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2
votes
0answers
107 views

A question regarding an analogue of the Kleene $T$-predicate for Koepke's ordinal computability

Does Koepke's notion of ordinal computability admit an analogue of the Kleene $T$-predicate? If so, is the existence of such a $T$-predicate independent of $ZFC$? Also, if one assumes the existence ...
6
votes
0answers
87 views

By replacing the Laver tables with nearly distributive fake Laver tables, can one produce algebras where the period of 1 grows fast?

I am now generalizing the notion of the classical Laver table and the fake Laver table to a larger class of algebras. A sequence of algebras $(\{1,...,2^{n}\},*_{n})_{n\in\omega}$ is said to be a ...
6
votes
0answers
115 views

How distributive are the fake Laver tables?

The Laver table $A_{n}$ is the unique algebra $(\{1,...,2^{n}\},*)$ such that $x*1=x+1$ for $1\leq x<2^{n}$, $2^{n}*1=1$, and $x*(y*z)=(x*y)*(x*z)$. Let's now replace the Laver table $A_{n}$ with ...
1
vote
0answers
181 views

How may we define a bijection from $\wp(\mathbb{Q})$ to $\mathbb{R}$ in $ZFC$? [closed]

Some expressed difficulty understanding that there are more members in $\mathbb{R}$ than in $\mathbb{Q}$ according to classical set theories because there between all real numbers is a rational ...
17
votes
0answers
401 views

The cofinality of $(\mathbb{N}^\kappa,\le)$ for uncountable $\kappa$?

For a partially ordered set $P$, a set $A\subseteq P$ is cofinal if for each element of $P$ there is a larger element in $A$. The cofinality of $P$, ${\rm cof}(P)$, is the minimal cardinality of a ...
9
votes
2answers
241 views

Two questions about the “grasp” cardinal function

For a topology $\mathcal{T}$ on a set $S$, where $\mathcal{T}$ does not have a finite base, I define the grasp $g(\mathcal{T})$ to be the least infinite cardinal $\kappa$ such that $\mathcal{T}$ has a ...
12
votes
2answers
834 views

On Hamkins' answer to a problem by Michael Hardy

Based on a post by Michael Hardy and Hamkins' answer to it Andreas Blass, Will Brian, Joel Hamkins, Michael Hardy and Paul Larson introduced a new cardinal characteristic of the continuum ...
4
votes
0answers
167 views

On the Axiom of Choice for Conglomerates and Skeletons

Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and ...
4
votes
0answers
112 views

Characterization of $L[T_{2n+1}]$ as a direct limit of mice

I am asking for a reference request/proof sketch for the result of Steel that characterizes $L[T_{2n+1}]$ as a direct limit of mice. Given that both $L[T_{2n+1}]$ and $M_{2n}$ have a $\Sigma_{2n+2}$ ...
2
votes
1answer
158 views

Set theoretic issue of localization of abelian categories

For a small abelian category in which every object is also a set, consider its localization with respect to a Serre subcategory (thus a quotient category), is it true that under this localization ...
24
votes
0answers
525 views

Where do uncountable models collapse to?

Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
12
votes
2answers
380 views

Woodin on Posner-Robinson for the hyperjump and sharp

The Posner-Robinson theorem states that, if $X$ is noncomputable, there is some $G$ such that $X\oplus G=G'$; that is, even though genuine jump inversion only works above $0'$, every (nontrivial) $X$ ...
11
votes
1answer
454 views

Can you have many independent reals?

Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that: $\Bbb P_\alpha$ is c.c.c. $\Bbb P_\alpha$ adds a real ...
3
votes
0answers
324 views

“Nicely” strong measure zero sets

This question is essentially an expanded version of the unanswered half of Two strengthenings of "strong measure zero". A set $X$ of reals is strong measure zero if, for any $f: ...
7
votes
1answer
334 views

Is every ordinal potentially definable?

