23
votes
Accepted
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
It is a very nice question, but unfortunately, this is impossible.
Each member $s\in S$ must be countable, since uncountable sets have accumulation points. And since the hierarchy is accumulating as ...
- 230k
21
votes
Accepted
Are there substantive differences between the different approaches to "size issues" in category theory?
From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is ...
- 40.8k
19
votes
Accepted
Proof/Reference to a claim about AC and definable real numbers
The argument, given to me by Hugh Woodin over a drink in Barcelona in September 2016, is a nice retort to "AC is obviously false since there cannot be a well-ordering of the real numbers". ...
- 38.5k
17
votes
Are there substantive differences between the different approaches to "size issues" in category theory?
Although people often talk as though it just doesn't matter which approach you use — perhaps all universes are alike? Let me prove that in a strict sense this is not true. The nature of the ...
- 230k
17
votes
Is there a minimal (least?) countably saturated real-closed field?
In Solution d'un problème d'Erdös, Gillman et Henriksen et application à l'étude des homomorphismes de $\mathcal{C}(K)$, Acta Math. Acad. Sci. Hungar. 30 (1977), no.1-2, 113–127 (EuDML), Jean Esterle ...
- 6,190
16
votes
Accepted
What gets to be called a "proper class?"
The term "class" is not a technical term with a universally definite meaning, but there are various established meanings in various contexts.
In ZFC the established usage as Wojowu mentions ...
- 230k
14
votes
Proof/Reference to a claim about AC and definable real numbers
Unfortunately, the claim you have stated is not true. Regardless of the axiom of choice, every real is definable from a countable sequence of ordinal parameters, since the real is definable from the ...
- 230k
13
votes
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
Here's a ZF proof that if $S$ is a chain of sets with $\bigcup S = \mathbb{R},$ then there is $X \in S$ which contains a countable set dense in some nonempty open set.
If there is $X \in S$ such that $...
- 4,054
11
votes
Accepted
Determinacy and Woodin cardinals
The result is true if you actually collapse the Woodin cardinal, not just the cardinals smaller than it. This follows from the results in Itay's chapter in the Handbook. See MR2768701, zbM1198.03057
...
- 31.9k
10
votes
Determinacy and Woodin cardinals
(This answers what was the original question, which was with $\mathrm{Coll}(\omega,{<\kappa})$ replacing $\mathrm{Coll}(\omega,\kappa)$.)
Hmm...doesn't this contradict Theorem 1.22 of "MICE ...
- 9,167
9
votes
Are there substantive differences between the different approaches to "size issues" in category theory?
One naïve difference is the handling of things like functor categories between large categories. Let $\mathcal{C}$ and $\mathcal{D}$ be large categories. If we want to consider something like $\...
- 9,107
9
votes
Accepted
Which are the hereditarily computably enumerable sets?
The complexity of h.c.e. sets is quite subtle as the structure of the target set affects our ability to offload complexity to the limiting process. The property of being an h.c.e. code is $Π^1_1$ ...
- 6,365
8
votes
Which are the hereditarily computably enumerable sets?
Re question 3: Not every arithmetic real is h.c.e.. In fact, every h.c.e. real is $\Sigma_2^\mathbb{N}$. For fix a program $e$ such that $x_e\subseteq\omega$. Let $n\in\omega$. Then $n\in x_e$ iff ...
- 9,167
8
votes
Accepted
Is it obvious that Global Choice holds for $\mathrm{ZF}^-$?
The statement "$\mathsf{ZF}^-$ has only countable sets" is false. $\mathsf{ZF}^-$ makes no commitment to the existence or not of uncountable sets. I suspect the theory you actually have in ...
- 20.7k
8
votes
Accepted
A function $f$ such that $j_U(f)(\kappa)=[\operatorname{id}]_U$ for all ultrapower embeddings $j_U$ with critical point $\kappa$
This is never true in the circumstances you request, quite apart from your uniformity requirement, since some $U$ admit no such $f$ at all.
The reason is that if $[\text{id}]_U$ is generated by $\...
