22
votes
Accepted
Is there a form of choice that can elude Kunen's inconsistency theorem?
Work of Usuba combined with work of Woodin shows that if there is a Reinhardt cardinal $\kappa$ that is a limit of Lowenheim-Skolem cardinals, then there is a forcing extension in which $\kappa$ ...
- 5,796
19
votes
Statements in differential geometry independent from ZFC
[Using the comments for context on undecidability/independence of ZFC]
A computably undecidable problem is whether or not a homology sphere has a metric of positive scalar curvature [Page 79 of ...
16
votes
Accepted
Universal property of the set of injections in the category of sets
$\newcommand{\Inj}{\operatorname{Inj}}\newcommand{\Set}{\mathrm{Set}}$Maps $X \to \Inj(A,B)$ correspond to monomorphisms $A \times X \to B \times X$ in the slice $\Set/X$, which can be thought of as “$...
- 16.9k
14
votes
Accepted
How “disconnected” can a continuum be?
The answer to all three of your questions is yes.
The cardinal $\mathrm{disc}([0,1])$ is discussed in this MO question of Taras Banakh. He calls this cardinal the Sierpiński cardinal and denotes it $\...
- 15.7k
12
votes
Accepted
Can the axiom of choice be proved with ZF+Tarski axiom?
Following the link found in the Wikipedia article about the Tarski–Grothendieck set theory, the required proof (by Tarski himself!) can be found beginning on p.181 of his article "On the well-...
- 5,005
11
votes
Does Urysohn's Lemma imply Dependent Choice?
In Versions of normality and some weak forms of the axiom of choice Paul Howard et al exhibit a model of MC (Multiple Choice) and not-DC, see page 381.
In that model Urysohn's Lemma (NU) holds, so it ...
- 8,365
10
votes
Minimum transitive models and V=L
This is not a full answer, but I found it interesting to notice that if we relax the c.e. requirement somewhat, then there is a sweeping positive answer.
Theorem. Every complete theory extending ZFC + ...
- 205k
10
votes
Accepted
Bernstein's proof of the continuum hypothesis
This is really a long comment. This paper has been reviewed twice by zbMATH: one by H. B. Curry, which is not informative; another by W. Ackermann, which is in German. The following is the (...
9
votes
Accepted
Consequences of foundation/regularity in ordinary mathematics (over ZF–AF)?
I think Frucht's theorem, the statement "every group is isomorphic to the automorphism group of a graph", is a great example of what you're looking for; from the statement, it'd be hard to ...
- 106
9
votes
Accepted
Minimum transitive models and V=L
Yes, I claim you can in fact get one whose minimum model is a set forcing extension of a segment of $L$. Let $L_\alpha$ be least modelling ZFC. Let $\mathbb{P}=\mathbb{P}^{L_\alpha}$ be Jensen's ...
- 6,154
9
votes
Accepted
Is it true that $\mathit{MA}(\omega_1)$ iff $\omega_1<\mathfrak{p}$?
Q1: No, see Between Martin's Axiom and Souslin's Hypothesis by Kunen and Tall. Note: Bell proved in The combinatorial principle $P(\mathfrak{c})$ that $\mathfrak{p}>\aleph_1$ is equivalent to $\...
- 8,365
8
votes
Elementary countable submodels in Gödel's universe
As Asaf and Joel have observed, the answer to your question is negative. However, there is a sense in which being an elementary submodel of $L_{\omega_1}$ is the only way to "persistently" ...
- 20.1k
8
votes
Elementary countable submodels in Gödel's universe
No, there are many instances of $L_\alpha\prec L_\beta$ without $L_\alpha\prec L_{\omega_1}$.
Here is one easy way to construct one.
Consider the smallest $\alpha$ that has an elementary extension $L_\...
- 205k
8
votes
Accepted
Can we have this sequence where choice fails and returns?
Sure.
Start with countably many inaccessible cardinals, $\kappa_n$, and now take the full support product adding $\kappa_n^+$ subsets to each $\kappa_n$. Then the $n$th model is the symmetric ...
- 36.6k
7
votes
Accepted
Is ordinal definability in terms of stages of cumulative size hierarchy equivalent to the usual one?
The answer is yes, in a very general way.
What I claim, first, is that the Lévy-Montague reflection theorem holds in ZF for any definable continuous cumulative hierarchical representation of the set-...
- 205k
7
votes
Accepted
Elementary countable submodels in Gödel's universe
Very clearly not. Take some countable elementary submodel $M_0$ of $L_{\omega_2}$, and take $M_1$ to be another one, but with $M_1$ a end extension of $M_0$. We can find such models by first finding ...
