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2 votes
1 answer
161 views

Are Cohen Generics Minimal Covers?

Are Cohen generics (in $2^\omega$) minimal covers? I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full ...
8 votes
2 answers
518 views

History of forcing over admissible sets

In his paper "Forcing in admissible sets", Ershov writes In unpublished lectures given at Novosibirsk State University in 1976-1977 on the theory of admissible sets, the author showed that it is ...
1 vote
1 answer
362 views

How are Koepke's ordinal computability and E-recursion related?

In Koepke's paper, "Turing Computations On Ordinals", one has the following (well-known) result: A set $x$ is ordinal computable from a finite set of ordinal parameters if and only if it is ...
4 votes
0 answers
182 views

Some questions on a paper of Gerald Sacks

I've been reading Sacks' Countable admissible ordinals and hyperdegrees as I'm interested in Theorem 5.3 of the paper: Let $M$ be a countable standard model of $\mathsf{ZF}$ and $V=L$. Suppose $\...
6 votes
1 answer
227 views

How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?

Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{...
7 votes
0 answers
313 views

An uncountable structure with unusual "relatively-computable shadow"

Below, all structures are infinite and in a finite language. Given a structure $\mathcal{A}$ with domain $\omega$, we conflate $\mathcal{A}$ with some reasonable encoding of its atomic diagram for ...
7 votes
0 answers
468 views

"Relative plausibility" of some infinitary theories

We work in $\mathsf{ZFC+V=L}$. Define a plausible theory to be a theory $T\subseteq\mathcal{L}_{\omega_1,\omega}$ in an $\omega_1$-finite language which is $\omega_1$-c.e. and $\omega_1$-finitely ...
10 votes
1 answer
411 views

The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...
8 votes
2 answers
752 views

Paul Cohen on genesis of method of forcing and mathematical similarities

We have on record Paul Cohen's comments on being inspired by issues of formalizing algorithms in number theory (this needs to be verified as per comment) as well as related remarks on computability. ...
3 votes
0 answers
203 views

Class forcing over E-closed sets

Short version: does anyone know of any good sources on class-forcing over E-closed, non-admissible sets? Longer version: A problem I'm working on has reached an interesting conclusion - I've managed ...
13 votes
1 answer
742 views

Is forcing computable?

By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...
4 votes
0 answers
262 views

Recursively Pointed Sacks Forcing and Preserving $\omega_1$

Let $\mathbb{P}$ denote recursively pointed Sacks forcing. This is forcing with recursively pointed perfect trees ordered by inclusion. A tree $T \subseteq {}^{<\omega}2$ is recursively pointed if ...
8 votes
2 answers
473 views

Does forcing with recursively pointed perfect trees add a Turing degree that is minimal over $V$?

A tree $T$ on $\omega$ is recursively pointed if it is recursive in each of its branches. We can consider a variant of Sacks forcing where the conditions are recursively pointed perfect trees ordered ...
6 votes
1 answer
416 views

An eventually different function adding no Solovay real nor dominating function?

Definitions I believe set theorists have studied all of the following three notions in the context of forcing extensions of a model of ZFC, $M$ (hopefully the terminology is the standard one). A ...