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Results tagged with set-theory
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user 23648
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
10
votes
1
answer
459
views
What is the least $\alpha$ such that $L_\alpha$ contains a non-measurable set
What is the least level of the constructable hierarchy that contains a non-measurable (Lebesgue) subset of $2^\omega$. If it makes a difference assume we are working inside L (V=L).
I'm pretty sure it …
- 50.6k
2
votes
1
answer
139
views
Are Cohen Generics Minimal Covers?
Are Cohen generics (in $2^\omega$) minimal covers?
I'm ultimately interested in this question for some more effective notion of forcing but I realized I wasn't sure how to show this even assuming full …
- 44.4k
6
votes
2
answers
262
views
Extending polynomial hierarchy above $\omega$
The arithmetic hierarchy is naturally extended to all ordinals via ordinal notations creating a hierarchy for all hyperarithmetic sets. The polynomial time hierarchy is defined analogously to the ari …
- 328
2
votes
Hyperarithmetically least elements in $\Pi^1_1$ sets
I'm pretty sure the claim isn't even true for every $\Pi^0_1$ class (working in $\omega^\omega$ or $\Pi^0_2$ if working in $2^\omega$). It's well known that one can produce a recursive tree in $\omeg …
- 3,029
2
votes
1
answer
132
views
Harrington's notes on McLaughlin/Arithmetically incomparable singletons
At one point I had copies of the handwritten notes Leo created about the McLaughlin conjecture and I know a similar set of notes exist titled Arithmetically incomparable arithmetic singletons. I've s …
- 3,029
1
vote
Harrington's notes on McLaughlin/Arithmetically incomparable singletons
Thanks to the individual who helped me out (not sure if they want to be publicly identified). Since I'm sure Leo wouldn't mind I'm posting a link to the McLaughlin notes here (I'll add the other one …
- 3,029
5
votes
Theorems in set theory that use computability theory tools, and vice versa
My favorite are the results in computability theory that rely on Martin's cone theorem (if A is sufficiently definable degree invariant set (Certainly if Borel but I think more) then either A or it's …
- 3,029
1
vote
0
answers
73
views
Standard terminology for node in tree with multiple children
Is there a standard terminology for a node in a tree that has multiple children?
For instance, in describing in perfect tree in $\omega^{< \omega}$ how would you describe the nodes that are extended b …
- 3,029
1
vote
Can this naïve-like set theory using acyclic membership be consistent?
I misread comprehension the first time. Now that I understand it correctly can't you construct the standard Russell set:
$\phi(y) = \lnot y \in^{*} y$
So let's ask if the resulting $x$ satisfies $x \ …
- 3,029
1
vote
Accepted
Is set of the indices of c.e.sets that cover a productive set also productive one?
No, it isn't. This is a consequence of the fact that we have a choice of infinitely many equivalent indices for each c.e. set. Let $E$ be a set of indices such that
$$(e, j \in E \land e \neq j) \i …
- 3,029
3
votes
On Applications of Forcing in Domain Theory
Note that forcing techniques are frequently used in computability theory, e.g., to establish the existence of degrees with certain properties. I refer the reader to Odifreddi's Handbook of Computabil …
- 3,029
3
votes
On independence and large cardinal strength of physical statements
In my opinion one of the best examples of a physical statement depending on a mathematically independent statement is provided by Malament-Hogarth machines (You can find a bunch of other links just by …
- 3,029
1
vote
1
answer
239
views
In NBG (ZFC + classes) if $P$ nonempty predicate of classes must $P$ have definable solution?
Let $A, B, C..X, Y$ range over potentially proper classes in NBG set theory (or adding class quantification to ZFC in obvious way) and $x,y,z$ range over sets.
Since I don't know the proper symbols …
- 3,029
4
votes
1
answer
568
views
Definition of HYP in $L_{\omega_1^{CK}}[a]$?
The structure $L_{\omega_1^{CK}}$ consists of only HYP sets (I believe) and HYP in this structure is the same as the actual hyperaritmetic sets. Now if I move to the structure $L_{\omega_1^{CK}}[a]$ …
- 1
6
votes
0
answers
248
views
$\omega$-models of $\mathbf{\Sigma^1_1}-DC$ and $\mathbf{\Delta^1_1}-CA$
So what is needed to demonstrate something (say like $L_{\omega_1^{CK}}[a]$ is a $\omega$-models of $\mathbf{\Sigma^1_1}$-$DC$ or $\mathbf{\Delta^1_1}$-$CA$? It's not like I don't understand what the …
- 32.5k