- 36.6k
7
votes
Accepted
Are there premice that are $\omega_1$-iterable but not $(\omega_1+1)$-iterable?
It is consistent (relative to large cardinals). There is an example given in Example 3.6 here. For a brief summary: the model is the minimal proper class mouse $S$ such that $\mathbb{R}^S$ is closed ...
- 6,154
6
votes
Complexity of infinitary satisfiability, part 2
Here is a partial answer; it deals with those admissibles $\kappa$ large enough to see that $\mathrm{cof}(|\kappa|)=\omega$ and such that $L_\kappa$ has largest cardinal $\theta$.
That is, let $\kappa$...
- 6,154
6
votes
Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
Any model of $ZF$+stationary proper class of $I3$ ordinals (whose consistency strength is at most $ZF$+$I2$) where $W_α$ is listing of $V_{κ}$ where $κ$ is $I3$ and $j_α$ the witness of $I3(V_κ)$ will ...
- 708
6
votes
Accepted
On the definition of small categories in SGA4
You’re correct: read literally, those definitions are mismatched, for the reasons you give. The solution is to fix the definition of “$U$-small category” to say that “$\newcommand{\C}{\mathcal{C}}\...
- 16.9k
5
votes
Accepted
What is the "iterated definability" limit of first-order logic?
If I understand the question properly (I'm not sure whether I do), then it looks like your conjecture for question 2 is correct, i.e. $L_{\beta_0}\cap\mathcal{P}(\omega)$. Here is a hastily written ...
- 6,154
5
votes
Is there a countably infinite closed interval in the lattice of topologies?
The special case where $\sigma=\{\emptyset,X\}$ is the trivial topology is easy to resolve. In this case, if $\tau$ is finite, then the interval $[\sigma,\tau]$ is finite. If $\tau$ is infinite, then ...
- 10k
5
votes
Accepted
About the relationship between the generalized continuum hypothesis and the axiom of choice
Possibility (1) holds; i.e. ZF + "for all limit ordinals $\lambda$, GCH2($\lambda$) holds" implies choice. For it implies that for cofinally many ordinals $\lambda$, $V_\lambda$ can be ...
- 6,154
5
votes
Classifying set theories whose standard models sharing the same ordinals are equal
For every c.e theory $T$ extending KP (Kripke-Platek) with a model $M$ of height $α<ω_1$, the intersection of all such $M$ is a subset of $L_{α^{+,\mathrm{CK}}}$. This holds since the existence of ...
- 5,437
5
votes
Minimum transitive models and V=L
Here is another partial result; it complements Joel Hamkins' answer. Note that in the following theorem, $T$ is not necessarily a c.e. theory.
Theorem. Suppose $T$ is an extension of $\mathrm{ZF} + \...
- 15.1k
5
votes
Accepted
If we add stratified\acyclic replacement to the wholeness axiom, would that increase its consistency strength?
The statement beginning "The rationale beyond" is not quite correct. The critical sequence can be constucted by exploiting the facts that $\bigcup j(x)$ will be assigned the same number as $...
- 1,004
5
votes
Accepted
Ramsey-like property with order condition
A coloring $c:[\kappa]^2\rightarrow\theta$ is subadditive of the first kind if for all $\alpha<\beta<\gamma<\kappa$, $c(\alpha,\gamma)\le\max\{c(\alpha,\beta),c(\beta,\gamma)\}$. It is ...
- 1,056
4
votes
Accepted
Is there a hereditary $\sigma$-space $X$ such that it is not $Q$-space?
Every $S_1(B_\Gamma,B_\Gamma)$ space is a $\sigma$-space, and the property $S_1(B_\Gamma,B_\Gamma)$ is hereditary for subsets (B. Tsaban and M. Scheepers, The combinatorics of Borel covers, Topology ...
- 2,624
4
votes
Accepted
When does the cardinality of a set equal the cardinality of an element of $V_\lambda$ for $\lambda$ being a limit ordinal?
The proposition holds for the limit ordinal $\lambda$ iff $V_\lambda$ is wellorderable.
For if $V_\lambda$ is wellorderable where $\lambda$ is a limit, then it proposition easily follows at $\lambda$. ...
- 6,154
4
votes
Does $\mathit{Aut}(\mathbb{R};+)$ have a copy in $L(\mathbb{R})$ granting large cardinals?
Assuming also CH, the answer to the more general question is yes,
there is a structure in $L(\mathbb{R})$ (in fact, just the set $\mathbb{R}$, with no additional structure), whose automorphism group ...
- 6,154
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