All Questions
62 questions
1
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2
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228
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Can this semi-constructible structure satisfy existence of a measurable cardinal?
If we add a primitive unary function symbol $\mathfrak L$ to the first order language of set theory.
Axiom of semi-constructibility: if $\phi^\alpha (y,x_1,\ldots,x_n)$ is a formula in which all and ...
- 11.3k
7
votes
1
answer
556
views
Does this ZFC+V=L like theory, have a limit on large cardinal properties?
Let $\sf T$ be a theory that has as axioms every axiom of $\sf ZFC$, and every theorem of $\sf ZFC + [V=L]$ that is neither provable nor disprovable by $\sf ZFC$, whose addition or addition of its ...
- 11.3k
7
votes
2
answers
311
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At what ordinal $\chi$ does $\mathrm{L}_\chi$ contain a surjection from $\omega$ to $\mathrm{L}_{\beta_0}$?
Let $\mathrm{ZF^-}$ be $\mathrm{ZF}$ minus power set, and let $\beta_0$ be the ordinal for ramified analysis so that $\mathrm{L}_{\beta_0}$ is the least $\mathrm{L}$-model of $\mathrm{ZF}^-$. Clearly, ...
- 3,574
1
vote
1
answer
112
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Constructible cardinality downslides and their consistency strengths?
Posting "Large cardinals and constructible universe" mentions that $\omega_1^L < \omega_1$ if we assume Ramsey cardinal.
My question can we have more downslides like for example $\omega_2^...
- 11.3k
14
votes
1
answer
619
views
Ordinal realizability vs the constructible universe
Koepke's paper Turing Computations on Ordinals defines a notion of "ordinal computability" using Turing machines with a tape the length of Ord and that can run for Ord-many steps, and shows ...
- 66.8k
4
votes
1
answer
158
views
Impact of coining $L$ in $\mathcal L_{\omega_1, \omega}$ on which large cardinals it can satisfy?
What happens to the constructible universe $L$ if we build it in the first of infinitary languages $\mathcal L_{\omega_1, \omega}$?
Would the usual limitation of $L$ not satisfying existence of a ...
- 11.3k
13
votes
1
answer
2k
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Are some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.
Are some interesting mathematical questions, ...
- 3,574
1
vote
0
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174
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How does the cardinality of a set and its powerset compare in the hereditarily rank-concordant constructible world?
Working in the constructible universe "$L$", if we define two kinds of ranks for any constructible set $x$, one being the ordinal index of the first $L_\alpha$ where $x$ appears as a subset ...
- 11.3k
5
votes
2
answers
432
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Models of second-order arithmetic closed under relative constructibility
I know little to nothing about second-order arithmetic and its subsystems. However, I would like to understand when a model of (a subsystem of) second-order arithmetic ($\mathsf{Z}_2$) is downward ...
- 2,286
2
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1
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269
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Inner model for KP and a Well-Ordering of the Reals
It is well known that Gödel proved the following theorem:
$\mathsf{ZFC + V=L}$ has a $\mathit{\Delta}^1_2$-good well-ordering of $\mathbb{R}$. (Gödel, Addison)
So:
Is there an inner model for KP/Z/....
- 695
1
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0
answers
163
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Can the Constructible Universe be built in absence of Unions and Power?
Can $L$ be built in
$\sf ZF$ $\sf-Regularity-Union-Power+ Boolean \ Union$?
We know that $L$ can be built in $\sf KP$, but here we don't have Set Union.
If the answer is to the negative, then would ...
- 11.3k
1
vote
0
answers
142
views
Must models of the following theory satisfying opposing infinitary sentences, satisfy opposing finitary sentences?
This is a follow-up to posting titled "Is this theory finitary first order complete?"
Recall the theory presented at that posting. Replace the size axiom by the following:
$\textbf{...
- 11.3k
2
votes
1
answer
160
views
Is this theory finitary first order complete?
If we coin a theory in $\mathcal L_{\omega_1, \omega}$ that begins with constructing pure true well founded finite sets, then the set of all true well founded hereditarily finite sets, then builds up ...
- 11.3k
14
votes
1
answer
695
views
Can $L$ be defined without parameters?
If we omit parameters in the definition of $L$ would the result still be $L$?
That is, we define a successor stage $L_{\alpha+1}$ in the constructible universe $L$, without including parameters; as:
$...
- 11.3k
2
votes
1
answer
243
views
Is stable ordinals in non-well-founded model the same as well founded models?
Let $BST$ be the axiom system $KP$ - $\Delta_0$ collection. For an ordinal $\alpha$, we say that $\alpha$ is $\varphi$-$\Sigma_n$-stable, if there is a $\beta>\alpha$ satisfies the formula $φ$ such ...
- 1,490
3
votes
1
answer
541
views
Are all constructible from below sets parameter free definable?
Lets take the intersection of the theory of $L_{\omega_1^{CK}}$ and $\sf ZF + [V=L]$, this is equivalent to the theory of constructability from below + limit stages.
Can this theory prove the ...
- 11.3k
7
votes
1
answer
346
views
Which one of the following two ordinals is larger?
