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Results tagged with lo.logic
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user 113405
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.
7
votes
Is the union of two conservative extensions of a theory conservative?
As Emil pointed out, assuming $\Sigma=\Sigma_1\cap\Sigma_2$, the union of 2 such theories is always conservative extension, Emil had proven it using the standard formulation Robinson’s joint consisten …
- 1,676
0
votes
Can having no more than countably many classes, be inferred from, having every class being c...
As stated in the comments, I misread the definition of the theory and assumed it is completely in infinitary logic, while in reality the ZFC+classes fragment is still in finitary logic, so the answer …
- 1,676
2
votes
Accepted
Which fragment of ZF does the class of all hereditarily predicatively definable sets capture?
HPD satisfy extensionality and regularity trivially.
It satisfy Pairing, as if $x,y\in HPD$ we can explicitly write down the definition of $(x,y)$ using only $x,y$ as parameters (and they have strictl …
- 1,676
5
votes
Accepted
Does n-well ordered choice schema imply the axiom of choice?
$2$-well ordered choice is enough to imply AC.
Let $α$ be any ordinal, and look at $\mathcal P^2(α)\setminus\{\emptyset\}$.
We have $x\in^2 \mathcal P^2(α)\setminus\{\emptyset\}⇒∃z⊆\mathcal P(α)\;(x\i …
- 1,676
6
votes
0
answers
164
views
Weak trichotomy principle in the absence of choice
It is well known that the trichotomy property of cardinals ($∀κ,λ\in\operatorname{Card}(κ<λ∨κ=λ∨κ>λ)$) is equivalent to the axiom of choice.
D. Feldman and M. Orhon had defined in [1] a generalization …
- 1,676
10
votes
2
answers
431
views
The additive structure of clusters of nonstandard models of arithmetic
Given $\frak M$ a countable nonstandard model of $\sf PA$ and let $a\in M$ be a nonstandard element. A "cluster around $a$" is the set of successors and predecessors of $a$, a cluster is a subset of $ …
- 1,676
12
votes
1
answer
533
views
Building the real from Dedekind finite sets
It is well known that the real numbers can be countable union of countable sets by starting with GCH and taking a finite support permutations while collapsing all of $\aleph_n$ for natural $n$.
The id …
- 1,676
11
votes
1
answer
768
views
Scott's trick without regularity
In ZF(C), one can easily get a class partition of $V$, we can even get an $\mathrm{Ord}$-partition using the Cumulative hierarchy: $P=\{V_{α+1}\setminus V_α\mid α∈\mathrm{Ord}\}$, such a partition let …
- 1,676
11
votes
Simpler proofs using the axiom of choice
There is an interesting class of problems that are provable in $ZF$ by "coincidence" in the sense that either (a fragment of) $AC$ holds and then it is provable from the proof in $ZFC$, or there speci …
14
votes
1
answer
489
views
Injection into a proper class and choice without regularity
In $\sf ZF$, we have that the axiom of choice is equivalent to:
For all sets $X$, and for all proper classes $Y$, $X$ inject into $Y$
and
For all sets $X$, and for all proper classes $Y$, $Y …
- 1,676
6
votes
Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
Any model of $ZF$+stationary proper class of $I3$ ordinals (whose consistency strength is at most $ZF$+$I2$) where $W_α$ is listing of $V_{κ}$ where $κ$ is $I3$ and $j_α$ the witness of $I3(V_κ)$ will …
- 1,676
3
votes
Accepted
Can we iteratively reflect on self elementary embeddable stages of the cumulative hierarchy?
$\newcommand\Ord{\mathit{Ord}}$The edit to the question has changed it enough so I think it deserves its own answer.
I assume that in reflection we have $\forall \vec{x} \in W_\alpha \, (\phi \to \ph …
- 1,676
4
votes
Accepted
Does cardinal definable choice imply AC?
Over ZF yes, it does.
Let $X$ be any family of nonempty sets, and let $κ$ be cardinal such that $ρ(κ)>ρ(X)$, then $V_{ρ(κ)}\setminus\{∅\}$ is cardinal definable, hence has a choice function that induc …
- 1,676
2
votes
How do these two principles of Foundation written in $\mathcal L_{\omega_1,\omega}$ compare?
Your axiom schema is equivalent to being an $\omega$-model.
Working inside an $\omega$-model, any counter example to your schema will be counter example of the axiom of foundation of ZFC, as you can u …
- 1,676
6
votes
Accepted
Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?
Any model of ${\sf ZFC}+V=\sf HOD$ has an elementary equivalent pointwise definable model.
If $M$ models $V=\sf HOD$, it has a parameter free definable well ordering, for each formula $φ$ consider the …
- 1,676