Questions tagged [set-theory]
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
1,112 questions
4
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A weakening of cardinal compactness - is it equivalent?
I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak ...
- 1,252
4
votes
1
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221
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Comparing bornologies for cardinal characteristics via Borel maps
This question is "take 2" of this older one, following a suggestion of Francois Dorais. Consider the following bornologies $\mathbb{D},\mathbb{E}$ on the set $\mathcal{N}$ of all functions ...
- 20.5k
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0
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241
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Is the lowenheim-skolem number of nth order logic larger than the corresponding number for 2nd order logic
According to this paper, by Vaananen, the $LS$ number for $2^{nd}$ order logic is given by "the supremum of $Π_{2}$-definable ordinals", where "The Lowenheim-Skolem number $LS(L)$ of $L$ is the ...
- 95
4
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2
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503
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"Potentially club" filters on $\omega_2$
Short version: what can we say about subsets of $\omega_2$ which - in a generic extension where $\omega_2$ is the new $\omega_1$ - contain a club?
We could of course generalize beyond $\omega_2$, but ...
- 20.5k
4
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2
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448
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Number of torsion-free abelian groups
Let $\mathfrak{c}$ be the cardinality of the continuum. How much Choice, if any, is needed to prove that there are $2^{\mathfrak{c}}$ distinct (mutually nonisomorphic) torsion-free abelian groups of ...
- 2,049
4
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518
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Can an ultrapower be undone by class forcing?
Suppose I have a transitive model $M$ of ZFC, and - in $M$ - $U$ is a measure on $\kappa$. Then the transitive collapse of the ultrapower of $M$ along $U$ is an inner model, $N\subset M$.
My question ...
- 20.5k
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472
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Kunen inconsistency with atoms
Kunen showed that there is no nontrivial $j: V \rightarrow_e V$. One might wonder what happens in $\mathsf{ZFC}$ with atoms.
Let's denote the universe by $U$. We aren't assuming that the atoms form ...
- 1,155
4
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227
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Cardinality of maximal chains in the poset of ultrafilters with Rudin-Keisler ordering
Let $\text{NPU}(\omega)$ be the set of non-principal ultafilters on $\omega$. The Rudin-Keisler preorder on $\text{NPU}(\omega)$ is defined by
$${\cal U} \leq_{RK} {\cal V} :\Leftrightarrow (\exists f:...
- 48.9k
4
votes
2
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220
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locally incomparable dense linear orderings extending $\langle \mathbb{R}, < \rangle$
This follows up on Incomparable dense linear orderings extending $\langle \mathbb{R},< \rangle$
and hopefully sparks more discussion.
Where $a<b$, say that the four “types” of nonempty bounded ...
- 449
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416
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"Lexicographic" ordering on ${\cal P}(\omega)$
For $A\neq B\in {\cal P}(\omega)$ we set $$\mu(A,B) = \min\big((A\setminus B)\cup (B\setminus A)\big).$$ We define $A < B$ if and only if $A \neq B$, and
$A = B\cap \mu(A,B)$ (that is $A$ is an ...
- 48.9k
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0
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435
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Can infinite bounded distibutive lattices be "arbitrarily wide"?
I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\...
- 48.9k
4
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1
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321
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Can planar set contain even many vertices of every unit equilateral triangle?
Is there a nonempty planar set that contains $0$ or $2$ vertices from each unit equilateral triangle?
I know that such a set cannot be measurable. In fact, my motivation is to extend a Falconer-Croft ...
- 19k
4
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A question about "local" versus "global" large cardinal axioms
The terms "local" and "global" when applied to large cardinal axioms seem to have a well understood intuitive meaning, although a formalized definition of them in (a meta-language for)ZFC might be ...
- 6,215
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1
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888
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About the relationship between the generalized continuum hypothesis and the axiom of choice
I was trying to get a short, intuitive proof of Sierpinski’s theorem (gch implies axiom of choice) and I could but only by using the following gch2 for the generalized continuum hypothesis gch.
gch: ...
- 481
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Does every non-empty set admit an (affine) scheme structure (in ZFC)?
This question is partially inspired by this question: Does every non-empty set admit a group structure (in ZF)?
It was also inspired by my desire to explain the importance of quotient morphisms when ...
- 8,529
4
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0
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172
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Ultracoproducts of C(X)-algebras
Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...
- 1,786
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689
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"Nicely" strong measure zero sets
This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero".
A set $X$ of reals is strong measure zero if, for any $f: \omega\...
