Skip to main content
Oscar Cunningham's user avatar
Oscar Cunningham's user avatar
Oscar Cunningham's user avatar
Oscar Cunningham
  • Member for 14 years, 9 months
  • Last seen this week
comment
Is there a universal way to force the Axiom of Choice to be true?
Haha, okay. So I do still need some more conditions if I want the construction to be natural in any sense.
comment
Is there a universal way to force the Axiom of Choice to be true?
But it would be nice if this was true whenever out original model obeyed choice. Do you know of an interpretation of ZFC in ZF that does this? i.e. Is there an interpretation of ZFC in ZF such that whenever you apply it to a model of ZF that happens to obey Choice the induced model is definably isomorphic to the original? That question is probably what I would have asked if I was trained to think in terms of logic rather than category theory.
comment
Is there a universal way to force the Axiom of Choice to be true?
Ignore my above comment, I hadn't drunk enough coffee yet (I was thinking that bi-interpretability was a relationship between models, whereas in fact it's a relationship between theories). Am I right in thinking that ZF has an obvious interpretation in ZFC (just take the whole model)? So what we're looking for is an interpretation of ZFC in ZF with some nice properties. In this case (where one interpretation is trivial) the definition of bi-interpretation says that every model of ZF should be isomorphic to the model of ZFC it induces. Clearly this is stronger than what we want.
comment
Is there a universal way to force the Axiom of Choice to be true?
Luckily I don't think we need a bi-interpretation. We're not looking for an adjunction between models of the set theory; we're looking for an adjunction between the categories of the models. This adjunction will induce a map from each model of ZF to the corresponding model of ZFC (or in the other direction depending on which way around the adjunction is). So we only need a morphism going one way rather than both. The category of models and interpretations looks like an interesting place to look for the adjunction in.
revised
Loading…
comment
Is there a universal way to force the Axiom of Choice to be true?
@AsafKaragila That's a good observation. If an adjunction existed there would be a unit (or counit) morphism between each model of ZF and the corresponding model of ZFC. Since this can't exist you've answered four parts of my twelve part question! (left or right adjoint $\times$ two kinds of object $\times$ three kinds of morphism)
comment
Is there a universal way to force the Axiom of Choice to be true?
@AsafKaragila I'm afraid I don't know exactly what you mean by "inclusion". From what I've found it seems that logicians sometimes consider various different notions of maps between set theories that are stronger than mere $\in$-homomorphisms but weaker than elementary embeddings. An answer in terms of any of these would be interesting, but from a category theoretic point of view you would want their properties to be strong enough to induce a functor between the corresponding toposes. So the image of a function between sets should be a function between the images of those sets.
comment
Is there a universal way to force the Axiom of Choice to be true?
@AsafKaragila Any model of ZF corresponds to a category whose objects are the sets and whose morphisms are the functions between sets defined in that model. This category is a topos, so a logical functor between models is just a logical functor between the corresponding toposes. I don't know if this has a good characterisations when thinking in terms of the membership relations.
awarded
revised
Loading…
revised
Loading…
Loading…
Loading…
comment
What information is lost in $X \to \mathrm{Sh}(X)$?
The nlab entry here suggests that a point of a locale is a frame homomorphism $O(L)\rightarrow O(1)$, where $1$ is the one point space $\{0\}$ with open sets $\{\{\},\{0\}\}$, rather than the Sierpinski space. Can you clarify?
comment
Perimeter-halving center of a convex shape
If you can calculate its position numerically then you can look it up in the Encyclopedia of Triangle Centers and see if it matches any of them.
awarded
comment
Can a row of five equilateral triangles tile a big equilateral triangle?
Did you hear anything back from Michael Reid?
comment
Anti-large cardinal principles
$PA$ is equivalent to $ZF$ with the axiom of infinity replaced by its negation. That fact sort of fits in here.
1
7 8
9
10 11
13