1
vote
1answer
70 views

On the number of countable models of complete theories of models of ZFC

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
2
votes
0answers
97 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
3
votes
2answers
109 views

Use of forcing to real line to make elements countable

Can we use forcing techniques to force the set of elements of the real line to be countable? If not can anyone show why it is not possible?
2
votes
2answers
134 views

Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$

I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf). My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf). First, my vague ...
1
vote
1answer
111 views

Quickie on Boolean valued models

Bell writes on page 21 (you may use the search in the preview to search for "21" to view the page): "..., we show that, for any complete Boolean algebra $B$, all the theorems of $ZFC$ are true in ...
3
votes
2answers
115 views

Question about passage in Halbeisen's book

I am looking at the following passage in Halbeisen's book "Combinatorial Set Theory" (p 260 at the bottom): What is the role of $\Phi$? It seems to me that a finite fragment is the same as a ...
2
votes
1answer
187 views

Can we prove the completeness of FOL based on forcing?

In David Marker's Model Theory, there is a exercise said " we can view a countable Henkin construction as a forcing construction ". But in the book the Henkin construction is used to prove the ...