We say that $\alpha$ is $\Sigma_n$-extendable (to $\beta$), if there is $\beta>\alpha$ such that $L_\alpha$ is a $\Sigma_n$ elementary submodel of $L_\beta$.
First ordinal: the least $\alpha_0$ ...
- 1,490
2
votes
1
answer
235
views
End elementary extension in infinitary logic of some $L_\alpha$ producing a $L_\beta$
Let $L_\alpha$ be some admissible level of the constructible hierarchy and $M \supseteq L_\alpha$ an extension of $L_\alpha$. I am looking for conditions under which $M \simeq L_\beta$. It is not ...
- 531
5
votes
2
answers
386
views
Terminology for ordinals whose constructible level is the least one satisfying some formula
An ordinal $\alpha$ is "meta-definable" by some formula $\varphi$ without free variables if:
$$
\begin{cases}
L_\alpha \models\varphi \\
\forall\beta < \alpha \, L_\beta \not\models \...
- 531
1
vote
0
answers
266
views
Is Jensen's covering lemma meaningful in a platonist's view?
The typical applications of fine structure theory are finding out the lower bounds of consistency strength of axiom systems. In such a proccess, we also constructs many combinatorial objects in core ...
- 1,490
4
votes
1
answer
439
views
What's the order type of the following set?
Fix a positive integer n. Assume $Lan=\{R_0,R_1,...,R_n\}$ be a language of first order logic, where every $R_i$ is a 2-ary relation symbol.
Assume $M$ is an Lan-model, where the underlying set is $...
- 1,490
4
votes
0
answers
253
views
Is this recursion theoretic analogue of a criterion of weakly compact cardinal accurate?
Jensen proved that, if V=L, and $\kappa$ is a regular cardinal, then if for any stationary $A\subseteq \kappa$, the set $\{\alpha\mid A \text{ is stationary below }\alpha\}$ is stationary in $\kappa$, ...
- 1,490
1
vote
1
answer
191
views
At which large cardinal, the theory of the minimal transitive model of ZFC starts proving its absence?
Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be ...
- 11.3k
3
votes
1
answer
303
views
When does $\Pi_2$-reflection on $X$ fail to imply iterated $\Pi_1$-reflection on $X$?
Let lowercase Greek letters denote ordinals. Recall from Richter and Aczel's "Inductive definitions and reflecting properties of admissible ordinals", for a set of formulae $\Gamma$ and a ...
- 2,031
1
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0
answers
170
views
Can Jensen's covering lemma be proven easier in generic extensions of L?
Jensen's covering lemma, stating that if there is no $0^\#$ in V, then some covering property holds true, has a very complex proof.
In any generic extension L[G] of L, $0^\#$ don't exist, so the ...
- 1,490
7
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2
answers
518
views
Can countable ordinals start gaps of every order in the constructible universe?
Define "$\alpha$ starts a gap of order $n+1$ and length $\beta$" iff $\mathcal P^n(\omega)\cap (L_{\alpha+\beta}\setminus L_\alpha)=\emptyset\land\forall\gamma\in\alpha: L_\alpha\setminus L_\...
- 524
9
votes
2
answers
426
views
Can local $0^\#$ exists in L?
Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
- 1,490
7
votes
1
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395
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If $L_\alpha \vDash ZFC$, then do we have $L_{\alpha+1} \vDash \alpha\text{ is inaccessible}$?
Here we choose the definition of "is a cardinal" as there is no surjective map from a smaller ordinal to it.
It's easy to prove that, if $L_{\alpha+1} \vDash\ \alpha\text{ is inaccessible}$, ...
- 1,490
5
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1
answer
407
views
Is this relation about elementary embedding transitive?
For ordinals $\alpha<\beta$, we say $\alpha<_{el}\beta$, if there is an elementary embedding with domain $L_\beta$ and critical point $\alpha$.
Is $<_{el}$ transitive?
- 1,490
6
votes
0
answers
202
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Consistency strength of Sy Friedman's result about admissibility spectrum
A result by Sy Friedman in his book "fine structure and class forcing", is that, assume $0^\sharp$ exists, there exists a real number R such that the ordinals admissible in R (called $\...
- 1,490
5
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1
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407
views
Height of diamond
Assume $V=L$.
Let $\alpha$ be the least ordinal such that there is a $\Diamond_{\omega_1}$-sequence in $L_\alpha$.
It's obvious that $\omega_1 < \alpha < \omega_2$.
Do we have some better ...
- 1,490
1
vote
0
answers
70
views
Can all sets in stratified L above some stage be proximate?
Define stratified $L$, denoted by $^S L$, as:
Let $S$ be the set of all stratified formulas in first order language of set theory.
Define:
${ }^S Def (X) = \{\{y \in X \mid (X, \in) \models \phi(y,z_1,...
- 11.3k
-2
votes
1
answer
294
views
What does the Concordant constructible universe model?
Define a ranking function $\cal R$ as:
$\mathcal{R}: V \to ON; \,\mathcal {R}(x)= \min \alpha \, \forall y \in x: \alpha > \mathcal {R}(y) $
Now the constructible rank $\mathcal R^c$ of a set $X$ ...
