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 ...
- 229k
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". ...
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16
votes
Accepted
Rigid non-archimedean real closed fields
Charles Steinhorn and I have answered this question positively by constructing a rigid non-archimedean real closed field of transcendence degree 2. Our preprint is now posted on arxiv.
https://arxiv....
- 3,460
16
votes
Is the field of constructible numbers known to be decidable?
See
Barry Mazur, Karl Rubin, Alexandra Shlapentokh, Defining $\mathbb{Z}$ using unit groups, Acta Arithmetica (Published online: 27 June 2024) DOI: 10.4064/aa230505-6-6
One of the corollaries of our ...
- 161
15
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 ...
- 229k
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 ...
- 229k
14
votes
Infinitary logics and the axiom of choice
What you're basically describing is the result of replacing, in the usual definition of $\mathsf{ZF}$, schemes ranging over first-order formulas by schemes ranging over formulas in a different logic $\...
- 20.8k
13
votes
Accepted
How are real numbers defined in elementary recursive arithmetic?
They aren't. Analysis requires a richer language. Note the particular restriction in Friedman's conjecture:
...whose statement involves only finitary mathematical objects (i.e., what logicians call ...
- 20.8k
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 $...
- 4,044
10
votes
Accepted
Which part(s) of this proof of Goodstein's Theorem are not expressible in Peano arithmetic?
Because your argument involves arithmetical classes at several points, as you noticed, it is not directly expressible in the first-order language of $\newcommand\PA{\text{PA}}\PA$, although as Noah ...
- 229k
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 ...
- 9,027
8
votes
Accepted
Am I doing a forcing argument here?
This is not quite an answer to the question you asked, but:
The argument, and any similar argument, can't work because the theorem to be proven is false. Let me quote the result:
Let $\mathcal C$ be ...
- 140k
7
votes
What is a "general" relation algebra?
There is one natural candidate that leaps to mind, namely the construal of $2^G$ as a relation algebra for each group $G$ $\ldots$
This construction will not answer your question, since each relation ...
- 12.8k
7
votes
Accepted
Is existence of this function on nonempty sets of Quine atoms consistent with ZF-Regularity?
Yes, this is consistent.
Here is one way to build a model. Start in a transitive model of ZFC with some $V_\kappa\prec V$. Now interpret a model $V[\![A]\!]$ with countably many urelements $A=\{a_n\...
- 229k
7
votes
Accepted
Negating fundamental axioms
In my limited experience (which may soon be changed! :P), merely negating "fundamental" axioms does not yield strong in-system consequences. The word "merely" is doing some work ...
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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 \{...
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5
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. ...
- 126
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 ...
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4
votes
What is a "general" relation algebra?
I don't know anything about the history and context of this idea of a relation algebra but the definition doesn't smell like "the right one" to me, and for a simple reason: why restrict ...
- 116k
4
votes
Accepted
Does Peano's axioms prove $\alpha$-induction for primitive recursive sequences for every concrete $\alpha < \varepsilon_0$?
I suppose you asked proving Goodstein's theorem for a given primitive recursive base sequence $\langle b_n\mid n\in\mathbb{N}\rangle$ (that is, the $(n+1)$th element of the Goodstein sequence is ...
- 2,910
4
votes
Is it consistent to have $\kappa$-Kurepa trees for some $\kappa$, but no Kurepa trees of other heights?
It seems to me that the following construction will work.
Start in $V$ with a Kurepa tree on $\omega_1$, and assume that there are a proper class of inaccessible cardinals above. By cutting off, if ...
- 229k
4
votes
Accepted
How can we define non-finitely axiomatizable extensions of set theories?
For any "reasonable" theory $T$, we can find a computable sequence of sentences $(\sigma_i^T)_{i\in\omega}$ such that
$T\cup\{\sigma_i^T: i\not=n\}\not\vdash\sigma_n^T$ for each $n$ (so the ...
- 20.8k
4
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 ...
- 45.5k
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 ...
- 229k
4
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 $...
- 229k
4
votes
Equivalence of omniscience principles for natural numbers and analytic omniscience principles for Cauchy real numbers
Let me try to tackle the LPO case. I will even try to show that it doesn't matter whether we assume our Cauchy sequences to have a modulus or not. (Please check me carefully because I know I've made ...
- 30.8k
3
votes
What goes wrong in Easton forcing if we don't just use regular cardinals?
What Andreas Blass fails to mention is that the forcing used to add subsets to a cardinal $\kappa$ will not only add subsets to $cf(\kappa)$ but even collapse $\kappa$ to $cf(\kappa)$ if $\kappa$ is ...
- 965
3
votes
Accepted
A question on the size of an admissible ordinal
Some references to literature: In "Reflection and Partition Properties of Admissible Ordinals" (Annals of Math. Logic vol. 22, iss. 3, 1982), Kranakis defines a $\Sigma_n$-admissible ordinal ...
- 1,531
3
votes
Accepted
Is discriminative choice provable in ZFC?
Yes. Enumerate $F$ as $\langle F_\alpha : \alpha<\kappa\rangle$, where $\kappa$ is a cardinal. Inductively pick $x_\alpha \in F_\alpha$ such that $x_\alpha$ is not $\phi$-equivalent to any $x_\beta$...
- 18k
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