Questions tagged [fine-structure]
Individual levels of inner models, projecta, mastercodes, definability, Jensen's covering lemma.
9 questions
15
votes
1
answer
765
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Is 0# still unique in ZFC without powerset?
Working in $ZFC$, the statement "$0^\sharp$ exists" is often liberally taken to be one of many known equivalent statements.
However, working in $Z_2$ or $ZFC^-$ (with collection, well-...
- 397
2
votes
1
answer
211
views
Ordering patterns of projecta by least witness
Let $J$ denote Jensen's modification of the constructible hierarchy. For an ordinal $\alpha$ and an $n\in\mathbb N^+$, let $\rho_n^{J_\alpha}$ denote the $\Sigma_n$-projectum of $J_\alpha$, the least $...
- 2,884
4
votes
1
answer
1k
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A doubt about the Gödel condensation lemma
To simplify the notation, assume $V=L$. We have $\lvert V_{\omega_{1}} \rvert=\aleph_{\omega_{1}}$ and $\lvert H(\aleph_{1})\rvert=\aleph_{1}$, so in particular $V_{\omega_{1}} \models \exists x \...
- 391
2
votes
0
answers
297
views
When is a $\Sigma_n$ Skolem hull a proper submodel?
For $M$ an amenable structure and $X \subset M$, the $\Sigma_n$ Skolem hull of $X$ is a $\Sigma_n$-elementary submodel of $M$. That is, as presentend in Devlin, Constructibility, pp. 85-88, for $h_n$ ...
- 531
9
votes
2
answers
688
views
On the utility of Silver machines
This question arises out of having Devlin's Constructibility [1] in my collection of books at home during the lockdown. Chapter IX of the book deals with Silver machines, which are presented as ...
- 7,487
6
votes
1
answer
411
views
What is a 'power admissible model'?
Q: What exactly is a power admissible model?
Background: Admissible models, introduced by Jon Barwise, form the building blocks of inner model theory. They are transitive models $\mathcal M = (M; \in)...
- 1,090
4
votes
1
answer
290
views
Mitchell, Steel. FSIT. Lemma 2.8: Is $k$-solidity actually needed?
Consider the following result (which is Lemma 2.8 in Mitchell and Steel's paper on Fine Structure and Iteration Trees):
Lemma 2.8 Let $\pi \colon \mathcal{H} \to \mathcal{M}$ be generalized $r \...
- 1,090
7
votes
0
answers
364
views
Does $\mathsf{fReR}_0$ prove the existence of the cartesian product of two sets
$\mathsf{fReR}_0$ is the set-theoretical system whose axioms consist of:
(1) Axiom of extensionality: $\forall z\in x\ (z\in y)\wedge\forall z\in y\ (z\in x)\rightarrow x=y$
(2) Axiom of empty set: $...
- 173
4
votes
2
answers
426
views
A question on Gandy-Jensen system and the rudimentary functions
Let $\mathrm{R}_0,\cdots,\mathrm{R}_8$ be the following functions:
$\mathrm{R}_0(x,y)=\{x,y\}$
$\mathrm{R}_1(x,y)=x-y$
$\mathrm{R}_2(x)=\bigcup x$
$\mathrm{R}_3(x,y)=x\times y$
$\mathrm{R}_4(x)=\...
- 173