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Results tagged with lo.logic
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user 121
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
What is the metamathematical interpretation of knot diagrams?
If you want to consider knot diagrams as finitistic algebraic objects, it is not hard to show that they can be encoded as sets. For example, you may choose to label crossings and line segments betwee …
13
votes
Direct construction of the integers
You could try base -2 representations, also called negabinary strings. These are finite strings drawn from the alphabet $\{ 0, 1\}$, starting with 1 (except when zero or empty, depending on your choi …
- 45.7k
16
votes
Set theories without "junk" theorems?
Many of these answers are quite satisfying, but I'd just like to emphasize that much of the confusion may come from overloading of symbols like "$\in$", "$\subset$", "$\cap$", and "$2$", that is, such …
- 45.7k
20
votes
Is there a theorem that says that there is always more than one way to "continue a finite se...
As others have mentioned, there can be many ways to continue a sequence, but it can be difficult to argue that a particular way is best. Kolmogorov complexity gives us a quantitative method to say th …
- 45.7k
4
votes
Basic results with three or more hypotheses
Any field that is algebraically closed, characteristic zero, and of continuum cardinality is ring-theoretically isomorphic to the complex numbers.
6
votes
Are $\mathbb{C}$ and $\overline{\mathbb{Q}}_p$ isomorphic?
I'm afraid this was a bit too long for a comment.
It seems that people don't object too much when we assert that $\mathbb{C}$ and $\overline{\mathbb{Q}_p}$ are isomorphic as sets or as abelian groups …
27
votes
What are the most attractive Turing undecidable problems in mathematics?
The problem of distinguishing two manifolds (up to homeomorphism, or even homotopy equivalence Edit: given a triangulation) is undecidable. This follows from the word problem applied to fundamental g …
10
votes
How many groups of size at most n are there? What is the asymptotic growth rate? And what of...
The asymptotics for groups are strictly speaking still open, since extensions of nonsolvable groups are apparently rather thorny. Edit:It seems my information is somewhat out of date (see Milne's ans …
- 45.7k
26
votes
4
answers
1k
views
Are there lightweight foundations for arbitrarily extendable objects?
My experience with foundations is rather scant, but I've run into some types of objects that seem to resist the sort of set-theoretic encoding schemes via Kurowski tuples that are rather common for ob …
- 45.7k