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3
votes
1answer
173 views

Adding Generic Reals to Forcing Extensions

I'm following the Jech's Multiple Forcing for a seminar group and I intend to show how to add one or some reals to extensions. I studied Solovay's model and I can see why learning how to add random ...
16
votes
2answers
454 views

Is the notion of fixed point property for topological spaces an absolute notion?

Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point. Is the notion of FPP for topological spaces an absolute notion? More ...
1
vote
1answer
343 views

Compactness and completeness in Gödel logic

The standard proof of the completeness theorem in first-order Gödel logic is based on a first-order countable language. I want to know that is there any proof of the completeness theorem in ...
10
votes
1answer
387 views

Ground Axiom and behaviors of continuum function

The Ground Axiom ($GA$) is the assertion that the universe of sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing $P\in W$. Is $GA$ consistent with any possible ...
8
votes
3answers
591 views

What set theoretical questions could never be answered by Turing machines of arbitrary cardinality?

Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of ...
1
vote
1answer
78 views

Is there such a sufficient condition for “$X$ is a stationary subset of uncountable regular $\kappa$” involving limit points?

If $X$ is a set of ordinals, define $f(X, \alpha)$ such that $f(X, 0) = X$, $f(X, \beta + 1) = f(X, \beta) \cap Lim(f(X, \beta))$, where $Lim(Y) = \{$limit points of $Y\}$, and $f(X, \gamma) = \bigcap ...
4
votes
1answer
213 views

Perfect set property implies $\omega_1$ is a limit cardinal in $L$

Specker proved in 1957 that if in $V$ every set of real numbers has the perfect set property, than in $L$, $\omega_1^V$ is actually a limit cardinal. The original proof is in German, and I've been ...
2
votes
0answers
102 views

Does the Lévy collapse obey this nice characterization? [duplicate]

This question is related to an issue in my answer to Monroe Eskew's question on the failure of Cantor-Bernstein for the Lévy collapse. Question. Is the Lévy collapse $\text{Coll}(\omega,\lt\kappa)$ ...
6
votes
1answer
169 views

Failure of Cantor-Bernstein for the Levy Collapse

Related to this question, is it possible to give an example of the failure of Cantor-Bernstein for complete embeddings of forcing notions involving the Levy Collapse $Col(\omega,<\kappa)$? Suppose ...
8
votes
1answer
143 views

Does being special on a club imply being special?

Let $T$ be an Aronszajn-tree, $C\subset \omega_1$ a club set and $f:\bigcup\limits_{\alpha\in C}T_\alpha\longrightarrow \mathbb Q$ a strictly increasing function (where $T_\alpha$ is the ...
5
votes
1answer
82 views

Property of $L$ Relating to Reflection

The idea of the question is whether it is ever possible that $L$ is so nice in the sense that $\{L_\alpha\}$ does not incorrectly "guess" a bigger inaccessible than $L$ really has, as long as ...
5
votes
1answer
214 views

A model of Krivine

In a paper by J.-L. Krivine, Modèles de ZF+AC dans lesquels tout ensemble de réels définissable en termes d'ordinaux est mesurable-Lebesgue [C. R. Acad. Sci. Paris Sér. A-B 269 (1969), A549–A552, ...
6
votes
1answer
183 views

Morley Phenomena for Special Families of Reals

Vaught's Conjecture is a dual form of Continuum Hypothesis in model theory. It asserts that for each complete consistent theory $T$ in a countable language if $I(T,\aleph_{0})>\aleph_{0}$ then ...
2
votes
1answer
205 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
147 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
142 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
117 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
184 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
290 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 ...
5
votes
2answers
223 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 ...
10
votes
3answers
841 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
278 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 ...
4
votes
1answer
227 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
188 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
174 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 ...
6
votes
4answers
286 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
285 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
231 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
114 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 ...
8
votes
2answers
231 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
476 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
420 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
285 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
170 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
226 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 ...
5
votes
1answer
138 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
219 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
156 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
198 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
118 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
166 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
306 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
153 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
834 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
238 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
164 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
184 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
859 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 ...
5
votes
1answer
244 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 ...