It is easy to see that, if $V\models\alpha>\omega_1^{CK}$, then $\alpha$ is not recursive in any forcing extension of $V$. The argument goes as follows: The relation "$\Phi_e=r$" is $\Pi^0_2$. ...
11
votes
0answers
300 views

Large cardinals arising from alternate set theories

My question is whether there are model-theoretic large cardinal axioms associated with $NFU$ - or more generally, with set theories other than $ZFC$. Large cardinal properties generally come in one ...
9
votes
0answers
147 views

Idea behind the proof of consistency of club filter of $\omega_1$ is ultrafilter + ZF + DC

I've been trying to understand Radin Forcing and some of its applications, one of which is the use of it to prove the consistency of ''Club filter of $\omega_1$ is an ultrafilter + ZF + DC''. However, ...
12
votes
1answer
225 views

Forcings that are not equivalent to Levy collapse

Assume GCH and that $\kappa$ is a regular uncountable cardinal. Let $\mathbb{P}$ be a separative, $<\kappa$-directed closed, nowhere trivial, $\kappa^+$-cc poset of size $\kappa^+$. Must ...
3
votes
0answers
95 views

A question regarding forcing in $NGBC^{-f}$+$BAFA$

Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can ...
19
votes
3answers
392 views

Which partitions of $[0,1]$ are collection of level sets of a real continuous function?

Let $f:[0,1]\to[0,1]$ be given. The level sets of $f$ (ie the collection of all sets of the form $\{x\in[0,1]:f(x)=y\}$, for each fixed $y\in[0,1]$) partition the domain of $f$. I am curious for set ...
0
votes
1answer
79 views

Prime ideals containing the finite members of ${\cal P}(\omega)$

Let ${\frak P}$ denote the collection of prime ideals containing the finite members of ${\cal P}(\omega)$, and order ${\frak P}$ by set inclusion. What is the cardinality of ${\frak P}$, and what's ...
15
votes
2answers
330 views

Can $L$ be thin?

I have recently been wondering if the following is consistent with ZFC: For every infinite ordinal $\alpha$: $|V_\alpha\cap L|=|\alpha|$. Intuitively, this states that for $L$ is very "thin", in ...
0
votes
1answer
91 views

Boolean algebras and free filters generated by chains

Suppose $\mathbf{B}$ is a complete Boolean algebra with an infinite domain $B$. Suppose $\mathbf{B}$ is atomic (i.e. every element is the supremum of some set of atoms). This algebra contains the ...
9
votes
0answers
153 views

From interpolation to separation

Lusin's separation theorem states that, if $A$ and $B$ are disjoint analytic subsets of a Polish space, then there is a Borel set $X$ separating them ($A\subseteq X$, $B\cap X=\emptyset$). Craig's ...
2
votes
1answer
377 views

Does Borel's proof for existence of normal numbers make an essential use of axiom of choice?

A normal number is a real number whose infinite sequence of digits in every base $b$ is distributed uniformly in the sense that each of the $b$ digit values has the same natural density $\frac{1}{b}$, ...
12
votes
0answers
217 views

A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
5
votes
1answer
155 views

$\kappa$-support iterations of $<\kappa$-strategically closed forcing

Let $\kappa$ be an uncountable regular cardinal, and suppose that $\langle \mathbb{P}_\alpha,\dot{\mathbb{Q}}_\alpha:\alpha<\delta\rangle$ is a $\kappa$-support iteration of ...
9
votes
0answers
148 views

Homogeneity of a variant of Prikry forcing

Prikry forcing is easily seen to be cone homogeneous (for any $p, q \in \mathbb{P}$, there are $p' \leq p, q' \leq q$ and an isomorphism $\Phi: \mathbb{P}/p' \simeq \mathbb{P}/q'$); in particular for ...
22
votes
5answers
2k views

Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of ...
5
votes
1answer
164 views

Does $ZF$+$LM$+$A_{\lt2^{\aleph_0}}$ imply $\lnot$$WCH$?

