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10
votes
1answer
439 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Harrington's unpublished note “The constructible reals can be anything”

Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being ...
0
votes
0answers
39 views

Subset of the integers with certain properties

How would one find the maximal $n$ such that there exists an $n$-subset $S$ of $\mathbb{Z}^+$ such that $\forall a\subseteq S, \sum_{a\in A}a$ is either a perfect square or a perfect cube, or can one ...
4
votes
1answer
262 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 ...
2
votes
3answers
222 views

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

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
0answers
71 views

How subset is a set is proved in ZF system? [on hold]

I guess that all subset is a set is guaranteed by the axiom of separation in ZF system. Otherwise the notion of power-set will not make sense. But I wander how it's proved. I guess that the prove ...
-5
votes
1answer
132 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 ...
3
votes
1answer
66 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 ...
5
votes
1answer
283 views
+100

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 $ ...
4
votes
1answer
228 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 ...
4
votes
1answer
152 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
142 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 ...
5
votes
2answers
335 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 ...
6
votes
2answers
572 views

Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

It's not hard to show that for any cardinal $\kappa$, there is no dense linear order without endpoints (DLO) of size greater than $2^{\kappa}$ that has a dense subset of size $\kappa$. But one can ...
0
votes
0answers
31 views

Is the pseudomenon a statement? [migrated]

I'm asking this because I'm teaching a class on paradoxes for kids, and I realized I have no idea what the answer to this question is. It is a research question in the pedagogical sense, I suppose. ...
4
votes
1answer
138 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 ...
7
votes
0answers
97 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 ...
0
votes
0answers
7 views

Upper bound on cardinality of a field [migrated]

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
4
votes
3answers
324 views

Is every set class generic over a given inner model?

In a paper by B. Mitchell, I stumbled into the following sentence: "In the summer of 1986 Woodin discovered the second of the forcing orders associated with a Woodin cardinal, the extender algebra. ...
14
votes
4answers
1k views

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 ...
7
votes
2answers
207 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 ...
8
votes
1answer
270 views

Canonical model for $\neg\mathsf{CH}$ and $\Omega$-logic

Recently I found this book by Woodin. In the introduction of it the author writes the following: The main result of this book is the identification of a canonical model in which the Continuum ...
4
votes
1answer
220 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 ...
4
votes
1answer
243 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 ...
7
votes
1answer
108 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 ...
2
votes
1answer
122 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 ...
15
votes
0answers
348 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 $G\subseteq P_x$ ...
6
votes
1answer
191 views

Solovay's Theorem on Partitions of Stationary Sets and Weak Choice Principles

There is a weak choice principle called $DC_\lambda$ which holds in $L(V_{\lambda+1})$ under the assumption of a non-trivial elementary embedding $$j:L(V_{\lambda+1})\prec L(V_{\lambda+1})$$ and it is ...
5
votes
1answer
131 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, ...
7
votes
2answers
182 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 ...
7
votes
2answers
482 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 ...
70
votes
9answers
11k views

Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? Completion of ZFC ) Background The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ ...
8
votes
1answer
217 views

Destroying the Mahloness of a cardinal with $\kappa$.c.c. forcing

Question: Is it possible to have a Mahlo cardinal $\kappa$ such that there is a $\kappa$.c.c. forcing that makes it non-Mahlo? If this is possible then this forcing must change the cofinality of all ...
9
votes
1answer
244 views

splitting subsets of cardinals

Suppose $\mathbb{P}$ is a separative partial order of uniform density $\kappa$, i.e. for all $p \in \mathbb{P}$, the least size of dense set below $p$ is $\kappa$. Does forcing with $\mathbb{P}$ add ...
14
votes
2answers
827 views

Decomposing $\mathbf{\Pi}^1_1$ sets into closed sets

It is well known that every $\mathbf{\Pi}^1_1$-set is a union of $\aleph_1$-many Borel sets. I wonder whether it can be improved under certain reasonable set theory axioms assumption. For example, ...
22
votes
7answers
2k views

What is the general opinion on the Generalized Continuum Hypothesis?

I'm community wikiing this, since although I don't want it to be a discussion thread, I don't think that there is really a right answer to this. From what I've seen, model theorists and logicians ...
6
votes
2answers
313 views

Question about Woodin's stationary tower

Suppose that $\delta$ is a Woodin cardinal and that $\kappa$ is the critical point of the generic embedding $j:V\rightarrow M$ after forcing with the stationary tower ($\kappa$ can be $\omega_1$ or ...
20
votes
2answers
1k views

Axiom of Choice: Ultrafilter vs. Vitali set

It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set, a set of ...
6
votes
1answer
209 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 ...
0
votes
0answers
184 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 ...
10
votes
2answers
222 views

Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...
5
votes
1answer
145 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 ...
8
votes
3answers
595 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 ...
13
votes
1answer
836 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, ...
7
votes
1answer
179 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 ...
5
votes
1answer
354 views

Comparability implies well-orderability?

I am trying to prove a small proposition that got me completely stumped, and I cannot find a single counterexample. (ZF) Suppose that $E$ is such that for every $A\subseteq\mathcal P(E)$ either ...
35
votes
1answer
1k views

When does $A^A=2^A$ without the axiom of choice?

Assuming the axiom of choice the following argument is simple, for infinite $A$ it holds: $$2\lt A\leq2^A\implies 2^A\leq A^A\leq 2^{A\times A}=2^A.$$ However without the axiom of choice this doesn't ...
0
votes
0answers
58 views

What is semantics of “type”? Do “types” of “type theory” semantically differ from “set” of set theory? [migrated]

"To be of a (certain type)" is a fundamental relationship for ontology and the computer science "ontologies" are in the core of Semantic Web (which is my interest). But I did not encounter a ...
5
votes
2answers
351 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 ...
10
votes
1answer
262 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
314 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 ...