Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with lo.logic
Search options answers only
not deleted
user 83901
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
Accepted
Extending models of topological set theory
The answer to both of your main questions is no assuming the existence of a weakly compact cardinal larger than $M$.
In A general construction of hyperuniverses Forti and Honsell showed how to build a …
- 13.2k
16
votes
Do the surreal numbers enjoy the transfer principle in ZFC?
In $\mathsf{ZFC}$ if any two proper class models of the theory of an infinite set are isomorphic, then global choice holds. This is because $V$ and $\mathrm{Ord}$ are both models of this theory and an …
- 13.2k
10
votes
Who needs Replacement anyway?
I don't know if this counts as 'ordinary' mathematics (given its historical connection to set theory), but we use replacement a fair amount in model theory. The two most prominent places are in the us …
- 13.2k
16
votes
If every definable class admits a group structure, must global choice hold?
ZFC proves that $V$ admits a definable group structure (and actually ZF does too). First note that the Schröder–Bernstein theorem still holds in the context of definable classes (i.e., for any definab …
- 13.2k
2
votes
Condition to guarantee that an inhabited and bounded set of reals has a supremum
The following (somewhat strong) property seems to be sufficient:
For any open sets $U,V \subseteq \mathbb{R}$, if $\forall x \in \mathbb{R}.\; x \in U \vee x \in V$, then either $S \subseteq U$ or $S …
- 35.5k
8
votes
Accepted
Is the set of ordinals in Double Extension Set Theory really a set?
Perhaps I'm misunderstanding something, but it should be the case that $\beta \in_2 \mathbf{S}_1$ is equivalent to $S_1(\beta)$. Therefore the defining formula of $\mathbf{ORD}$ can be written as
$$\m …
- 13.2k
31
votes
Lists as a foundation of mathematics
Andreas Blass has already provided a good reference in the literature, but unfortunately I cannot read German, so I've had to make do with writing my own answer.
As you observed, you're clearly not go …
- 13.2k
7
votes
Weakest theory over which "all sets are measurable" has consistency strength?
This is more of an extended comment than an answer, but I think it's an important point to make. Also, as I discuss at the end it is technically an answer.
You are never going to have a constructively …
- 13.2k
6
votes
Is the set of permissible numbers of models of various cardinalities computable?
This is really an addendum to Alex's answer. I wrote a program in SageMath (using GAP) that computes these numbers, so I was able to expand Alex's lists considerably. Each of these lists should be com …
- 13.2k
12
votes
What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
This is perhaps more of an extended comment than a real answer, but I do think it goes a long way towards answering these kinds of questions.
The set-theoretic result referred to as Shoenfield absolut …
- 13.2k
8
votes
Quantifier complexity of definition of compactness
Often the way you prove that something isn't formalizable in first-order logic is (ironically enough) with a compactness proof. This is how you show, for instance, that there isn't a first-order theor …
- 13.2k
9
votes
Accepted
Can a non-standard model $M$ of $\mathsf{ZF}$ contain an internally infinite set that is ext...
The answer is yes. Let $M$ be a countable transitive model of $\mathsf{ZF}$ containing an amorphous set $A$ (i.e., every subset of $A$ is either finite or co-finite).
Let $T$ be the theory consisting …
- 13.2k
25
votes
Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?
In the topos we construct in our paper there is a surjection/epimorphism from the natural numbers to the Dedekind reals. In the model of CZF you mention (and in the effective topos) the Dedekind reals …
- 12.9k
10
votes
Accepted
"Surjective cardinals" - using surjections rather than injections to define isomorphism clas...
It is a studied concept. I'm not sure what it's called but it's often defined with the empty set as a special case to deal with the issue Gro-Tsen mentioned. I've seen it notated $A \leq^\ast B$ to di …
- 35.5k
14
votes
Can ultraproducts avoid all "factor structures"?
I realized there's an easier example that doesn't need a measurable cardinal.
Consider the language $\def\Lc{\mathcal{L}}\Lc$ which consists of unary predicates $U_n$ for each $n<\omega$. Consider the …
- 13.2k