All Questions
Tagged with axiom-of-choice model-theory
21
questions
3
votes
1
answer
226
views
Is Morse-Kelley set theory with Class Choice bi-interpretable with itself after removing Extensionality for classes?
Let $\sf MKCC$ stand for Morse-Kelley set theory with Class Choice. And let this theory be precisely $\sf MK$ with a binary primitive symbol $\prec$ added to its language, and the following axioms ...
- 10.2k
7
votes
0
answers
165
views
"Minimal-ish" Dedekind-finite cardinalities of models
Throughout, we work in $\mathsf{ZF}+$ "There is an infinite Dedekind-finite set."
Say that a Dedekind-finite cardinality $\kappa$ is $\Sigma^1_1$-isolated iff there is some first-order ...
- 20.7k
2
votes
0
answers
75
views
Is "strongly unbounded logics are unbounded" equivalent to "no descending sequence of cardinals"?
This question is motivated by Vaananen's paper Generalized quantifiers in models of set theory. Say that a (set-sized, regular) logic $\mathcal{L}$ is
unbounded if there are $\mathcal{L}$-sentences $\...
- 20.7k
8
votes
1
answer
438
views
Automorphisms of projective spaces, and the Axiom of Choice
It is known that upon not accepting the Axiom of Choice (AC), there exist models of ZF in which there are projective spaces (over a division ring) with a trivial automorphism group. (This is a truly ...
- 4,503
7
votes
1
answer
369
views
How hard is it to get "absolutely" no amorphous sets?
A beautiful and surprising (to me at least) result around the axiom of choice is that, while full $\mathsf{AC}$ is preserved by forcing, a model of $\mathsf{ZF}$ + "There are no amorphous sets&...
- 20.7k
7
votes
0
answers
307
views
Is there a "nice" inner model for $\mathsf{ZF}$ + a Dedekind-finite infinite set of reals?
Below, given a formula $\varphi$ which $\mathsf{ZF}$ proves defines a set of reals and an inner model $W$, I'll write "$\varphi^W$" and "$L(\varphi^W)$" for "$\{x:W\models\...
- 20.7k
4
votes
0
answers
283
views
Is this "finite-ish combinatorics" reflection principle consistent?
This question is an attempt to chisel away at this earlier question of mine, which in retrospect may be rather intractable. Throughout, we work in $\mathsf{ZF}$.
Briefly (see the linked question for ...
- 20.7k
6
votes
0
answers
350
views
How much "finitary combinatorics" can be emulated by an infinite Dedekind-finite set?
Throughout, we work in $\mathsf{ZF}$. Let $[n]=\{1,...,n\}$. Given a set $X$ and a first-order sentence $\varphi$, let $M_X(\varphi)$ be the set of isomorphism types of models of $\varphi$ with ...
- 20.7k
39
votes
3
answers
3k
views
Can one show that the real field is not interpretable in the complex field without the axiom of choice?
We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number ...
- 230k
2
votes
1
answer
222
views
Second-order strong minimality and amorphousness, take 2
Recently I asked a question about whether a second-order analogue of strong minimality could correspond to amorphous satisfiability (= having a model whose underlying set cannot be partitioned into ...
- 20.7k
7
votes
2
answers
673
views
Can second-order logic identify "amorphous satisfiability"?
Recall that a set is amorphous iff it is infinite but has no partition into two infinite subsets. I'm interested in the possible structure (in the sense of model theory) which an amorphous set can ...
- 20.7k
9
votes
1
answer
360
views
Do saturated models require choice?
Let $T$ be a first-order theory, and suppose we want to build a saturated model $\mathbb U$ of $T$. That is, we want a model $\mathbb U$ of cardinality bigger than $|T|$, saturated in its own ...
- 62.6k
7
votes
1
answer
282
views
Is the hereditary version of this weak finiteness notion nontrivial?
Say that a set $X$ is $\Pi^1_1$-pseudofinite if every first-order sentence $\varphi$ with a model with underlying set $X$ has a finite model. The existence of infinite $\Pi^1_1$-pseudofinite sets is ...
- 20.7k
13
votes
1
answer
432
views
Is this notion of finiteness closed under unions?
This was asked and bountied at MSE without success.
Throughout, we work in $\mathsf{ZF}$.
Say that a set $X$ is $\Pi^1_1$-pseudofinite if for every first-order sentence $\varphi$, if $\varphi$ has a ...
- 20.7k
10
votes
1
answer
424
views
What is first-order logic with Dedekind-finite sets of variables?
The usual set up of first-order logic is with an infinite reservoir of variables which we can use in formulas. This is one of the annoying reasons why we need to put $\aleph_0$ into the cardinal ...
- 38.5k