- 20.5k
3
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249
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Independence through forcing vs generic collapses
Are there known statements in $V_{ω+ω}$ independent through forcing after $\mathrm{Col}(ω,<κ_1)*\mathrm{Col}(κ_1,<κ_2)*\mathrm{Col}(κ_2,<κ_3)*...$ where $κ_1<κ_2<κ_3<...$ are ...
- 7,545
3
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2
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480
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A Baire subset of reals that is not Suslin measurable
EDIT: The definition of a Suslin measurable set I wrote here is incorrect. It should be that $\mathcal{S}$ contains the field (or algebra) of open subsets of ${}^\omega\omega$ (or, in other words, it ...
- 1,442
3
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1
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648
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Subcovers without a choice set
Let $X\neq \emptyset$ be a set. We say ${\cal C} \subseteq {\cal P}(X)\setminus\{\emptyset\}$ is a cover if $\bigcup {\cal C} = X$. A subset $D\subseteq X$ is a choice set for ${\cal C}$ if $|D\cap c| ...
- 48.9k
3
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1k
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About Grothendieck Universe and Tarski's A and A' Axioms
A-The addition of the Grothendieck Universe Axiom (for every set x, there exists a set y that is a universe and contains x as member element) to ZFC (ZFC+GU) is considered as giving an almost good ...
- 2,655
3
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2
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432
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When is a filter generated by a (countable) chain?
In any partial order $(P,\leq)$ it is easy to see that every chain generates (i.e., by taking the upwards closure) a filter, and any filter built as a result of the Rasiowa-Sikorski lemma in forcing ...
- 3,115
3
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1
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188
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A sequence of cardinal characteristics constructed with hypergraph coloring
Let $[\omega]^\omega$ denote the collection of infinite subsets of $\omega$.
A hypergraph $H=(V,E)$ consists of a set $V$ and a collection of subsets $E \subseteq {\cal P}(V)$. A coloring is a map $c: ...
- 48.9k
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2
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3k
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What is the smallest cardinal number of a set that requires the axiom of choice to prove that it exists and is non-empty?
Let C(x) be a formula belonging to the language of ZFC in which the variable "x" and no
other variable occurs free. Suppose that (a sentence of this language equivalent to) the
following statement, is ...
- 6,215
3
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1
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144
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Levels of L resembling each other, take 2
(Everything below is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\}...
- 20.5k
3
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334
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Preservation of measurable cardinals in mild extensions
I would like some help with the proof of the preservation of measurable cardinals in mild extensions, as I am a bit new with forcing.
By mild extensions, I mean the generic extension produced from a ...
3
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0
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231
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Is $\sf ZFC + Classes$ finitely axiomatizable?
$\sf ZFC + Classes$ is a bi-sorted theory with lower cases standing for sets and upper cases for Classes; axioms include all $\sf ZFC - Extensionality$ axioms written in lower case, and the following ...
- 11.3k
3
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262
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"Matryoshka" sets and the Axiom of Choice
Consider the following two very similar statements in ${\sf ZF}$:
(Mat_1) There is a set $A$ a map $\alpha: \omega \to {\cal P}(A)$ such that for all $n\in \omega$ we have $\alpha(n+1) \subseteq \...
- 48.9k
3
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1
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366
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Can you formulate a theory stating that a truth predicate does not exist for first order set theory?
A truth predicate for first order set theory would allow you to determine the truth of statements in first order set theory. A definition is given here.
My question is, can you formulate a statement ...
- 6,353
3
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1
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264
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Usual technical term for replacing a set by the set of singletons of its members?
What is a standard technical term in axiomatic set theory for the operation which sends a given set $A$ to the set $A':=\{\{a\}\colon a\in A\}$?
(Replacement implies that $A'$ is a set.)
Some ...
- 6,051
3
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1
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163
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Bounds for a covering number of the circle group $\mathbb T$ by some its small subgroups
$\newcommand{\w}{\omega}\newcommand{\A}{\mathcal A}\newcommand{\F}{\mathcal F}\newcommand{\I}{\mathcal I}\newcommand{\J}{\mathcal J}\newcommand{\M}{\mathcal M}\newcommand{\N}{\mathcal N}\newcommand{\x}...
- 5,409
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1
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Is there an analogue of Shoenfield's absoluteness theorem, but for $\mathrm{On}$?
From wikipedia:
Shoenfield's absoluteness theorem shows that $\Pi^1_2$ and $\Sigma^1_2$
sentences in the analytical hierarchy are absolute between a model $V$
of ZF and the constructible ...
- 3,793
3
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0
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99
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Counting Eilenberg/Schutzenberger-type definitions of pseudovarieties
See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that ...