- 11.3k
8
votes
1
answer
627
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Is $\mathsf{ZFC+V=L}$ consistently $\omega$-complete?
This was previously asked and bountied on MSE:
For brevity, let $T$ be $\mathsf{ZFC+V=L}$.
Say that an extension of $\mathsf{ZFC}$ is $\omega$-complete iff it has exactly one $\omega$-model up to ...
- 20.5k
3
votes
1
answer
325
views
If we have a class like $L$ but allowing a set number of unbounded quantifiers, is it strict superset of $L$?
The definition of $L$ only permits bounded quantifiers. If we allow a certain number of unbounded quantifiers, does this result in a strict superset of $L$? For example:
$$
\operatorname{Def}^{\...
- 6,353
5
votes
1
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294
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End-extension in Gödel's constructible universe
Given two ordinals $\alpha < \beta$, considering the subsets of Gödel's constructible universe, one say that $L_\beta$ is a $\Sigma_n$ end-extension of $L_\alpha$ (and $L_\alpha$ is an $\Sigma_n$ ...
- 531
6
votes
3
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837
views
Is the power set axiom essential for constructing L?
Take ZFC, remove axiom of Power set, and put instead of it the following axiom:
Axiom of Successor Cardinals: $\forall \kappa\, \exists x \, \forall \alpha \, ( \alpha \leq \kappa \to \alpha \in x)$
...
- 11.3k
4
votes
1
answer
592
views
Is ZFC interpretable in a kind of an extended form of second order arithmetic?
Informally the following theory is a kind of extension of second order arithmetic, where numbers are not limited to naturals, instead here we have formation of further numbers by setting limits on ...
- 11.3k
7
votes
1
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440
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Capturing the $\omega_1^{\mathrm{CK}}$-th stage of Gödel's constructible hierarchy
For an ordinal $\alpha$, let $L_\alpha$ be the $\alpha^{th}$ set of Gödel's constructible hierarchy and let $\omega_1^{\mathrm{CK}}$ be the first non-recursive ordinal or the first admissible ordinal ...
- 531
4
votes
0
answers
144
views
Can this reflective class theory interpret ZFC?
Reflective Set Theory $\mathsf{RfST}$ is formulated in first order predicate logic with extra-logical primitives of equality $``="$, membership $``\in"$, and a single primitive constant symbol $V$ ...
- 11.3k
0
votes
0
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150
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Is "ZF+ V=L" an upper limit theory for cardinal decidability (per its strength)?
{EDIT: this posting has been edited, the additional text is in italics}
If $\varphi, \psi$ are two parameter free formulas in the language of set theory $T$ such that there is a theorem of $T$ that ...
- 11.3k
1
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1
answer
203
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What is the strength of this strict constructible iterative hierarchy?
Begin with the empty set then construct the set of the empty set, then construct the set of all subsets of the latter set, then at each level of construction construct the next level as the set all ...
- 11.3k
8
votes
1
answer
864
views
Is the smallest $L_\alpha$ with undefinable ordinals always countable?
Let $\mathfrak{t}$ be the least ordinal such that $L_{\mathfrak{t}}$ has undefinable ordinals; i.e. there is an $\alpha<\mathfrak{t}$ such that $L_{\mathfrak{t}}$ cannot define $\alpha$.
This ...
- 1,252
10
votes
1
answer
452
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Stability for the Gödel and Jensen hierarchies
Notations: Let $L_\alpha$ stand for the Gödel constructible hierarchy ($L_0=\varnothing$ and $L_{\alpha+1} = \mathrm{def}(L_\alpha)$ is the set of definable subsets of $L_\alpha$ and $L_\delta = \...
- 32.5k
7
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1
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398
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How similar are large cardinals, over $L$?
EDIT: Joel's answer shows that no $\Sigma_2$ large cardinal property will do the job - however, $\Pi_2$ properties (such as unfoldability and its relatives) may still be useful.
Throughout this ...
- 20.5k
1
vote
1
answer
428
views
Further research on $\mathrm L_{\infty}$
In the mathoverflow question , "Godel's Constructible Universe in Infinitary Logics (A Possible Solution to $HOD$ Problem), Prof Hamkins answered user46667's question 2
What is $\mathrm L_{\infty}$?...
- 6,132
2
votes
0
answers
147
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When do wide initial segments ruin admissibility?
Suppose $\alpha$ is admissible and $\beta<\alpha$. We know that $L_\alpha$ is an admissible set (by definition), but adding subsets of $\beta$ to $L_\alpha$ might break admissibility: while set ...
- 20.5k
3
votes
1
answer
144
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Levels of L resembling each other, take 2
(Everything below is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\}...
- 20.5k
6
votes
1
answer
267
views
Fine structure question: when do levels of $L$ look "a lot" like each other?
(Everything is assuming $V=L$.)
Fix an uncountable regular cardinal $\kappa$, and let $$E_\kappa=\{\mu<\kappa: \mbox{there is an elementary substructure of $L_\kappa$ isomorphic to $L_\mu$}\},$$ ...
- 20.5k
8
votes
1
answer
623
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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 $\Pi^0_2$.
The ...
- 20.5k