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.
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On independence and large cardinal strength of physical statements
The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics.
Maybe after all ...
27
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5
answers
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What are the known implications of "There exists a Reinhardt cardinal" in the theory "ZF + j"?
This is, alas, in large part a series of questions on unpublished work of Hugh Woodin; it's also quite frivolous if Reinhardt cardinals turn out inconsistent.
Definitions:
Call $\kappa$ an $I-1(\...
25
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1
answer
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Surreal exponentiation -- are the varying definitions equivalent? If not, is there agreement on which ones are better?
The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
25
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4
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What is the definition of a large cardinal axiom?
In different books one can find different implicit definitions for a large cardinal axiom.
My question is that which one of these definitions are more popular or standard amongst set theorists?
Any ...
25
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4
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The Chocolatier's game: can the Glutton win with a restricted form of strategy?
I have a question about the Chocolatier's game, which I had
introduced in my recent answer to a question of Richard
Stanley.
To recap the game quickly, the Chocolatier offers up at each stage
a finite ...
24
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2
answers
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Aleph 0 as a large cardinal
The first infinite cardinal, $\aleph_0$, has many large cardinal properties (or would have many large cardinal properties if not deliberately excluded). For example, if you do not impose ...
24
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2
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Short proof of $\frak p=t$
It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods.
I've heard rumors that there was a proof which was purely set ...
23
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2
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What is the largest Laver table which has been computed?
Richard Laver proved that there is a unique binary operation $*$ on $\{1,\ldots,2^n\}$ which satisfies $$a*1 \equiv a+1 \mod 2^n$$
$$a* (b* c) = (a* b) * (a * c).$$
This is the $n$th Laver table $(A_n,...
23
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5
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What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?
I am interested to know the full range of possibilities for the cofinality type of cuts in an ordered field and in other structures, such as nonstandard models of arithmetic.
Definitions. ...
22
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2
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If ZFC has a transitive model, does it have one of arbitrary size?
It is known that the consistency strength of $\sf ZFC+\rm Con(\sf ZFC)$ is greater than that of $\sf ZFC$ itself, but still weaker than asserting that $\sf ZFC$ has a transitive model. Let us denote ...
20
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2
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What is a Choice Principle, really?
This question is quite soft, and I apologize in advance if it borderline off-topic.
When working in theories between ZF and ZFC the term "choice principle" is heard quite often. For example:
$\quad$ ...
20
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3
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Cohen reals and strong measure zero sets
A set of reals $X$ is $\textit{strong measure zero}$ if for any sequence of real numbers $ ( \epsilon_n ) _{n \in \omega }$ there is a sequence of open intervals $ ( a_n ) _{n \in \omega }$ which ...
20
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4
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Find a "natural" group that contains the quotient of the infinite symmetric group by the alternating subgroup
Let $S_\infty$ the group of permutations of $\mathbb{N}$. It can be shown that there is no homomorphism $S_\infty \to \mathbf{Z}/2$ extending the sign on the finite symmetric groups. Is it possible to ...
19
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3
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Large categories vs. $\mathrm{U}$-categories: why is the loss of category-theoretic information inessential?
I've asked a related question about nine months ago here, however, apparently, I lacked expertise to ask the precise question I want to ask here, as I wish to revisit the matter of universes. I hope ...
19
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2
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Can we take a supremum over all Hilbert spaces?
In my paper On the optimal error bound for the first step in the method of cyclic alternating projections, I defined functions $f_n:[0,1]\to\mathbb{R}$,
$n\geqslant 2$, by
$$
f_n(c)=\sup\{\|P_n\dotsm ...
18
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10
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Examples of ZFC theorems proved via forcing
This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it ...
16
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1
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A contradiction in the Set Theory of von Neumann–Bernays–Gödel?
Thinking on the theory NBG (of von Neumann–Bernays–Gödel) I arrived at the conclusion that it is contradictory using an argument resembling Russell's Paradox. I am sure that I made a mistake in my ...
