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2
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1answer
190 views

Forcing Notions with Unknown Real/Cardinal Preserving Situations

Question. Is there any set forcing notion $\mathbb{P}$ in one of the following categories? (a) $\mathbb{P}$ preserves cardinals but it is still open whether $\mathbb{P}$ adds reals or not. (b) ...
5
votes
1answer
142 views

Function Approximation in c.c.c Forcings without AC in Ground Model

Consider the following basic theorem. Theorem. If $M$ is a c.t.m of ZFC and $\mathbb{P}$ a c.c.c forcing notion in $M$ and $G$ a $\mathbb{P}$ - generic filter on $M$ then for all $A,B$ in $M$ and for ...
0
votes
1answer
137 views

Absoluteness of definability (2)

Let $M,N$ be transitive models of set theory and $M \subset N$. Assume that in $M$ we prove that there exists an object $X$ that satisfies some set $T$ of first order properties and which is definable ...
2
votes
1answer
110 views

How many elementary equivalent models are unifiable by ultrapower?

Definition. A class $\mathcal{C}$ of pairwise elementary equivalent $\mathcal{L}$-structures is unifiable by ultrapower if there is an index set $I$ and an ultrafilter $F$ on it such that $\forall ...
9
votes
4answers
1k views

Are there non-diagonal proofs for Cantor's continuum and Godel's incompletness theorems?

There is a formal definition for the notion of a formal proof. Question 1. Is there any formal definition for the notion of a diagonal formal proof? Consider the following theorems both proved by ...
9
votes
1answer
173 views

Obtaining a lightface pointclass from a boldface one

Define a pointclass to be: boldface inductive-like if it is $\mathbb{R}$-parameterized, has the scale property, and is closed under $\wedge$, $\vee$, $\forall^\mathbb{R}$, $\exists^\mathbb{R}$, and ...
1
vote
3answers
281 views

Can the structure of an ultrafilter determine the structure of its ultrapower?

Usually we work with ultrafilters as pure sets without any structure. Q1. Is there any important notion of structure on an ultrafilter? Q2. Is there any non-trivial notion of structure on ...
4
votes
2answers
217 views

What are benefits of foundations with more than one sort of objects?

Amongst different foundations of mathematics, $ZF$ and $NF$ are talking about "sets" but $MK$ and $GB$ are talking about two sorts of objects "sets" and "classes". What are benefits of studying the ...
9
votes
3answers
734 views

Is it ever a good idea to use Keisler-Shelah to show elementary equivalence?

The most useful way I know to show that two structures are elementarily equivalent is Ehrenfeucht-Fraisse games. These are quite nice and intuitive, and even when I can't use them to solve my problem ...
11
votes
1answer
270 views

Independence results about independence results

Define $Ind(ZFC,\sigma)$ to be the assertion "The sentence $\sigma$ is independent from ZFC". I am looking for theorems in the form $Ind(ZFC, Ind (ZFC,\sigma))$. Are there such theorems in set ...
2
votes
0answers
106 views

Re-interpreting vector spaces in a choice-less model of ZF as modules over a regular ring in ZFC

I am searching a module $M$ over a (von Neumann) regular ring $A$ ($\forall a\in A$, $\exists x\in A$: $axa=a$) with two properties: (1) every finitely generated submodule of $M$ is projective ...
5
votes
1answer
182 views

A ZFC construction to get a proper extension which is a $\omega_1$-model

In $V$, let me call a set theory structure A is a $\omega_1$ model if the $\omega_1$ of $A$ is the same as the $\omega_1$ in $V$ (up to isomorphism). The question I would like to ask is the following: ...
6
votes
1answer
163 views

Skolem Hulls in $H_{\omega_2}$

I put this on stack exchange over a week ago with no answer, so let's try here. Consider a model of the form $\mathfrak{A} = (H_{\omega_2}, \in, \lhd, f_0, f_1, ...)$, some expansion of ...
5
votes
4answers
247 views

Complete Boolean algebra not isomorphic to a $\sigma$-algebra

Does there exist a complete Boolean algebra that is not isomorphic to any $\sigma$-algebra? If so, what is an easy or canonical example or construction?
6
votes
2answers
268 views

What is the shape of large cardinal tree in implication strength order?

