All Questions
18 questions
23
votes
1
answer
3k
views
Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)
Gödel's constructible universe ($L$) is defined using definable power set operator in first order logic ($\mathcal{L}_{\omega ,\omega}$). One can produce such a universe in infinitary logics in ...
user46667
14
votes
3
answers
777
views
When is $A$ "$L$-ish" whenever $B$ is "$L$-ish"?
My question is about a variant of the usual notion of relative constructibility, $\le_c$ (which an earlier version of this question confusingly denoted "$\le_L$"), in set theory.
Fix a ...
- 20.9k
14
votes
2
answers
426
views
Consequences of existence of a certain function from $\omega_1$ to $\omega_1$
In his book [1], Paul Larson remarks (Remark 1.1.22) that in L there is a function $h:\omega_1\rightarrow\omega_1$ such that for any countable elementary submodel $X$ of $V_\gamma$ (where $\gamma$ is ...
- 7,125
13
votes
1
answer
971
views
V=HOD & The Height of the Large Cardinal Tree
As we know the assumption $V=L$ adds a restriction on the height of the large cardinal tree. Also there is a strict border like $0^{\sharp}$ exists such that all large cardinal axioms which are ...
user43940
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
8
votes
1
answer
623
views
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 ...
- 20.9k
7
votes
1
answer
395
views
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
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
1
answer
398
views
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 ...
- 20.9k
6
votes
0
answers
202
views
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 $\...
- 1,490
6
votes
0
answers
451
views
Inaccessible cardinals and the perfect set property for coanalytic sets
I am wondering who proved the following fact:
($\ast$) If $\omega_1$ is not inaccessible in $L$, then there is an uncountable coanalytic set of reals without a perfect subset.
I have been unable to ...
- 5,471
5
votes
1
answer
489
views
Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?
What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible $...
- 4,985
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
3
votes
1
answer
236
views
smallest ordinal $\alpha$ such that $L \cap P(L_\alpha)$ is uncountable
Let $V$ denote the von Neumann universe and $L$ Gödel's constructible universe. For any set $X$, let $P(X)$ denote the power set of $X$.
Assume that $0^\sharp$ exists (and ZFC).
What is the smallest ...
- 4,985
3
votes
0
answers
123
views
At which large cardinal property this second order ordinal arithmetic stops?
Language: Second order logic, with as usual predicates written in upper case, and objects in lower case. Let $<$ be a primitive constant binary relation symbol.
Equality between objects is ...
- 11.3k
2
votes
0
answers
147
views
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 ...
- 20.9k
1
vote
2
answers
228
views
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
1
vote
1
answer
112
views
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