- 20.5k
3
votes
1
answer
849
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cardinality of product modulo direct sum
let $(X_i)_{i \in I}$ be an infinite family of sets with $|X_i| \geq 2$. we define an equivalence relation on $X = \prod_{i \in I} X_i$ by $x \sim y \Leftrightarrow \{i : x_i \neq y_i\}$ is finite. ...
- 63.1k
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0
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238
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Move one element of finite set out from A in plane
Suppose we are given two sets, $S$ and $A$ in the plane, such that $S$ is finite, with a special point, $s_0$, while neither $A$ nor its complement is a null-set, i.e., the outer Lebesgue measure of $...
- 19k
3
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946
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On statements provably independent of ZF + V=L
Are there any known statements that are provably independent of $ZF + V=L$? A similar question was asked here but focusing on "interesting" statements and all examples of statements given in that ...
- 33
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2
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672
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Partition calculus question
For $m,n,k < \omega$, consider the equation
$X \to (\omega \times k)^{m}_{n}$
What is the smallest $X$ known to satisfy it?
Baumgartner-Hajnal theorem gives a satisfactory answer for $m=2$, but ...
- 341
3
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1
answer
71
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Minimal subcoverings of a cover with finite intersection
Let $X\neq \emptyset$ be a set, and let ${\cal U}$ be a collection of subsets of $X$ such that
$\bigcup {\cal U} = X$, and
$U_1\neq U_2\in {\cal U}$ implies $|U_1\cap U_2| < \aleph_0$.
Is there ${...
- 48.9k
3
votes
1
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960
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Extensions of fast-growing hierarchy
In recent weeks, I have been fascinated by the possible extensions of the fast-growing hierarchy. But is there a way to define it for all recursive ordinals? I saw a statement of this sort on ...
- 3,738
3
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3
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3k
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Well-ordered cofinal subsets [closed]
Let $(P, \leq)$ be a total ordering (some of you prefer the name linear order). Can we find a subset $R\subseteq P$ which is well ordered (with respect to $\leq\upharpoonright R$) and cofinal in $P$, ...
- 113
3
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1
answer
192
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Co-analytic $Q$-sets
A subset $A\subseteq \mathbb{R}$ is said to be a $Q$-set if every subset $B\subseteq A$ is $F_\sigma$ wrt the subspace topology on $A$. For example $\mathbb{Q}$ is a $Q$-set. The first time I have ...
- 2,286
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1
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231
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Under which conditions the domain of the surjective function $f:[a,b]\times[c,d]\to[0,1]^{2}$ can be split s.t. the restrictions are bijective?
This is a follow-up question to this.
Since it is not always possible to construct such partition, I would like to know if there are additional restrictions which we could impose so that the wanted ...
user127351
3
votes
1
answer
511
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Is this version of van der Waerden's Theorem consistent with ZFC?
One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...
- 18.6k
3
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1
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802
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Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2
While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...
- 95
3
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1
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294
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Implications between different covering properties of spaces
Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$ and $X\notin {\cal C}$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set,...
- 48.9k
3
votes
1
answer
248
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A question on recursion in Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection
$\Sigma_{3}KP\omega$ be Kripke-Platek set theory with infinity and $\Sigma_{3}$-separation and $\Sigma_3$-collection. What strengthening of Barwise's Definition by $\Sigma$ Recursion (Theorem 6.4 on ...
- 3,574
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1
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492
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Does the partition principle imply (DC)?
For sets $x, y$ we write $x\leq y$, if there is an injection $\iota: x \to y$, and we write $x \leq^* y$ if either $x = \emptyset$ or there is a surjection $s: y \to x$. In ${\sf (ZF)}$ we have that $...
- 48.9k
3
votes
2
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233
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${\frak b}$ and ${\frak d}$ defined with $\leq$ instead of $\leq^*$
Let $\omega^\omega$ denote the collection of all functions $f:\omega\to\omega$. For $f,g\in\omega$ we define
$f\leq g$ if $f(n)\leq g(n)$ for all $n\in\omega$;
$f\leq^* g$ if there is $N\in\omega$ ...
- 48.9k
3
votes
0
answers
123
views
At which large cardinal property this second order ordinal arithmetic stops?
Language: Second order logic, with as usual predicates written in upper case, and objects in lower case. Let $<$ be a primitive constant binary relation symbol.
Equality between objects is ...
- 11.3k
3
votes
2
answers
880
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Veblen function with uncountable ordinals & beyond
Disclaimer: I am not a professional mathematician.
Background: I have been researching large countable ordinals for awhile & I think the Veblen function is particularly eloquent. My understanding ...
- 81