15
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0
answers
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Toposophy vs Set theoretical multiverse philosophy
Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
15
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2
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Is platonism regarding arithmetic consistent with the multiverse view in set theory?
A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...
15
votes
1
answer
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Does there exist an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?
Is it consistent with ZFC that there exists an uncountable separable metric space $X$ such that every subset of $X$ is a Borel set?
If the continuum hypothesis holds, or more generally $2^{\aleph_{0}}...
15
votes
3
answers
705
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Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?
This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...
14
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3
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Questions about $\aleph_1-$closed forcing notions
"Foreman`s maximality principle" states that every non-trivial forcing notion either adds a real or collapses some cardinals. This principle has many consequences including:
1) $GCH$ fails everywhere,...
14
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1
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Can one exhibit an explicit Kuratowski infinite set without invoking Replacement?
The customary formulation of the Axiom of Infinity within Zermelo-Fraenkel set theory asserts the existence of an inductive set: a set $ I$ with $\varnothing\in I$ such that $x\in I$ implies $x\...
13
votes
1
answer
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Unpublished works of Woodin on SCH and Radin forcing
There are many unpublished results of Hugh Woodin on ''singular cardinals hypothesis'' and '' Radin forcing''. Some of his results are published later by others, but it seems that there are still many ...
13
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1
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For what sets $X$ do there exist a pair of functions from $X$ to $X$ with the identity being the only function that commutes with both?
It is not too difficult to show that if $X$ is an infinite set, then there exists a two-element subset of the group $\operatorname{Sym}(X)$ with trivial centralizer iff $\lvert X\rvert \leq \lvert\...
13
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2
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What are Moschovakis cardinals?
The question is exactly that of the title: what are Moschovakis cardinals?
Background. In a recent answer to the question, "Are there examples of statements that have been proven whose consistency ...
12
votes
1
answer
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Definability of ground model
I have seen mentioned that by a result of Laver, the ground model is definable in any set forcing extension (using parameters).
Does the same hold for class forcing? If it does, in order to establish ...
11
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2
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708
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What can the degrees of constructibility be?
If $r, s\in\mathbb{R}$, we say $r$ is constructible relative to $s$ - and write $r\le_cs$ - if $r\in L[s]$. Modding out by the induced equivalence relation $\equiv_c$, we get a partial order, the ...
11
votes
1
answer
618
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A new cardinal characteristic (related to partitions)?
In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic.
By a partition ...
11
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1
answer
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Is there a suitably generalized Baire property for topological spaces of arbitrary cardinalities?
Is there some suitable generalization to the notion of Baire property for topological spaces of arbitrary cardinalities which satisfies the following condition:
The meager sets are sets which are ...
10
votes
5
answers
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Measurable cardinals under Axiom of Determinacy
I seem to remember reading somewhere that ZF+AD proves that $\omega_1$ and $\omega_2$ are measurable cardinals.
Is that right?
If so, can someone [point me to or give here] a [sketch or proof] of ...
10
votes
1
answer
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Reals added after Cohen forcing
Let $V_1$ be a generic extension of $V\models GCH$ obtained by adding $\aleph_{\omega}-$many Cohen reals. Then we have the following:
1- In $V_1$ there are $\aleph_{\omega+1}-$many reals,
2- In $V_1$...
10
votes
2
answers
780
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Category theory from MK class theory perspective?
I'm looking for a text that treats category theory from the perspective of MK class theory.
MK is already very well-designed and equipped for the type of abstraction that occurs in category theory, ...
9
votes
7
answers
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Uncountable family of infinite subsets with pairwise finite intersections
I am searching for a constructive proof of the following fact: If $X$ is an infinite set, there exists an uncountable family $(X_\alpha)_{\alpha \in A}$ of infinite subsets of $X$ such that $X_\alpha \...
9
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3
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Why isn't there more interest in "large powerset axioms"?
