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Results tagged with lo.logic
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user 75
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
For a partition of $\mathbb{R}$ into countably infinite sets, must there be an almost-disjoi...
The existence of such a family of almost-disjoint selectors which is merely uncountable is equivalent to CH. Indeed, suppose we have such a family; we can shrink it to a family of size $\aleph_1$. T …
- 31.2k
7
votes
Is there a non self-referencing non-computable function?
I don't have a specific example, but here's a way to think about it. Virtually all functions are not computable--there are uncountably many functions, but only countably many are computable. Neverth …
- 31.2k
5
votes
1
answer
590
views
Löwenheim-Skolem for many-sorted theories
Let $L$ be a many-sorted first order language, and let $\kappa$ be an infinite cardinal which is greater than or equal to the number of function and relation symbols in $L$. Let $T$ be a complete the …
- 31.2k
17
votes
Accepted
What do models where the CH is false look like?
I think your reading is wrong. Set theorists have studied all sorts of additional axioms, some implying CH, some being strictly weaker than CH, and many contradicting CH. My understanding is that most …
- 31.2k
9
votes
Accepted
Does Cantor-Bernstein hold for classes?
Ignoring set-theoretic technicalities of formulating the question properly, I see no reason that the usual proof of Schroder-Bernstein wouldn't work.
(Set-theoretic technicalities: In the standard la …
- 31.2k
18
votes
Circular, or missing, definition in set theory?
This answer doesn't really have any ideas that are not already present in Noah Schweber's answer, but there are some points that I feel should be made more forcefully. In particular, I'd like to focu …
- 31.2k
13
votes
Accepted
Does "$|{\cal P}_2(X)| = |X|$ for $X$ infinite" imply ${\sf (AC)}$?
Yes, the usual proof that $|X|^2=|X|$ for all $X$ implies AC works for (S) as well. In detail, let $A$ be any infinite set, let $H$ be its Hartogs number (the least ordinal that does not inject into …
- 31.2k
12
votes
Accepted
"set of all irreducible representations of a group", set-theoretic issues
It is easy to show that the cardinality of the underlying set of any irreducible representation is bounded in terms of the cardinality of $\mathcal{W}_F$, so can you just take your favorite set $S$ of …
- 31.2k
4
votes
Accepted
For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?
As explained in the comments, if $k$ is a field, then $k$-algebra homomorphisms $k^X\to k$ are in bijection with $|k|^+$-complete ultrafilters on $X$ (that is, ultrafilters closed under $|k|$-fold int …
8
votes
Does k(X) have a k-basis for every set X, without AC?
As a generalization of David Speyer's argument, here is a proof that if $k(X)$ always has a basis, then the axiom of choice for finite sets of bounded cardinality holds. In fact, to get the axiom of …
- 31.2k
8
votes
Darboux property of non-atomic sigma-additive nonnegative measures equivalent to the AC?
This theorem follows from Dependent Choice, and thus is strictly weaker than the Axiom of Choice. Here is a proof using only DC. Fix $X\in\Sigma$ such that $\mu(X)>0$ and let $a\in(0,\mu(X))$. We w …
- 31.2k
75
votes
What are some reasonable-sounding statements that are independent of ZFC?
This isn't an answer but an argument that there isn't really a good answer. Having done a good amount of set theory and seen how you prove some of these statements to be independent, I tend to be rat …