- 10.5k
6
votes
1
answer
228
views
Can we computably escape infinitely many functions (allowing partiality)?
Let $(p_i)_{i\in\omega}$ be a uniformly computable sequence of partial functions (i.e. the partial function $q(i,x)=p_i(x)$ is computable) such that infinitely many $p_i$s are total. Must there be a ...
- 20.9k
-2
votes
1
answer
218
views
If existence of a pre-isomorphism implies existence of an isomorphism, would AC follow?
Let a surjection $f: M \to N$ be called a pre-isomorphism on membership, if and only if:
$\begin{align} \forall x \in M \,\forall y \in M \,\exists x' \in M \exists y' \in M : \ & f(x')=f(x) \...
- 11.3k
9
votes
0
answers
265
views
Consequences of recent claims of Ordinal Analysis of $Z_2$
Recently Toshiyasu Arai submitted "An ordinal analysis of $\Pi_{N}$-Collection" and Henry Towsner submitted "Proofs that Modify Proofs", both of which claim ordinal analysis of ...
- 161
7
votes
0
answers
265
views
Herbrand's consistency proof
Jacques Herbrand's thesis "Investigations in proof theory: The properties of true propositions" (or in the original French "Recherches sur la théorie de la démonstration", with the ...
- 161
11
votes
1
answer
415
views
Examples of natural algebraic irreflexive relations
To motivate the question, consider the theory of rings.
Define $x \parallel y$ to mean $\exists w \exists z .((x - y) z = w (x - y) = 1)$, or in words, "$x - y$ is a unit".
Then $\parallel$ ...
- 15.9k
5
votes
1
answer
157
views
Intersection cardinalities in MAD families
Let $\newcommand{\o}{[\omega]^\omega}\o$ denote the collection of infinite subsets of the set of nonnegative integers $\omega$. We say ${\cal A}\subseteq \o$ is almost disjoint if $A\cap B$ is finite ...
- 48.9k
8
votes
1
answer
385
views
Is "every infinite set of strictly subnumerous sets is supernumerous to its union" equivalent to AC?
Is the following sentence equivalent to $\sf AC$ over the rest of axioms of $\sf ZF$?
For each infinite set $X$: if for all $y \in X$ we have $|y| < |X|$, then $| \bigcup X|\leq |X| $?
Note: ...
- 11.3k
3
votes
0
answers
250
views
Action (of a graded monoid) required
Reference request: Did the construction below appear anywhere before? Any mentions of it or especially any links to something commonly known would be really helpful. I feel that it might be related to ...
- 321
18
votes
1
answer
554
views
When can we add choice to a model of ZF
For countable transitive models of ZF, is existence of a ZFC extension with the same height a first order property?
In other words, is there a statement $τ$ (in the language of set theory) such that ...
- 7,545
4
votes
2
answers
134
views
Properties of all relatively computable branches
I'm probably just missing something obvious but suppose that $T \subset 2^{< \omega}$ is a perfect tree with no terminal nodes (what about just $[T]$ non-empty?). If $Y \leq_{T} X$ for all $X \...
- 3,029
1
vote
0
answers
106
views
The proposition associated with a set
Given a set $U$ and a set $A \subseteq U$, is there an accepted symbol for the proposition $p$ over the universe $U$ such that for each $x \in U$, $p(x)$ is the assertion that $x \in A$? (The symbol $...
- 19.7k
1
vote
0
answers
92
views
Proof for non-existence of short integer program for squares
We do not know if $P=NP$ or not or if there is a superfast integer mutiplication algorithm. But I do not think either assumption is necessary to answer this question.
Is there a way to show within an ...
- 13.9k
8
votes
1
answer
343
views
What topos-theoretic construction lies behind the “symmetric model” construction (used to refute AC) in Set Theory?
Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this ...
- 32.5k
1
vote
2
answers
228
views
Can this semi-constructible structure satisfy existence of a measurable cardinal?
If we add a primitive unary function symbol $\mathfrak L$ to the first order language of set theory.
Axiom of semi-constructibility: if $\phi^\alpha (y,x_1,\ldots,x_n)$ is a formula in which all and ...
- 11.3k
2
votes
0
answers
88
views
Intuitionistic countermodels in which $u \leq v \implies M_u \leq M_v$
In Fitting's Intuitionistic Logic Model Theory and Forcing, the following theorem is proven:
If $X$ is a formula with no universal quantifiers and $\not\vdash_I X$, then there is a countermodel $(\...
- 139