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Results tagged with descriptive-set-theory
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Descriptive Set Theory is the study of definable subsets of Polish spaces, where definable is taken to mean from the Borel or projective hierarchies. Other topics include infinite games and determinacy, definable equivalence relations and Borel reductions between them, Polish groups, and effective descriptive set theory.
5
votes
Accepted
Dependent choices (DC) in ${\bf HOD}(\mathbb{R},X)$, where $X$ is a set of reals
Consider the "singular Solovay-style model" of John Truss from
Truss, John, Models of set theory containing many perfect sets, Ann. Math. Logic 7, 197-219 (1974). ZBL0302.02024.
In that model the fo …
- 39.8k
4
votes
Is it inconsistent for a model of set theory to contain its own first order theory?
Not more than having large cardinals.
If $V_\kappa$ is a model of $\sf ZF$, it contains all the reals and therefore its own theory. It just doesn't know it. It's not a first order definable real the …
- 39.8k
5
votes
How much choice is needed to prove this statement?
Note that these are all countable sets, so what you'd have here is a sequence of countable sets of reals indexed by $\omega_1$.
This is certainly inconsistent with $\sf AD$, since from such a sequence …
- 39.8k
15
votes
Accepted
Can $\mathbb{R}$ be partitioned into dedekind-finite sets?
YES WE CAN!
Suppose that there is an infinite Dedekind-finite set of real numbers $A$ (e.g. Cohen's first model). Simple cardinal arithmetic shows that, $$|\Bbb R|\leq|\Bbb R\times A|\leq|\Bbb{R\time …
- 39.8k
5
votes
Accepted
$\mathtt{PSP}$ holding only for sets of cardinality $\mathfrak{c}$
Here are the answers you're looking for:
No.
No.
Yes, no large cardinals needed! (Which explains the previous two answers.)
Look no further than John Truss' paper:
Truss, John, Models of set theory …
- 39.8k
3
votes
"Potentially club" filters on $\omega_2$
Without choice, the Levy collapse, being $\omega_1^{<\omega}$ ordered by reverse inclusion, is well-orderable so every set of ordinals has a canonical name, and the usual choice-arguments continue to …
- 39.8k
11
votes
Cardinal arithmetic under determinacy
Starting from $L(\Bbb R)$, we can take a symmetric extension which preserves $\sf DC$ and adds an $\omega_1$-amorphous set, just somewhere far above $\Theta$.
So we cannot prove that (1) or (2) hold f …
- 39.8k
14
votes
Accepted
Pathological behavior of Borel sets?
Joel speaks on the case where the real numbers are a countable union of countable sets. The Feferman-Levy model is a strange model indeed.
However, I find the Truss construction to be even weirder. T …
- 39.8k