The n-Category Café

Thanks, fixed the typo.

I’ve always understood “it is consistent with P that Q” to mean just “P+Q is consistent”. In particular, note that this is equivalent to “P does not disprove Q” (whereas if P is inconsistent, then it does disprove Q — as well as proving it, of course). But maybe a logician will show up and set us straight.

Maybe it will clarify things if we take P to be a theory that’s known to be inconsistent, like set theory with Unrestricted Comprehension.

Would you want to say “it is consistent with ZFC+UC that $0=1$”?

Would you want to say “it is consistent with ZFC+UC that $0\neq 1$”?

Note that since ZFC+UC is inconsistent, it proves both $0\neq 1$ and $0=1$. Do you want a theory to be “consistent with” a statement whose negation it proves?

Looks like the wikipedia page on consistency agrees with you:

If T is a theory and A is an additional axiom, T + A is said to be consistent relative to T (or simply that A is consistent with T) if it can be proved that if T is consistent then T + A is consistent.

On the other hand, in particular cases wikipedia does sometimes adhere to my usage instead, like at continuum hypothesis:

The existence of an inner model of ZF in which additional axioms hold shows that the additional axioms are consistent with ZF, provided ZF itself is consistent.

and large cardinal:

A necessary condition for a property of cardinal numbers to be a large cardinal property is that the existence of such a cardinal is not known to be inconsistent with ZFC and it has been proven that if ZFC is consistent, then ZFC is consistent with the statement that “no such cardinal exists.”

If your guess about $W$-sets is right (which seems plausible), it suggests that a nonempty hyper-set (with its initial well-ordering) should satisfy $W = \aleph_{\omega^W\cdot V}$ for some $V$. In particular, any fixed point of the operation $W\mapsto \aleph_{\omega^W}$ would be one. I would expect such fixed points to exist in ZFC by the fixed-point lemma for normal functions, so in particular they needn’t be weakly inaccessible. But they don’t seem quite the same as aleph fixed points either.

You may be thinking of a normal function. However, I think the impression you’re getting doesn’t last much past the Mahlos. Larger large cardinals, like weakly compacts and especially measurables, aren’t really any kind of fixed point, as far as I know.

I think that once one gets up to measurable cardinals, that is, up towards the large large cardinals, they can start to be defined as critical points of nontrivial elementary embeddings. And I suspect that all large cardinals above measurables can be defined like this.

up towards the large large cardinals, they can start to be defined as critical points of nontrivial elementary embeddings.

It’s good that you bring this up, because I’m conscious that this kind of definition is over the horizon of what I’m covering in this series.

Right at the start of Part 1, I wrote this:

My primary interest is not actually in large cardinals themselves. What I’m really interested in is exploring the hypothesis that everying in traditional, membership-based set theory that’s relevant to the rest of mathematics can be done smoothly in categorical set theory.

One could spend entire lifetimes going through set theory research and attempting to recast it in categorical terms. (The question is not really whether everything can be done categorically, but whether things look smooth and natural.) And personally, I don’t intend to spend very much of a lifetime doing this kind of thing.

There’s always going to be something just over the horizon. As far as this series is concerned, there are two such things I’m particularly aware of. One is the style of large cardinal axiom that you mention. The other is PCF. I don’t know how either of these things look in categorical set theory.

Is there a good categorical statement of V=L?

I hope it’s equivalent. I haven’t seen it written like this before either, but I was trying to phrase it in a simple way without introducing notions like “club sets”.

How about this: let $X$ be Mahlo and $L$ unbounded in $K(X)$. Let $L' \subseteq K(X)$ consist of those $Y\lt X$ such that $L$ is unbounded below $Y$; then it suffices to show that $L'$ is closed and unbounded.

For unboundedness, if $Y\lt X$ then since $L$ is unbounded, there is a $Z_0$ with $Y\lt Z_0\lt X$ and $Z_0\in L$. By the same reasoning, there is a $Z_1$ with $Z_0 \lt Z_1\lt X$ and $Z_1\in L$. Proceed inductively and let $Z = \sup_{n\in\mathbb{N}} Z_n$; then $Z\in L'$ and $Y \lt Z\lt X$.

For closedness, suppose $Y\lt X$ is a limit point of $L'$, which is to say that $L'$ is unbounded below $Y$. We want to show $Y\in L'$, which is to say that $L$ is unbounded below $Y$. Now by assumption, for any $Z\lt Y$ there is a $W\in L'$ with $Z\lt W\lt Y$. But $W\in L'$ means $L$ is unbounded below $W$, so there is a $U\in L$ with $Z\lt U\lt W$, hence $Z\lt U\lt Y$. Since this holds for any $Z\lt Y$, it follows that $L$ is unbounded below $Y$ as desired.

Fantastic; thanks!

A couple of comments, more for the record than because I think you hadn’t noticed already:

• In the proof of unboundedness, where we put $Z = \sup_{n \in \mathbb{N}} Z_n$, we’re implicitly using the fact that $X$ has unbounded cofinality (which it does because it’s inaccessible, hence uncountable and regular).

• The proof that $L'$ is closed is a special case of the proof that in an arbitrary topological space, the Cantor–Bendixson derivative of any subset is closed. Presumably that’s why you chose the notation $L'$.

  (In my comment to which you were replying, I messed up the definition of the closed sets in $K(X)$, though I got the closure operator right. I’ve fixed it now.)

Mike wrote:

I haven’t seen it written like this before either, but I was trying to phrase it in a simple way without introducing notions like “club sets”.

Yes, that’s great. It took me a long time to get round to learning the definition of Mahlo set, for exactly that reason.

What would happen is that I’d get the urge to learn the definition of Mahlo and I’d go to Wikipedia. The Wikipedia page on Mahlo cardinals would tell me that the definition depends on the definition of stationary set. The Wikipedia page on stationary sets would tell me that that depends on the definition of club set. The Wikipedia page on club sets would tell me that it’s just an abbreviation for “closed and unbounded”. Now “closed” and “unbounded” sound like things that should be easy enough, but the definitions of those were phrased in a von-Neumannized way that would have to be deciphered.

So: in order to learn the definition of Mahlo, I’d first have to convert the definitions of closed and unbounded into structural terms I could understand, then digest what they mean, then put them together, then read the definition of stationary set, then digest what that means, then read the definition of Mahlo cardinal, and only then, finally, start digesting that too. And the prospect of doing all that would make me say to myself “you know what? maybe I’m not so interested in Mahlo cardinals after all.”

So, having a direct definition like the one you’ve given is great. I thought of adding it to the Wikipedia page, but unless someone can find a reference, I guess it falls foul of their “no original research” rule.

Thanks for adding those clarifications; I could have been more explicit about those facts. (Your link to the CB derivative is missing a parenthesis.)

I thought of adding it to the Wikipedia page, but unless someone can find a reference, I guess it falls foul of their “no original research” rule.

The usual thing I do in a situation like that is add it to the nLab instead!

If the definition is on the nLab, it can then be cited on WP, I feel, since the nLab seems to be considered a reliable secondary source (to the point where there is a citation template, IIRC). In any case, the nLab is going to be citing this blog post, making it a secondary source ;-)

If the definition is on the nLab, it can then be cited on WP, I feel, since the nLab seems to be considered a reliable secondary source (to the point where there is a citation template, IIRC).

That’s bad form from Wikipedia, since the nLab does allow for original research: for example, the article on absorption category which the original author admitted to being original research in the discussion page, and Urs Schreiber explicitly saying original research is allowed on the nLab in the ensuing discussion here

Yes, that’s what I should have said. Thanks for catching it.