From: ah170@FreeNet.Carleton.CA (David Libert) Subject: Re: The biggest number? Date: 10 Aug 2000 08:46:48 GMT Newsgroups: sci.math Summary: [missing] Jonathan Hoyle (Jonathan.Hoyle@kodak.com) writes: >>> aleph_(aleph_(aleph_(aleph_(...ad.inf...(aleph_0)...ad.inf...)))) >>> is bigger. > > Even that (I believe) can be beaten by the first strongly inaccesible > cardinal. Yes. That notation above could be most reasonably taken to represent the supremum of all finite iterations of aleph subscripting. This is the least fixed point of the aleph function. ZF proves this cardinal exists. The least inaccessible if it exists is larger than this. > By the way, is there a known size order for these very large cardinals? > From first inaccessible to first Mahlo to first measurable, etc. Thanks > in advance. Yes the cases you just mentioned are ordered as you listed them: least inaccessible < least Mahlo < least measurable. Working up the large cardinal hierarchy this is the pattern for a long time, and it is easy to expect it to always happen. At very high levels things can get strange though. After measurable are strongly compact and then supercompact. ZFC proves every supercompact is strongly compact and every stongly compact is measurable. Also ZFC proves that the least measureable is strictly less than the least supercompact. So against this background Magidor had his surprising identity crisis paper. This was forcing over ground models with appropriate large cardinals to start. So I don't remember the exact details, but it was something along these lines: Assume you have a countable model to start of ZFC + kappa is a stongly compact cardinal. (Or did he need supercompact to even do this much?) Anyway, you allow for the possibility that kappa has many measurable cardinals below it. Now force over that, by forcing Magidor defines. This destroys the measurability of all the measurables below kappa. Being measurable means having a certain kind of ultrafilter. Those ultrafilters will still exist in the forcing extension as sets, but the forcing extension no longer models that they are the right sort of ultrafilters. In general, passing from a smaller model to a larger model, large cardinal properties can be lost. This is what happens here to all the cases of measurability below kappa. On the other hand, in the forcing extension, Magidor arranges that kappa is still strongly compact. So in this forcing extension kappa is both the least measurable and the least strongly compact. Next Magidor does a second construction. Start with a new ground model of ZFC + lambda is supercompact. Force over this, destroying the strong compactness of any cardinals less than lambda, but retaining the supercompactness of lambda. So in this model lambda is both the least stongly compact and the least supercompact. So in other words, if there is a supercompact lambda then there is ZFC provably a measureable less than lambda. Let mu be the least measurable, so mu < lambda. ZFC proves if there is a supercompact then there are strongly compacts, so given mu, lambda as above, kappa the least strongly compact must have mu <= kappa <= lambda, and mu < lambda. Where does kappa sit within the nontrivial interval [mu, lambda] ? Magidor's first model has kappa = mu, his second model has kappa = lambda. So anything is possible. I have forgotten the exact title of Magidor's paper but it includes the phrase "identity crisis" :) . Then another strange point mentioned a while back in sci.math. A further large cardinal property is huge. Huge would normally be considered a stronger property than supercompact, in the sense for example that ZFC + there exists a huge cardinal |- Con(ZFC + there exists a supercompact cardinal) . In fact, ZFC proves that if nu is a huge cardinal, then V_nu, the set of sets of rank < nu, models ZFC + there is a proper class of supercompacts. This looks at first like the usual picture of increasing cardinal strength, like inaccessible to Mahlo to measureable for example. But it is more complicated. Given a huge cardinal nu, the so-called "supercompacts" below nu in V_nu need not be true supercompacts. V_nu thinks they are, but that is only because V_nu quantifies only over itself, missing the higher sets above V_nu that expose those so called "supercompacts" for the imposters that they are. Having these V_nu certified "supercompacts" below nu is no guarantee that there really are any actual supercompacts below nu. And in fact in some cases such a failure is exactly what happens. ZFC + there exists a huge cardinal + there exists a supercompact cardinal |- the least huge cardinal < the least supercompact cardinal So, in a model with both supercompacts and huges, letting nu be the least huge, we have exactly that situtation, that all the V_nu - "supercompacts" below nu were fake supercompacts. So according to consistency strength huge is a stronger property than supercompact, but according to size of least example, supercompact is bigger. This does not happen at the low levels, as far as I know anyway. Ie everything I have ever heard of below measurable. There consistency strength and least example agree. Well one point along these lines. This strangeness of the last example is based on the possibilty of fake "supercompacts". A property has to be sophisticated before such fakery is possible. For the low level properties P, measureable and below, if rho1 is a cardinal < rho2, rho2 inaccessible, and V_rho2 |= P(rho1), then really P(rho1). Ie the V_rho2 's are smart enought to properly identify the small large cardinal properties. When you get to higher larger cardinal properties like supercompactness, the V_rho2 's can be fooled. -- David Libert (ah170@freenet.carleton.ca) 1. I used to be conceited but now I am perfect. 2. "So self-quoting doesn't seem so bad." -- David Libert 3. "So don't be a morron." -- Marek Drobnik bd308 rhetorical salvo IRC sig