2
votes
3answers
247 views

Upper bound on cardinality of a field

Is there an upper bound on the cardinality of a field? The "biggest" fields I know are the field of real numbers, or the field of complex numbers. Is there a field with cardinality greater than ...
2
votes
1answer
68 views

Number of models for some theory

Let $\mathcal L = \{ E(\_,\_) \}$ and $T$ be the $\mathcal L$-theory that says that $E$ is an equivalence relation with an infinite number of infinite classes. (I find this statement not clear, ...
5
votes
1answer
176 views

In ZF, does there exist an ordinal of provably uncountable cofinality?

Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
2
votes
2answers
136 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
53 views

Uncountable models for a language $L_Q$

$L$ is first-order language with identity and $L_Q$ a language obtained by adding to $L$ the quantifier $Q$. Definition of $Q$: If $P$ is a formula and $x$ a variable, $QxP$ is a formula of ...
2
votes
1answer
59 views

Possible typo in Just/Weese's set theory

In Just Weese on page 197 there are the following corollaries: Regarding Corollary 24: Is this a typo and should say "$CON(ZF) \not\rightarrow CON(ZF + \exists \text{ "a strongly ...
1
vote
1answer
134 views

Proving Axiom Schema of Replacement holds in $H_\lambda$

I think I proved the following, can you tell me if my proof is correct? Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
4
votes
1answer
118 views

Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$?

Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite ...