All Questions
12 questions from the last 7 days
17
votes
1
answer
1k
views
Are integers conservatively embedded in the field of complex numbers?
I am looking for a reference to the fact that $\mathbb{Z}$ is conservatively embedded into the field $\mathbb{C}$ of complex numbers, that is anything in $\mathbb{Z}$ which is definable in $(\mathbb{C}...
- 301
5
votes
1
answer
153
views
Does there exist a section of $\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ that is "nearly Boolean"?
The following might be a somewhat esoteric question:
Does there exist an infinite cardinal $\kappa$ and a section $f$ of the quotient map $\pi:\mathcal{P}(\kappa)\to\mathcal{P}(\kappa)/(\text{fin})$ (...
- 2,830
8
votes
0
answers
207
views
Can $\mathbb{C}$ have a "doppelganger" in $L(\mathbb{R})$ with countable automorphism group?
Working in $\mathsf{ZFC}$ + large cardinals (a proper class of Woodins, to be precise), is there a field $F\in L(\mathbb{R})$ such that $V\models F\cong\mathbb{C}$ and $L(\mathbb{R})\models\vert\...
- 20.5k
3
votes
1
answer
181
views
Can one say that there are equal numbers of sets satisfying formulas in Second Order Arithmetic?
Is there a way of saying in second order arithmetic that the number of sets $X$ such that $\phi$ equals the number of sets $X$ such that $\psi$, where $\phi$ and $\psi$ are formulas with $X$ free, and ...
- 2,555
0
votes
1
answer
70
views
Is there a characterization of monoids that distribute over each other?
Let $(M, e_1, \times_1, e_2, \times_2)$ be an algebraic structure such that
$(M, e_1, \times_1)$ and $(M, e_2, \times_2)$ are monoids
$x \times_1 (y \times_2 z) = (x \times_1 y) \times_2 (x \times_1 ...
- 631
4
votes
0
answers
160
views
Are the natural powers of two conservatively embedded in $\mathbb{C}$?
This is a followup to this question.
Consider $\mathbb{C}$ as a structure - in the sense of first-order logic - with the graphs of addition and multiplication. Let $\mathcal{X}$ be the substructure ...
- 20.5k
7
votes
0
answers
133
views
On the optimal strength of Goodstein's theorem
Goodstein's theorem is a famous example of an arithmetical statement that is unprovable in $\mathsf{PA}$ but provable in a stronger theory. It is well-known that Goodstein's theorem implies the ...
- 3,042
0
votes
0
answers
122
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
1
vote
0
answers
62
views
Why do we require that all successors model this formula?
I'm reading Fitting's Intuitionistic Logic, Model Theory and Forcing. This occurs in Chapter 7.15.
The aim is to prove that a certain intuitionistic model is an intuitionistic model of ZF. I ...
- 149
1
vote
0
answers
67
views
Construction of the smallest nucleus above a prenucleus: what does this proof tell us?
While reading Hyland's paper on the effective topos [retyped version here] in the L. E. J. Brouwer Centenary Symposium, specifically prop. 16.3, I realized that the following proposition is implicit:
...
- 32.5k
2
votes
0
answers
66
views
Consistency of Sigma-V-2 uniformization with AD
Is ZF + AD consistent with: For every real $r$, every true $Σ^V_2(r)$ statement has a $Δ^V_2(r)$ example?
DC is provable in ZF + every true $Σ^V_2$ statement has a $Δ^V_2$ example (i.e. witness). ...
- 7,545
3
votes
0
answers
55
views
Does there exist a multi-valued "monotone" and "compact" map from a Boolean algebra to the "free" part of $\mathcal{P}(\kappa)$?
This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested:
Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...
- 2,830