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 ...

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Question on the consistency of Zermelo set theory minus specification and extensionality [closed]

Let $W=Z^{-}-Specification$ where $Z^{-}=Z-Extensionality$ and Z is Zermelo set theory. What is known about models of $W$ or $W^{+}=W+Extensionality$?
5
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0answers
107 views

A question about ordinal definable sets of real numbers revisited [duplicate]

Citing (almost) A question about ordinal definable real numbers If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is ...
7
votes
0answers
109 views

Can $Ded(\kappa)$ be a supremum?

Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$. It ...
1
vote
1answer
212 views

Problem of book Kunen [closed]

Suppose $P$ is a notion of forcing in $M$ such that $\left | P \right | \leq \omega_{1}$ and $P$ is ccc. Suppose further $\Diamond$ holds in $M$. How does one show that $\Diamond$ also holds $M[G]$?
4
votes
1answer
191 views

The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$. One important example of left distributive algebras arises ...
3
votes
3answers
300 views

Cardinality of $C^*([0,1])$ [closed]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?
3
votes
1answer
92 views

Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion. Fix some dense embedding $ i \colon P ...
4
votes
1answer
324 views

Set-theoretic tautologies

Let us consider unquantifed formulas of a set theory (for example, NBG), more precisely, the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set and the class of all ...
4
votes
1answer
194 views

forcing square with small conditions

In the paper, Large cardinals and definable counterexamples to the continuum hypothesis, Foreman and Magidor mention a way to force $\square_{\omega_1}$ with countable conditions. (This is used in ...
4
votes
2answers
170 views

Borel Sets in Sacks Generic Extension

Let $\mathbb{S}$ denote Sacks forcing. This is forcing with perfect trees or equivalently forcing with uncountable Borel subsets of ${}^\omega 2$ with the relation $\subseteq$. Let $G \subseteq ...
6
votes
2answers
448 views

Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...
5
votes
1answer
310 views

Embeddings of forcing notions - preserve properness?

Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M $ ...
9
votes
1answer
190 views

Without AC, which implications between the different definitions of amenability still hold?

More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF ...
3
votes
1answer
159 views

Is it compatible with ZF to assume that every amenable discrete group is finite?

The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated ...
4
votes
1answer
261 views

How do we know if Vaught's Conjecture is Absolute?

Please note that this might be some confusion on my part about the work surrounding Vaught's conjecture. First of all, Vaught Conjecture states that if a first-order complete theory $T$ in a ...
8
votes
2answers
244 views

A Weakening of the Tree Property

If $f$ and $g$ are two functions, define $f \sim g$ if they differ only finitely often on their common domain. The following property of a large cardinal arose from a problem in model theory. I am ...
-5
votes
1answer
139 views

Does an arbitrarily selected infinite number of integers form a set? [closed]

A serial of arbitrarily selected infinite number of integers S = {I1, I2, ...} (not sorted) is a subset of the set of all the integers. But is it a set? If yes, could we determine whether the ...
4
votes
1answer
273 views

Question about “Coding the universe”

The following is a result which I know as a weak form of Jensen's coding lemma$^*$ (first published in the book "Coding the universe"; also see http://www.jstor.org/stable/2273986): For any class ...
2
votes
1answer
139 views

Kunen's inconsistency concerning $L$

A famous result by Kunen regarding elementary embeddings states that there is no such embedding from $V$ onto itself which would be non-trivial. It's clear that if $V=L$ then there is also no ...
7
votes
1answer
124 views

Strongly compact cardinal with bad covering properties

This is a continuation of the question covering properties of strongly compact embedding. Recall that a cardinal $\kappa$ is $\nu$-strongly compact cardinal if there is an elementary embedding ...
15
votes
0answers
385 views

If all reals are generic, is the set of reals generic?

Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent: For every $x\in\Bbb R^V$ there is some $P_x\in W$ such that for some $G\subseteq P_x$ ...
10
votes
1answer
604 views

Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
6
votes
2answers
580 views

Is there one binary operation foundational for set theory?

The membership relationship "$\epsilon$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\epsilon$". Naturally, the question arises ...
5
votes
1answer
140 views

A realcompact analogue of the Baire category theorem

Let $\frak{m}$ be the least measurable cardinal. A space $X$ is realcompact if it is homeomorphic to a closed subset of some product $\mathbb{R}^I$. Let $X$ be realcompact with $P_\frak{m}$ topology, ...
8
votes
2answers
230 views

Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...
0
votes
1answer
274 views

One or two questions about so-called “absolute” set theories

Nearly fifty years ago Takeuti called attention to a phenomenon that occurs in connection with the construction of set theories such as ZF that result in a hierarchy of sets (indexed by ordinal ...
6
votes
1answer
230 views

Does V=L imply transitive containment over, say, Z?

In Zermelo set theory, the axiom of constructibility V=L seems to imply every set has a transitive closure. But does that argument have to assume transitive closure in the first place, to get an ...
5
votes
1answer
154 views

Covering properties of strongly compact embedding

Let $\kappa$ be a $\mu$-strongly compact cardinal, which means that there is an elementary embedding $j:V\rightarrow M$, with critical point $\kappa$ such that $M$ is well founded (even closed under ...
7
votes
1answer
271 views

Forcing, cuts, and Dedekind-finite cardinalities

Tl;dr version: there are two natural classes of cuts in the nonstandard model of arithmetic consisting of the Dedekind-finite sets (if, in fact, they constitute such a model); both these classes are ...
14
votes
1answer
899 views

How much of GCH do we need to guarantee well-ordering of continuum?

