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Results tagged with lo.logic
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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.
3
votes
Accepted
Are Cohen Generics Minimal Covers?
Indeed, this has been answered very negatively in the literature:
Abraham, Uri; Shore, Richard A., The degrees of constructibility of Cohen reals, Proc. Lond. Math. Soc., III. Ser. 53, 193-208 (1986). …
- 44.4k
3
votes
Am I doing a forcing argument here?
It looks like the "logic aspects" of the argument boil down to using compactness. [However, this appears to be moot since the argument may have flaws pointed out by Will Sawin.]
First, given a bound …
0
votes
When can a function defined on $[a, b] \cup [b, c]$ be constructively extended to a function...
This is mostly a long comment, not really an answer.
If one slightly modifies the closed interval notation as follows:
$$[a,b\Vert = \{ x : a \leq x \not\gt b \}$$
$$\Vert b,c] = \{ x : b \not\gt x \l …
- 44.4k
6
votes
Is the Ordering Principle equivalent to a selection principle?
Here is a variation on the same theme as Joel David Hamkins' answer.
Theorem. The following are equivalent over ZF set theory:
Every set admits a linear order.
For every set $X$, there is a function …
7
votes
Accepted
Formulas that are valid simultaneously in a power set Boolean algebra $B$ and the 2-element ...
The class of formulas you're looking for contains all quasi-identities:
$$a_1 = b_1 \land \cdots \land a_k = b_k \to a = b$$
where $k \geq 0$ and $a,a_1,\ldots,a_k,b,b_1,\ldots,b_k$ are terms formed u …
- 44.4k
6
votes
Accepted
Comparing bornologies for domination/escaping
Note that $\mathfrak{b}=\mathfrak{d}$ is equivalent to the existence of a $<^\ast$-increasing sequence $(f_\alpha)_{\alpha<\mathfrak{d}}$ which is cofinal in $(\mathcal{N},{<^\ast})$, where $f <^\ast …
- 44.4k
8
votes
Accepted
Are we sure the calculus of inductive constructions and ZFC plus countably many inaccessible...
The situation is a bit subtle. One can interpret CIC in any model of ZFC with infinitely many inacessibles. However, interpreting ZFC in CIC is more subtle. First one needs to assume the law of exclud …
- 44.4k
8
votes
How much induction does a p-adic valuation need?
This is not intended as an answer but rather as a long-winded explanation of what having a 2-adic valuation means for really weak theories such as Q and open induction, which is much too long for a co …
- 44.4k
6
votes
Accepted
The strength of "There are no $\Pi^1_1$-pseudofinite sets"
My "hunch" in the comments to the question appears to be correct! This model comes from Howard, Paul E.; Yorke, Mary F., Definitions of finite, Fundam. Math. 133, No. 3, 169-177 (1989). ZBL0704.03033. …
- 44.4k
0
votes
In choiceless constructivism: If $f'=0$ then is $f$ constant?
This answer is incorrect since the function $f\colon\mathbb R \to\mathbb R$ is not computable. That said, it is possible that a similar idea could provide a counterexample.
There is a computable cl …
- 44.4k
7
votes
Accepted
Every complex number has a square root via LLPO without weak countable choice
Yes it is but there is no extensional square root function unless we also have LPO.
Note that the squaring function is a bijection from $Q_{+} = \{x + iy \mid x \geq 0, y \geq 0\}$ onto $H_{+} = \{x …
- 44.4k
14
votes
Accepted
Uncountable disjoint closed coverings of $[0,1]$
This is question has a long and interesting history, which is discussed in Arnie Miller's paper cited below. The first construction of a model of ZFC + $\aleph_1 < 2^{\aleph_0}$ where $[0,1]$ can be p …
- 44.4k
2
votes
Set of perfect subsets of a Borel set
This appears not to be the case: there is a $F_\sigma$ set $B$ such that $S_B$ is not Borel. This is optimal since the bullets in the question explain how $S_B$ is Borel when $B$ is $G_\delta$.
There …
- 44.4k
28
votes
Cauchy reals and Dedekind reals satisfy "the same mathematical theorems"
Here is a way to do this in ZFC. Similar ideas work in a bunch of other contexts.
First, given any set $A$ in the universe of sets $V$ we can form the set theoretic universe $V(A)$ by mimicking the c …
- 44.4k
7
votes
Accepted
Axiomatizations of arithmetical parts of theories
A somewhat general method has been explained by Azriel Lévy
[Axiomatization of induced theories, Proc. Am. Math. Soc. 12, 251-253 (1961); ZBL0178.31603; MR0122702] The method does not give axioms that …
- 44.4k