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Results tagged with lo.logic
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user 2233
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.
4
votes
MSO2-expressible graph properties unexpressible in MSO1
Nice question. I am also keen to learn of more examples.
Here are some more details on the Hamiltonian cycle example. This is expressible in $\mathsf{MSO_2}$ since testing if a graph is connected …
- 32.1k
2
votes
Basic results with three or more hypotheses
Poincaré Conjecture. If you are (A) a 3-manifold, (B) closed, and (C) simply-connected, then you are (D) the 3-sphere.
7
votes
Uncountable family of infinite subsets with pairwise finite intersections
I think that this was answered by Andres Caicedo in a comment to an answer to this question.
I quote:
Given an infinite sequence of 1s and 2s, its initial segments are numbers (written in decim …
- 32.1k
37
votes
When are two proofs of the same theorem really different proofs
There is a very nice paper of Wagon, which can serve as a sort of case study. The paper presents fourteen different proofs of the following theorem.
Theorem.
If a rectangle $R$ is tiled by rectangle …
- 32.1k
13
votes
2
answers
1k
views
Ultrafilters vs Well-orderings
This question was actually asked by John Stillwell in a comment to an answer to this question. I thought I would advertise it as a separate question since no one has yet answered and I am also curiou …
- 32.1k
10
votes
Objects which can't be defined without making choices but which end up independent of the ch...
I think homology theory is a good example of this phenomenon. Namely, to construct the simplicial homology groups it is necessary to first choose a triangulation of the space. In the end, it turns o …
2
votes
Infinite games: are they well defined?
The Axiom of Choice implies that there exist infinite games in which neither player has a winning strategy. Thus, the Axiom of Choice is incompatible with the Axiom of Determinacy. An example of a g …
5
votes
Accepted
The number of $3$-CNF formulas in $n$-variables and the fraction of satisfiable ones
Regarding the fraction of satisfiable 3-CNF formulas in $n$ variables, it is widely believed that there is a phase transition that occurs depending on how many clauses there are compared to the number …
- 32.1k
96
votes
16
answers
34k
views
Most 'unintuitive' application of the Axiom of Choice?
It is well-known that the axiom of choice is equivalent to many other assumptions, such as the well-ordering principle, Tychonoff's theorem, and the fact that every vector space has a basis. Even tho …