It's well known that, if GCH holds, then every cardinal can be well-ordered. However, I'm sure we don't need full power of GCH to prove it for specific cardinal, e.g. continuum. I have been wondering, ...
11
votes
1answer
291 views

What is the large cardinal strength of the assertion that every $\kappa$-complete filter on $\kappa$ extends to a $\kappa$-complete ultrafilter?

It is well-known that an uncountable regular cardinal $\kappa$ is strongly compact if and only if every $\kappa$-complete filter on any set extends to a $\kappa$-complete ultrafilter on that set. The ...
9
votes
2answers
355 views

How necessary is Godel's Condensation Lemma

It seems that the Godel's Condensation Lemma is typically used to show that certain constructible sets will appear by some stage of the construction of $L$. For example in the proof that CH holds in ...
4
votes
2answers
275 views

Maximum cardinality of a filtered limit of finite sets

Let $(I,<)$ be a directed, partially ordered set. Consider an inverse system $(S_i)_{i \in I}$ of finite sets, i. e. a functor $S:I^{op}\to \mathbf{FinSet}$. What is the maximum possible ...
2
votes
0answers
92 views

Does $\forall X \in L_{\kappa}: P(X) \subset L_{\kappa}$ hold in $L$

While working on an exercise in Jech's Set Theory, I tried to prove that if $V=L$, then $\forall X \in L_{\kappa}: P(X) \subset L_{\kappa}$, where $k$ is any cardinal. I was hoping someone could ...
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votes
1answer
165 views

A general question on nonnegative integer sequence [closed]

Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$. Let $B=\mathbb Z_{\ge 0}-A$. Define $\ 2A= \{a+b : a \in A,\ b \in A\}$. Define $\ 2B=\{a+b : a \in B,\ b \in B\}$. Then the set ...
5
votes
2answers
376 views

How strong is limitation of size + generalized continuum hypothesis?

This is a question about ZFC (or maybe NGB), but it is motivated by Randall Holmes' article on alternative set theories, especially his elaborations on TST, bounded Zermelo set theory and pocket set ...
0
votes
0answers
76 views

Rank function and closure operator for a set system

I would like to trace the concepts "rank function" and "closure operator" back to some structures as primitive as possible. For a set system $(E,F)$ which is an independence system or a greedoid, I ...
0
votes
1answer
157 views

forcing and set theory [closed]

Let $\kappa , \lambda , \theta$ be infinte cardinals de $M$ represents a c.t.m, and $P=Fn( \kappa\ \times \omega,2)$. Show that $(\lambda^{\theta})^{M[G]}\leq((max( \kappa, \lambda)^{\theta})^{M}$. ...
6
votes
0answers
199 views

Does Sageev's result need an inaccessible?

In 1981, building on work by Ellentuck in 1974, Sageev showed ("A model of ZF + there exists an inaccessible, in which the Dedekind cardinals constitute a natural non-standard model of arithmetic," ...
3
votes
1answer
244 views

What axioms (other than choice) have a taming effect on the ordering of cardinalities?

Axiom of choice arranges all cardinalities into a well-ordered chain but without it their ordering can be wild in general ZF models, e.g. two cardinalities may not even have inf or sup. However, ...
1
vote
2answers
227 views

Valid statement in submodel? (Consistency lemma in Cohen's CH book)

In Paul Cohen's Set Theory and the Continuum Hypothesis (Page 71) there is a lemma with the assumption that $\exists x\, P(x)$. The ''proof'' there, uses the following argument: Intuitively the ...
5
votes
1answer
190 views

Ordinal definable sets of reals in the Solovay

To be precise, let $\Omega$ be an inaccessible cardinal in $L$ and let N be the Solovay model defined by the Levy-collapse in this case. Then $\Omega$ is $\aleph_1$ in $N$. How many different OD ...
10
votes
1answer
439 views

Does Turing determinacy imply full determinacy?

The axiom of Turing determinacy is a weakening of the full axiom of determinacy, $AD$, in which only games with payoff sets which are $\equiv_T$-invariant are demanded to be determined. In "Turing ...
3
votes
2answers
425 views

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 ...
2
votes
1answer
156 views

The number of countable models [duplicate]

Let $\mathcal{L}$ be a countable first order language. For a natural number n, can we find a complete $\mathcal{L}$-Theory $T$ which has exactly n non-isomorphic countable models ?
8
votes
2answers
313 views

When is $A$ “$L$-ish” whenever $B$ is “$L$-ish”?

My question is about a kind of relative constructibility in set theory. Fix a countable transitive model $W\models ZFC$ which is much bigger than $L^W$. There is a natural way within $W$ to compare ...
8
votes
1answer
281 views

Is $\ell^\infty$ Polishable?

Consider $\ell^\infty$ as a subspace of the Polish space $\mathbb{R}^\omega$. It is easy to check that $\ell^\infty$ is not Polish in the subspace topology, as it is countable union of the compact ...
3
votes
1answer
155 views

Are normal ultrafilters generated by conditional closure systems?

Suppose that $\kappa$ is a cardinal, $X$ is a set with $|X|>\kappa$, and $\mathcal{U}\subseteq P(P_{\kappa}(X))$ is a normal ultrafilter. We say that a collection $C\subseteq P_{\kappa}(X)$ is a ...
2
votes
1answer
168 views

Do models of ZFC have arbitrarily large descent values?

Let $M$ denote a well-founded set-sized model of ZFC. The descent value of $M$ will be defined as the value of $n$ returned by the following process. Initialization. Let $n$ equal $0$ and $X$ equal ...
10
votes
1answer
303 views

Elements of the method of forcing in some papers of N. N. Luzin

In the paper Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...