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Results tagged with set-theory
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user 113405
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.
14
votes
1
answer
489
views
Injection into a proper class and choice without regularity
In $\sf ZF$, we have that the axiom of choice is equivalent to:
For all sets $X$, and for all proper classes $Y$, $X$ inject into $Y$
and
For all sets $X$, and for all proper classes $Y$, $Y …
- 1,676
12
votes
1
answer
533
views
Building the real from Dedekind finite sets
It is well known that the real numbers can be countable union of countable sets by starting with GCH and taking a finite support permutations while collapsing all of $\aleph_n$ for natural $n$.
The id …
- 1,676
11
votes
1
answer
768
views
Scott's trick without regularity
In ZF(C), one can easily get a class partition of $V$, we can even get an $\mathrm{Ord}$-partition using the Cumulative hierarchy: $P=\{V_{α+1}\setminus V_α\mid α∈\mathrm{Ord}\}$, such a partition let …
- 1,676
8
votes
Accepted
Large cardinal near inconsistencies
I would argue that a "restricting versions" of large cardinals are such.
Starting from the top down, we have the inconsistent Berkeley cardinals:
$κ$ is Berkeley if for every transitive $M\ni\kappa$ …
- 1,676
6
votes
Why we need to choose direction in the "marry the arrows" algorithm?
This exact point was discussed in that paper few paragraph before your quote:
Imagine for a moment that the strings of arrows represent streets—
circular drives, in the case of necklaces, and long bo …
- 1,676
6
votes
Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
Any model of $ZF$+stationary proper class of $I3$ ordinals (whose consistency strength is at most $ZF$+$I2$) where $W_α$ is listing of $V_{κ}$ where $κ$ is $I3$ and $j_α$ the witness of $I3(V_κ)$ will …
- 1,676
6
votes
0
answers
164
views
Weak trichotomy principle in the absence of choice
It is well known that the trichotomy property of cardinals ($∀κ,λ\in\operatorname{Card}(κ<λ∨κ=λ∨κ>λ)$) is equivalent to the axiom of choice.
D. Feldman and M. Orhon had defined in [1] a generalization …
- 1,676
6
votes
Accepted
Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?
Any model of ${\sf ZFC}+V=\sf HOD$ has an elementary equivalent pointwise definable model.
If $M$ models $V=\sf HOD$, it has a parameter free definable well ordering, for each formula $φ$ consider the …
- 1,676
5
votes
1
answer
169
views
Preserve unbounded sets between different cofinality
Working in ZFC, let $\kappa,λ$ be cardinals with $\kappa>λ$, and assume that $\kappa$ is regular.
We say that a function $F:\kappa^n→λ$, for some finite $n$, is preserving unbound, if for all $a⊆\kap …
- 1,676
5
votes
Accepted
Does n-well ordered choice schema imply the axiom of choice?
$2$-well ordered choice is enough to imply AC.
Let $α$ be any ordinal, and look at $\mathcal P^2(α)\setminus\{\emptyset\}$.
We have $x\in^2 \mathcal P^2(α)\setminus\{\emptyset\}⇒∃z⊆\mathcal P(α)\;(x\i …
- 1,676
4
votes
Accepted
Does cardinal definable choice imply AC?
Over ZF yes, it does.
Let $X$ be any family of nonempty sets, and let $κ$ be cardinal such that $ρ(κ)>ρ(X)$, then $V_{ρ(κ)}\setminus\{∅\}$ is cardinal definable, hence has a choice function that induc …
- 1,676
4
votes
1
answer
206
views
Exactly how much (and how little) can partial ordered sets (classes) embed to the cardinalities
In the paper "Convex Sets of Cardinals", Truss mentioned a result of Jech:
If $M$ is a countable transitive model of ZFC, and $(P,<)∈M$ is a poset, then there exists a Cohen extension of $M$ such tha …
- 1,676
4
votes
Accepted
Automorphisms of vector spaces and the complex numbers without choice
This is not a full answer, but it is too long to be a comment.
Let $B(F)$ for field $F$ be the statement "every vector space over $F$ has a basis" and let $AL19(F)$ be the statement "for every vector …
- 1,676
3
votes
Accepted
Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
$\newcommand\Ord{\mathit{Ord}}$The edit to the question has changed it enough so I think it deserves its own answer.
I assume that in reflection we have $\forall \vec{x} \in W_\alpha \, (\phi \to \ph …
- 1,676
2
votes
Accepted
Which fragment of ZF does the class of all hereditarily predicatively definable sets capture?
HPD satisfy extensionality and regularity trivially.
It satisfy Pairing, as if $x,y\in HPD$ we can explicitly write down the definition of $(x,y)$ using only $x,y$ as parameters (and they have strictl …
- 1,676