In what follows, I will use the following acronyms: $AS$:= Freiling's Axiom of Symmetry $LM$:="Every set of reals is Lebesgue measurable." $WCH$:="every uncountable subset of $\mathbf R$ can be put ...
10
votes
1answer
414 views

Cardinality of definable sets of reals

Throughout this question we assume ZFC. If CH holds, then the following is obvious: (S) Every definable infinite subset of $\mathbb R$ has size either $\aleph_0$ or $2^{\aleph_0}$. (It's true ...
4
votes
1answer
221 views

A question regarding a common critique of Freiling's Axiom of Symmetry

(In what follows, Freiling's Axiom of Symmetry is simply the following: ($A_{\aleph_0}$) :( $\forall$$f$: $\mathbf R$ $\rightarrow$$\mathbf ...
21
votes
2answers
731 views

Antirandom reals

This is a crossposting of http://math.stackexchange.com/questions/1446602/anti-random-reals, which has not gotten any answers; after thinking about the problem, I've become more convinced that it ...
16
votes
1answer
473 views

Two strengthenings of “strong measure zero”

A set $X\subseteq\mathbb{R}$ is strong measure zero if, for every sequence $(\epsilon_i)_{i\in\mathbb{N}}$ of positive reals, there is a sequence $(I_i)_{i\in\mathbb{N}}$ of open intervals covering ...
14
votes
2answers
544 views

Who needs RCS iterations?

According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more ...
8
votes
2answers
380 views

Ordering of large cardinals by cardinality

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic) Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type ...
10
votes
0answers
236 views

When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$ satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$. ...
6
votes
1answer
285 views

Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...
5
votes
1answer
153 views

almost disjoint ladder system on $\omega_2$

Suppose $\langle s_\alpha : \alpha \in \omega_2 \cap \mathrm{cof}(\omega_1) \rangle$ is a sequence such that each $s_\alpha$ is an increasing cofinal map from $\omega_1$ to $\alpha$. Is it possible ...
18
votes
2answers
537 views

History of set-class distinction

I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class: Who was the first mathematician to make this ...
2
votes
1answer
204 views

Wholeness Axiom and Ultimate L

From what I understand: The Wholeness Axiom(s) is/are the "ultimate axioms of infinity", bordering on inconsistency with ZFC. Ultimate L (Completion of ZFC) attempts to extend the orderly world of ...
6
votes
1answer
146 views

Different ways of making $HOD$ far from $V$

There are different criteria for building a model $V$ of $ZFC$ which is far from its $HOD$, for example: $(A)$ Cardinality criteria: For this in a joint work with James Cummings and Sy Friedman, we ...
9
votes
2answers
760 views

Non null Turing antichain

This interesting question resulted from a query of Mushfeq: In ZFC, can we find a non null set of pairwise Turing incomparable reals?
8
votes
2answers
362 views

Some “axiom of choice” and “dependent choice” issues

I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these. If I understand correctly, mathematicians tend to be quite happy working with ...
21
votes
0answers
533 views

Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...
6
votes
0answers
181 views

Remote points in $\beta X$

It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space ...
16
votes
1answer
474 views

Can we find CH in the analytical hierarchy?

Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out ...
15
votes
1answer
403 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
7
votes
1answer
173 views

Forcing Extension of Countable Linearly Iterable Structures

Let $V$ satisfy there exists a measurable cardinal. Let $\kappa$ be a measurable cardinal and $U$ be the normal measure on $\kappa$ witnessing this. Let $\mathbb{P}$ be a forcing of size less than ...
5
votes
0answers
65 views

Is there an analytic $\mathrm{P}$- ideal on $\omega$ which is not $\Sigma^0_2$ and not $\Pi^0_3$-complete?

Soleski proved that for any analytic $\mathrm{P}$-ideal on $\omega$ is $\Pi^0_3$. The usually example , such as the density zero ideal $Z_0$ is $\Pi^0_3$-complete, $I_{\frac{1}{n}}$ is ...