8
votes
0answers
146 views

Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem: Theorem. Assuming the consistency of infinitely many strong cardinals, one ...
4
votes
4answers
804 views

How short can we state the Axiom of Choice?

How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical ...
5
votes
0answers
138 views

Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$. As a C$^*$-algebraist, I ...
7
votes
1answer
225 views

Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...
6
votes
1answer
161 views

continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...
5
votes
0answers
96 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 ...
6
votes
0answers
98 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 ...
2
votes
1answer
144 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 ...
4
votes
1answer
302 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 ...
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
240 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
259 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
116 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
366 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
565 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
526 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 ...
7
votes
2answers
193 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 ...
6
votes
1answer
216 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 ...
7
votes
1answer
219 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 ...
13
votes
1answer
844 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, ...
10
votes
1answer
271 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 ...
6
votes
0answers
190 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
225 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, ...
0
votes
2answers
216 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 ...
10
votes
1answer
421 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 ...
2
votes
1answer
138 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
303 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 ...
10
votes
1answer
297 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, ...
6
votes
1answer
189 views

Stationary sets in HOD

My questions concern the following quote from “The HOD Dichotomy”, page 8. "… notice that $\ cof(\omega)\cap\lambda$ belongs to $HOD$ even though it might mean something else there. Also, ...
5
votes
1answer
321 views

On Consistency of an Existence

Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement: For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with ...
6
votes
0answers
159 views

Co-Heyting Valued Models of Paraconsistent Set Theory

I've been trying to do some forcing arguments in intuitionistic ZF using Heyting valued models where the Heyting algebra I'm using is actually a bi-Heyting algebra (both a Heyting algebra and a ...
12
votes
2answers
306 views

Producing no non-constructible reals

The following is stated without proof in Shelah's book "Cardinal arithmetic" (page 276), and is attributed to Uri Abraham: Suppose that $L[A], L[B]$ have no non-constructible reals and that ...
2
votes
1answer
207 views

Partial interpretation of an iteration

Suppose that $\langle\mathbb{P_\alpha,\dot Q_\beta}\mid \beta<\delta,\alpha\leq\delta\rangle$ is a system of iterated forcing. Let $\dot a$ be a name in $\mathbb P_\delta$, and let $G_\alpha$ be a ...
6
votes
1answer
221 views

Characterization of intermediate submodels of generic extensions

Question 1: Suppose that $V[G]$ is a set generic extension of $V$ by some forcing notion $P\in V,$ and suppose that $W$ is a model of $ZFC, V \subset W \subset V[G].$ Can we find a forcing notion ...
2
votes
0answers
307 views

Topological proof of a result in Logic

I proved the result below using logic. My questions: Can this theorem be proved by purely topological means? Do you know any theorems that either can be used to prove the same result, or which give ...
9
votes
2answers
779 views

Is there any research on set theory without extensionality axiom?

In practice (say, in computer science), collections with many "labels" ("identities"), or collections which occur in many copies, are more frequently used than sets. Such collections do not satisfy ...
5
votes
2answers
788 views

Why can't mathematics be formalised in terms of classes rather than sets? [closed]

I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...
21
votes
1answer
561 views

Is there a model of ZF set theory with a set that does not inject into the cardinals?

Question. Is there is a model of ZF set theory with a set $X$ that does not inject into the cardinals? I use the term "cardinal" here in the ZF sense, so they are not necessarily well-orderable. To ...
2
votes
1answer
212 views

Absolutness of $\Pi_1^1$ statements

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe. But it is also known that $\Pi_1^1$ statements are ...
5
votes
0answers
339 views

Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...
19
votes
1answer
611 views

Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that (a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...
0
votes
1answer
79 views

Question about the consistency of assuming (via axiom) that $\kappa < \nu$ for certain pairs of cardinal numbers provably satisfying $\kappa \leq \nu$

Call an ordered pair of formulae $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ in the language of $\{\in\}$ unproblematic iff ZFC proves that for all $\tilde{a}$ and all cardinal numbers ...
10
votes
1answer
322 views

Does small forcing preserve CH?

Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...
14
votes
6answers
2k views

What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$). Their non-existence is consistent with axioms of usual mathematics. It is provable that some of ...
3
votes
2answers
209 views

Sufficient Condition for Defining $\in$

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ with two binary relational symbols $\in , \in'$ and $ZFC$ as a $\{\in\}$-theory. If we define $\in'$ using $\{\in\}$-formula $\varphi(x,y)$ ...
1
vote
1answer
220 views

A Special Pair of Formulas

Consider the first order language ‎$‎‎‎\mathcal{L}=\{\in,\subseteq\}‎$ and ‎$‎‎\{\in\}$-theory ‎$\text{ZFC}$.‎ ‎Is ‎the‎re a formula ‎$‎‎\psi ‎(x,y)‎ \in \{\subseteq\}-Form‎$ ‎with ‎the ‎following ...
26
votes
1answer
771 views

Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...
10
votes
2answers
561 views

Questions about Prikry forcing and Cohen forcing

I have some questions. The first one is about the product of Prikry's forcing. Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...
6
votes
1answer
212 views

Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...
9
votes
1answer
282 views

Inner model in which every uncountable cardinal is large

The following is known: $(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$. My question is that: Are there any large cardinal ...