There are two natural orders on large cardinal axioms. (a) Consistency strength order $\sigma \leq_C \theta \Longleftrightarrow ZFC\vdash Con(ZFC+\theta)\longrightarrow Con(ZFC+\sigma)$ (b) ...
6
votes
2answers
209 views

Weakly Compact Cardinal and Iterability

In $\textit{Set Theory}$ by Jech 1978 edition, in the proof of Lemma 32.5 which you can hopefully see at the Google book link. In the course of the proof using the tree property, he produces from any ...
1
vote
2answers
107 views

Absoluteness of definability

Let $M,N$ be models of set theory such that $M$ is a submodel of $N$. Assume that there exists a formula of set theory which defines some set $X^M$ in $M$ (say $X^M$ is a subset of $\mathbb{R}$ which ...
7
votes
2answers
220 views

Vaught's conjecture for partial orders

In ``Steel, John R. On Vaught's conjecture. Cabal Seminar 76–77, pp. 193–208'' the following is proved: Theorem. Let $\phi\in L_{\omega_1,\omega}.$ If every model of $\phi$ is a tree, then $\phi$ has ...
4
votes
1answer
456 views

Origin of “Woodin cardinal”

Sorry if this is a completely stupid question (I'm a not a set-theorist, though I've been doing some reading in the subject), but I was wondering, specifically, about the exact provenance of the name. ...
6
votes
2answers
386 views

Is the forcing relation defined for mathematical formulas?

Meta-matematical formulas of the language of set-theory (which are not sets, but just sequences of signs) should not be confused with mathematical ones (i.e. formulas coded as sets, e.g. finite ...
4
votes
2answers
260 views

Forcing for Arbitrary First Order Theories

Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...
1
vote
0answers
159 views

Ultrapowers of ultrapowers

Suppose that you have some structure $S$, and you want to construct an ultrapower of cardinality $\kappa$ to obtain $S^*_\kappa$. Then, say you want to construct a new ultrapower from $S^*_\kappa$, ...
3
votes
2answers
209 views

Antichains and the Knaster Property

This may be a naive question, but I'll pose it. Is there an example of a notion of forcing $\mathbb{P}$ that has the $\kappa$-c.c. which is not also $\kappa$-Knaster Property also is not ...
6
votes
1answer
123 views

Intermediate submodels which do not satisfy AC

The following is known: Theorem. Suppose $V[G]$ is a generic extension of $V$ by a set forcing, and let $N$ be a model of $ZFC$ with $V\subseteq N\subseteq V[G].$ Then $N$ is a generic extension of ...
1
vote
0answers
210 views

If there is a Reinhardt cardinal, then there is one universe? [closed]

If there is a nontrivial elementary embedding $j:V \to V$, then there is a universe which contains all the large cardinals. Is there such a universe? Does this imply there is one universe from ...
4
votes
1answer
139 views

What does the set of cardinals admitting a k-additive measure look like?

Consider an infinite cardinal $\kappa$. Is it the case that the existence of a $\kappa$-additive measure on some infinite set implies the existence of such a measure on every infinite set of size ...
1
vote
1answer
190 views

Question about HOD

Consider the full Solovay model $N=M[G]$ of set theory. Let HOD be the set of hereditarily ordinal definable elements of $M[G]$. It is known that in $N$ every set of reals definable from ordinals ...
0
votes
0answers
113 views

Does the following result hold in the full Solovay model (2)

This is a sequel to my previous question with the same title. Let $N:=M[G]$ be the full Solovay model. And let $R$ be an archimedean complete real closed field in $M[G]$ (so $R$ has cardinality ...
2
votes
0answers
149 views

How many Dedekind-finite sets can $\mathbb{R}$ be partitioned into?

Building off Asaf Karagila's answer to my previous question (Can $\mathbb{R}$ be partitioned into dedekind-finite sets?) on partitioning $\mathbb{R}$ into strictly Dedekind-finite sets: (1) What ...
8
votes
1answer
290 views

Consistency of P1 on Kunen

It's the first time I'm posting here so I don't know if I really should put this question here... I tried to post it on math.stackexchange, but a friend told me I would get better results by posting ...
-1
votes
1answer
150 views

Algebra generated by a tree [Edit] [closed]

Suppose that $(T,\leq)$ is a partially ordered set, we say $T$ is a tree* if for every $i\in T$, $\{s: s\in T, s\leq t\}$ is a well-founded chain. What I need to know is: Can the algebra ...
11
votes
4answers
818 views

Weak forms of the Axiom of Choice

Let $n\geq 2$ be a natural number and consider the following: $AC(n)$: For each family $\{X_i\}_{i \in I}$ of $n$-element sets the product $\prod_{i\in I}X_i$ is non-empty. Is it known that for ...
8
votes
1answer
221 views

Can $\mathbb{R}$ be partitioned into dedekind-finite sets?