By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For ...
9
votes
1
answer
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Complexity of $L[\mathrm{cf}]$
Assuming large cardinal axioms, which real numbers are in $L[\mathrm{cf}]$, where $\mathrm{cf}$ is the cofinality function on ordinals?
$L[\mathrm{cf}]$ is the minimal inner model that 'knows' the ...
9
votes
2
answers
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Are there known examples of sets whose power set is equal in size to power set of larger sets only in absence of choice?
The question of existence of sets $x,y$ such that
$$|x|<|y| \wedge |P(x)|=|P(y)|$$
is known to be independent of $\text{ZFC}$!
But are there known examples of sets fulfilling the above condition
...
8
votes
2
answers
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Is there one binary operation foundational for set theory?
The membership relationship "$\in$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\in$". Naturally, the ...
8
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3
answers
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Tractability of forcing-invariant statements under large cardinals
It is usual to mention theorems of the kind:
Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi \...
8
votes
4
answers
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Are there $2^{\aleph_0}$ pairwise non-isomorphic Boolean algebra structures on $\omega$?
Is there a collection of $2^{\aleph_0}$ pairwise non-isomorphic countable Boolean algebras?
Equivalently, are there $2^{\aleph_0}$ pairwise non-homeomorphic closed subsets in the Cantor space?
8
votes
1
answer
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A criterion for second countability
Let $(X,\tau)$ be a topological space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming form $\mathcal{E}$ and $\tau$ are the same. ...
7
votes
0
answers
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How distributive are the fake Laver tables?
The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$.
Let's now replace the Laver table $A_{n}$ with ...
7
votes
1
answer
484
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Is $\in$-induction provable in first order Zermelo set theory?
Are there models of first order Zermelo set theory (axiomatized by: Extensionaity, Foundation, empty set, pairing, set union, power, Separation, infinity) in which $\in$-induction fail?
I asked this ...
7
votes
2
answers
882
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Mutually generics
Given posets $P,Q\in M$, I would like to know under what circumstances there are mutually generic filters $G\subseteq P$ and $H\subseteq Q$ (generic over $M$). Also, what are the characterizations of ...
7
votes
1
answer
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Can the Cohen forcing collapse cardinals?
Let $\kappa$ be a regular cardinal, and let $\mathbb{P} = Add(\kappa,1)$ be the standard forcing notion for adding a new subset of $\kappa$ using partial function from $\kappa$ to $2$ with domain of ...
6
votes
1
answer
468
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Sets of reals and absoluteness
Schoenfield's absoluteness states that if $\phi$ is $\Sigma^1_2$ then $V\models \phi$ iff $L\models \phi$. The set of reals in $L$ is $\Sigma^1_2$ and it is the largest countable $\Sigma^1_2$ set of ...
5
votes
0
answers
214
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Improving a Lindstrom-y fact about $\mathcal{L}_{\omega_1,\omega}$?
See e.g. the last section of Ebbinghaus/Flum/Thomas for the relevant background on abstract model theory. Below, all languages are finite for simplicity. "$HC$" is the set of hereditarily ...
5
votes
2
answers
695
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What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
In the same spirit of this question:
How much of mathematical General Relativity depends on the Axiom of Choice?
I want to go radically further ahead and ask for what remains of mathematical general ...
4
votes
1
answer
190
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The property of the dense subfilter of a selective ultrafilter
Let us define the density of subset $A\subset\omega$ :
$$\rho(A)=\lim_{n\to\infty}\frac{|A\cap n|}{n}$$
if the limit exists. Let $\mathcal{F_1}=\{A\subset\omega~|~\rho(A)=1\}$. $\mathcal{F_1}$ is the ...
4
votes
0
answers
333
views
Ordinal analysis and nonrecursive ordinals
Ordinal analysis is typically described as characterizing recursive ordinals in a theory $T$, but there is a sense in which it can characterize all $T$-ordinals, even those that are nonrecursive.
...