- 230k
7
votes
Hereditarily countable sets in Antifounded ZF
Update. This answer does not answer the question that was asked, since Jech is using what had seemed to me as an idiosyncratic definition of hereditary countable. But upon reflection, I find his ...
- 230k
7
votes
Is $\mathrm{ZF}^-$ consistent with an Axiom of Countability?
Short version: no, not in any way I can see.
First of all, there's a language issue: since a surjection $\mathbb{N}\rightarrow V$ would be a proper class, it's not clear how to even state your ...
- 20.7k
7
votes
Accepted
Large almost disjoint family on $\mathbb{N}$ with property $\mathbf{B}$
One of the standard examples of an almost disjoint family of cardinality $\mathfrak c$ is the set of paths through the complete binary tree $2^{<\omega}$ (identified with $\omega$ via your favorite ...
- 72.4k
6
votes
Accepted
Axiomatic strength of the cumulative hierarchy
This is an edited and improved answer; see edit details at the end.
We can obtain the whole of ZF using a single, natural, scheme.
I will keep the definition of level from Button 2021, as cited above. ...
- 136
6
votes
Extending normal filters
As Joel suggested in the comments, there are always normal filters that do not extend to normal ultrafilters. For example, let $F$ be the filter on $\kappa$ of all sets containing the set $S$ of ...
- 2,319
6
votes
Accepted
Natural functions outside $\sf PA$?
Sure, but this is really a fact about structures rather than theories. For example, $\mathsf{ZFC}$ can define the function sending $n$ to the least natural number not definable in the language of ...
- 20.7k
5
votes
Accepted
Extending normal filters
In general, no, because $\kappa$ might not be $\lambda$-supercompact, even if it is $\lambda$-strongly compact. The two large cardinal notions are not provably equivalent (although it is an open ...
- 230k
5
votes
How strong is the Schröder–Bernstein theorem where one set is the natural numbers?
I have a slight strengthening of OP’s result. Assume the Schröder–Bernstein Theorem for $\mathbb{N}$.
Lemma (Creating Subject): for all propositions $P$, there exists a function $f : \mathbb{N} \to \{...
- 263
5
votes
Accepted
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
You've shown how to prove, in PA, the statement "the Goodstein sequence starting with $p$ terminates" for any given $p$. But once $p$ is given, that statement has a proof in PA that just ...
- 72.4k
5
votes
Accepted
What determines non-finite axiomatizability of a class extension of a set theory?
As explained with more details in https://mathoverflow.net/a/87249, every sequential theory that proves the induction schema for all formulas in its languages is reflexive (even uniformly essentially ...
- 45.7k
5
votes
Accepted
About having one axiom schema for ZFC motivated after the iterative conception of sets?
I won't engage with the level terminology, but I believe your question is answered by the following observation.
Theorem. ZFC is equiconsistent with the theory ZFC + there is a closed unbounded class $...
- 230k
5
votes
Accepted
Can PA define functions related to higher theories?
Yes, this function is obviously definable in PA and PA proves it is total. You are defining the tower of theories by recursion, which PA can do, and taking the Rosser sentence of each theory, which PA ...
- 230k
5
votes
At what ordinal $\chi$ does $\mathrm{L}_\chi$ contain a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?
As a precoda to Gabe's answer, it's worth noting that $\beta_0$ is in fact the first ordinal $\alpha$ which "starts a gap," i.e. such that $L_{\alpha+1}\models$ "$\alpha$ is uncountable....
- 20.7k
4
votes
Heuristic interpretations of the PA-unprovability of Goodstein's Theorem
To my way of thinking, the arguments you mention seem not to distinguish sufficiently between the content of Goodstein's theorem as a universal claim $\forall p$ and Goodstein's theorem as a ...
- 230k
4
votes
Modification of Lemma 0 in Hajnal's paper "Embedding finite graphs into graphs colored with infinitely many colors"
This can be done by modifying the basic inductive construction of the Rado graph. The general idea is to enumerate all of the potential $\epsilon$ functions, and add $\kappa$ new vertices realizing ...
- 3,254
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