Assuming $ZF$ itself is consistent, it is consistent that there are sets $D$ which are infinite but cannot be placed in bijection with any of their proper subsets; such sets are called "strictly ...
4
votes
1answer
159 views

About subposet of Levy collapse

Let $\lambda>\kappa$ and $\operatorname{Coll}(\kappa, \lambda)$ be the poset collapsing $\lambda$ to $\kappa$. Pick a subposet $P$ which is $\lt\kappa$-closed and of size $\lambda$. Can we say that ...
4
votes
1answer
180 views

Climbing quickly up $L$

This question is motivated by Joel David Hamkins' answer to Godel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem), in which he shows that, if we replace ...
15
votes
1answer
820 views

Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)

Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
6
votes
1answer
234 views

Which large cardinals are upward reflecting?

Let the first order formulas $p(x)$ and $wi(x)$ assert "$x$ is a large cardinal of type $p$" and "$x$ is weakly inaccessible" respectively. The large cardinal type $p$ is upward reflecting if ...
9
votes
1answer
185 views

Class Forcing and Genericity: Predense sets vs Dense classes

In short my question is: why do we use definable classes in the definition of genericity for class forcing, instead of predense sets. To elaborate, in Sy's book and indeed other sources on the ...
9
votes
1answer
222 views

Suslin trees in ccc ideals

A countably complete ideal $I$ on a set $Z$ ideal is c.c.c. when there is no uncountable family of pairwise disjoint $I$-positive subsets of $Z$. If such an ideal exists, then there exists a weakly ...
1
vote
1answer
160 views

Different Methods for Forcing Total Failure of Generalized Continuum Hypothesis

It seems there are few papers on total failure of Generalized Continuum Hypothesis. As an example Merimovich in 2007 constructed a model of ZFC such that GCH fails everywhere assuming consistency of ...
8
votes
3answers
274 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 ...
1
vote
0answers
82 views

Dedekind–MacNeille completion of ordered abelian monoids

It's known that the Dedekind–Macneille completion of an ordered Abelian group necessarily is not an ordered Abelian group (and it is an ordered Abelian monoid). I want to know that what happened ...
21
votes
4answers
843 views

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 ...
7
votes
1answer
171 views

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 ...
5
votes
3answers
550 views

PFA: A New Godel's Program & A New Large Cardinal Ladder (Updated)

We know $PFA$ implies $2^{\aleph_0}=\aleph_2$. Q1. What does $PFA$ say about other values of continuum function? Does proper forcing axiom carry any further information about values of continuum ...
6
votes
3answers
495 views

Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?
1
vote
0answers
78 views

About Con($\mathfrak{u}>\mathfrak{i}$)?

I need some help looking for a paper. It is mentioned in a joint paper of M. Goldstern and S. Shelah that Shelah has proved the consistency of $\mathfrak{i}<\mathfrak{u}$. I was looking for this ...
7
votes
1answer
227 views

Are superstrong stronger than strongly compact cardinals? (or vice versa)

In the last part of Kanamori's excellent "The Higher Infinite" there is a small diagram about the strength and consistency strength of some major large cardinal axioms. Below supercompact cardinals ...
5
votes
1answer
104 views

Shelah's proof that proper forcing preserves P-points

In Proper and Improper forcing, VI.5; Claim 5.1 part 1 is the following: If $F$ is a P-point in $V$, $P$ is a proper forcing notion and $\Vdash_P `` F$ generates an ultrafilter" Then the ultrafilter ...
5
votes
3answers
188 views

In $L$, does there exist a definable non-principal ultrafilter on $\mathbb{N}$

The axiom of constructibility $V=L$ leads to some very interesting consequences, one of which is that it becomes possible to give explicit constructions of some of the "weird" results of AC. For ...