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Results tagged with lo.logic
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user 17064
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.
18
votes
1
answer
366
views
Proof as a Σ₁ approximation to truth: what about higher degrees?
Let us posit that the goal of mathematics is to study mathematical truths, and let us stick to arithmetic for simplicity: so let $\mathscr{T} \subseteq \mathbb{N}$ be the set of (Gödel codes of) true …
- 32.5k
12
votes
1
answer
296
views
Does this "mixable" property have a standard name in constructive mathematics?
While thinking about constructive mathematics, I stumbled on the following notion, and I would like to know if it has a standard name, a simpler equivalent, or has appeared in the literature:
Say tha …
- 32.5k
4
votes
1
answer
195
views
Does second-order Heyting arithmetic have the disjunction and existence properties?
Consider full second-order Heyting arithmetic, axiomatized in two-sorted first-order intuitionistic logic (with “number” and “class” variables) by the usual Peano axioms (with induction being stated q …
- 32.5k
9
votes
1
answer
333
views
Are finitely enumerated and subfinite sets Dedekind-finite?
The context of this question is constructive mathematics, such as in the internal logic of a topos with natural numbers object, or in IZF.
Let us say that a set $X$ is:
finite when there exists a nat …
- 32.5k
5
votes
1
answer
314
views
Does the Rieger-Nishimura lattice over a subset of $\mathbb{R}^k$ stabilize?
Notation: If $U,V$ are open subsets of a topological space $X$, let us write $U\Rrightarrow V$ for the Heyting operation: the largest open subset $W$ of $X$ such that $U\cap W \subseteq V$ (i.e., the …
- 32.5k
10
votes
1
answer
403
views
Examples of Kreisel-Putnam topological spaces
Let us say that a topological space $X$ is a Kreisel-Putnam space when it satisfies the following property:
For all open sets $V_1, V_2$ and regular open set $W$ of $X$, if a point $x\in X$ has a …
- 32.5k
10
votes
1
answer
366
views
Fibers of the morphism from the free Heyting algebra to the free Boolean algebra
For $k\in\mathbb{N}$, let $H_k$ be the free Heyting algebra on $k$ variables $p_1,\ldots,p_k$ and $B_k$ be the free Boolean algebra on the same $k$ variables. Thus, $B_k$ has $2^{2^k}$ elements (corr …
- 32.5k
4
votes
1
answer
353
views
Is there a correspondence between principles of omniscience and computability classes?
My question will be speculative and therefore a little vague.
I wonder if attempts have been made to define a correspondence between, on the one hand, limited principles of omniscience that can be add …
- 32.5k
11
votes
2
answers
707
views
Is there a modern account of Veblen functions of *several* variables?
Veblen $\phi$ functions extend the $\xi \mapsto \phi(\xi) := \omega^\xi$ and the $\xi \mapsto \phi(1,\xi) := \varepsilon_\xi$ functions on the ordinals by repeatedly taking fixed points (I won't repea …
- 32.5k
7
votes
2
answers
469
views
What is the strength of “if $c≥0$ then $[0,c] = c·[0,1]$” in constructive math (w.r.t., LPO,...
Context: This question is about constructive mathematics, such as in the internal logic of a topos with natural numbers object, or in IZF. (I wish to avoid the axiom of countable choice if possible, …
- 32.5k
3
votes
2
answers
501
views
Various notions of Turing reduction for partial functions
If $f$ and $g$ are partial functions $\mathbb{N} \to \mathbb{N}$, define six preorder relations $f \preceq g$ as follows:
$f \mathop{\preceq_{\mathrm{S}}} g$ ("$f$ is strict/Sasso reducible to $g$") …
6
votes
1
answer
178
views
Variable elimination for propositional formulas in Heyting algebras
By an (intuitionistic) propositional formula $\varphi(x_1,\ldots,x_n)$ I mean a formula built up from a (finite) number of variables $x_1,\ldots,x_n$ using connectors $\top, \bot, \land, \lor, \Righta …
- 32.5k
7
votes
1
answer
135
views
Invertibility and comparison to zero in the MacNeille sections (bounded extended reals)
(The following three paragraphs are given for context. Readers already aware of the terminology can skip to “the problem” below.)
In a spatial topos $\mathop{\textbf{Sh}}(X)$ the MacNeille sections ( …
- 32.5k
6
votes
1
answer
301
views
What is the power of the “anti-halting” oracle?
Let me first ask the question, and then, as it may seem a bit cryptic, explain how it comes up (and whence the “anti-halting oracle” in the title):
Notations: we write $\langle m,n\rangle$ for a stand …
- 32.5k
5
votes
1
answer
116
views
Understanding the definition of a (computably / continuously) “transparent” function
The following definitions of a “transparent function” are essentially taken from references [1] (where it is called a “jump operator”), [2] and [3], except that the variation “primitively recursively …
- 32.5k