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Results tagged with lo.logic
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user 113213
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
2
votes
0
answers
41
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). Al …
- 7,545
4
votes
0
answers
97
views
Partial uniformization under AD
Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of real …
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5
votes
1
answer
613
views
Non-atomic probability measures on N
One can intuitively imagine picking a random natural number and ask to what extent the intuition can be axiomatized.
Using the axiom of choice, there is a total finitely additive (monotonic) averaging …
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3
votes
0
answers
82
views
Existence of symmetric total measures
Is it consistent that there is a total finitely additive measure $μ$ on $ℝ$ extending the Lebesgue measure such that for every Borel Lebesgue-measure-preserving bijection $f$ of $ℝ$, $∀α∈Ord \, ∀s∈Ord …
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7
votes
0
answers
184
views
A version of determinacy for all sets
Under ZF + AD, some games are undetermined because of lack of choice. In fact, the axiom of choice is equivalent to determinacy of games of length 2. However, we can ask whether the lack of choice m …
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4
votes
0
answers
175
views
Fine structure without choice
In set theory, are there approaches to fine structure that give fine-structural models that do not satisfy the axiom of choice?
We can build fine-structural models above a given set (such as $\mathbb …
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6
votes
0
answers
153
views
$Δ^1_3$ reals in transitive models
Every real number is $Δ^1_2$ in some $ω$-model of ZFC\P. I am looking for analogous statements for $Δ^1_3$ definability and transitive models, where the situation is more subtle.
What is the consiste …
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18
votes
1
answer
532
views
When can we add choice to a model of ZF
For countable transitive models of ZF, is existence of a ZFC extension with the same height a first order property?
In other words, is there a statement $τ$ (in the language of set theory) such that f …
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4
votes
1
answer
223
views
AD and simultaneous well-orderability principle
Is the axiom of determinacy (AD) consistent with the following choice principle, and if yes, does it hold in $L(ℝ)$ under AD:
Simultaneous well-orderability: For every function $f:P(Ord)→\text{Wellord …
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7
votes
0
answers
184
views
Infinite cardinals and learnability of probability distributions
Two players play as follows. Player one chooses a secret finitely supported probability distribution $P$ on $ω_k$ (or another known set with $\aleph_k$ elements), and randomly takes $n+1$ samples usi …
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3
votes
0
answers
180
views
Periodicity in the cumulative hierarchy
Under Reinhardt cardinals in ZF, the cumulative hierarchy exhibits a periodicity in that for large enough $λ$, certain properties of $V_λ$ depend on whether $λ$ is even vs odd. See Periodicity in the …
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8
votes
0
answers
216
views
Large cardinals beyond choice and HOD(Ord^ω)
Are Reinhardt and Berkeley cardinals (and even a stationary class of club Berkeley cardinals) consistent with $V=\mathrm{HOD}(\mathrm{Ord}^ω)$ ?
It seems natural to expect no, but I do not see a proof …
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3
votes
0
answers
180
views
Weak extender models for supercompactness without choice
Assume ZFC and a supercompact cardinal $κ$. Is it consistent that there is a weak extender model $N⊨\text{ZF}$ for supercompactness of $κ$ such that the axiom of choice and well-ordering of $P_κ(λ)^N …
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4
votes
0
answers
161
views
Consistency of definability beyond P(Ord) in ZF
Is it consistent with ZF that the satisfaction relation of $L(P(Ord))$ is $Δ^V_2$ definable? More generally, is it consistent with ZF that there is a $Δ^V_2$ formula (taking $α$ as a parameter) that …
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2
votes
0
answers
112
views
Uniformization and functions on Turing degrees
Assuming Martin's Conjecture on functions between Turing degrees, is AD + DC consistent with existence of an $f:\mathcal{D}_t → \mathcal{D}_t$ of rank $Θ$ ?
$\mathcal{D}_t$ is the set of Turing degree …
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