ho.history-overviewset-theorysoft-question
It seems like Gödel didn't use the letter $L$ for his model before his book "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis with the Axioms of Set Theory", which is probably the first place it got used.
Do anyone of you know why he used the letter $L$? It does seem like a bit ad hoc in the book, where he names some functions $J_i$, then some other functions $K_i$ and then ends up defining $L$. But why $J_i$ then?
(I'm sorry if this is not suited for MO)
A Grothendieck universe is known in set theory as the set Vκ for a (strongly) inaccessible cardinal κ. They are exactly the same thing. Thus, the existence of a Grothendieck universe is exactly equivalent to the existence of one inaccessible cardinal. These cardinals and the corresponding universes have been studied in set theory for over a century.
The Grothendieck Universe axiom (AU) is the assertion that every set is an element of a universe in this sense. Thus, it is equivalent to the assertion that the inaccessible cardinals are unbounded in the cardinals. In other words, that there is a proper class of inaccessible cardinals. This is the axiom you sought, which is exactly equivalent to AU. In this sense, the axiom AU is a statement in set theory, having nothing necessarily to do with category theory.
The large cardinal axioms are fruitfully measured in strength not just by direct implication, but also by their consistency strength. One large cardinal property LC1 is stronger than another LC2 in consistency strength if the consistency of ZFC with an LC1 large cardinal implies the consistency of ZFC with an LC2 large cardinal.
Measured in this way, the AU axiom has a stronger consistency strength than the existence of any finite or accessible number of inaccessible cardinals, and so one might think it rather strong. But actually, it is much weaker than the existence of a single Mahlo cardinal, the traditional next-step-up in the large cardinal hierarchy. The reason is that if κ is Mahlo, then κ is a limit of inaccessible cardinals, and so Vκ will satisfy ZFC plus the AU axiom. The difference between AU and Mahloness has to do with the thickness of the class of inaccessible cardinals. For example, strictly stronger than AU and weaker than a Mahlo cardinal is the assertion that the inaccessible cardinals form a stationary proper class, an assertion known as the Levy Scheme (which is provably equivconsistent with some other interesting axioms of set theory, such as the boldface Maximality Principle, which I have studied a lot). Even Mahlo cardinals are regarded as rather low in the large cardinal hierarchy, far below the weakly compact cardinals, Ramsey cardinals, measurable cardinals, strong cardinals and supercompact cardinals. In particular, if δ is any of these large cardinals, then δ is a limit of Mahlo cardinals, and certainly a limit of strongly inaccessible cardinals. So in particular, Vδ will be a model of the AU axiom.
Rather few of the large cardinal axioms imnply AU directly, since most of them remain true if one were to cut off the universe at a given inaccessible cardinal, a process that kills AU. Nevertheless, implicit beteween levels of the large caridnal hiearchy are the axioms of the same form as AU, which assert an unbounded class of the given cardinal. For example, one might want to have unboundedly many Mahlo cardinals, or unboundedly many measurable cardinals, and so on. And the consistency strength of these axioms is still below the consistency strength of a single supercompact cardinal. The hierarchy is extremely fine and intensely studied. For example, the assertion that there are unboundedly many strong cardinals is equiconsistent with the impossiblity to affect projective truth by forcing. The existence of a proper class of Woodin cardinals is particularly robust set-theoretically, and all of these axioms are far stronger than AU.
There are natural weakenings of AU that still allow for almost all if not all of what category theorists do with these universes. Namely, with the universes, it would seem to suffice for almost all category-theoretic purposes, if a given universe U were merely a model of ZFC, rather than Vκ for an inaccessible cardinal κ. The difference is that U is merely a model of the Power set axiom, rather than actually being closed under the true power sets (and similarly using Replacement in place of regularity). The weakening of AU I have in mind is the axiom that asserts that every set is an element of a transitive model of ZFC. This assertion is strictly weaker in consistency strength thatn even a single inaccessible cardinal. One can get much lower, if one weakens the concept of universe to just a fragment of ZFC. Then one could arrive at a version of AU that was actually provable in ZFC, but which could be used for most all of the applications in cateogory theory to my knowledge. In this sense, ZFC itself is a kind of large cardinal axiom relative to the weaker fragments of ZFC.
In a more recent paper of Friedman
Friedman, Harvey M., Concept calculus: much better than, Heller, Michael (ed.) et al., Infinity. New research frontiers. Based on the conference on new frontiers in research on infinity, San Marino, August 18–20, 2006. Cambridge: Cambridge University Press (ISBN 978-1-107-00387-3/hbk). 130-164 (2011). ZBL1269.03008,
the author defines a mathematically precise system MBT (much better than) and proves it and ZF have mutual interpretability. This establishes that if either is consistent they both are. These axioms have some of the flavor of IP and PP, but of course these axioms are not implied by IP and PP.
At the end of the paper Friedman claims a to be published result. STAR is defined as:
There exists a star. I.e., something which is better than something, and much better than everything it is better than.
We have shown that MBT + STAR can be interpreted in some large cardinals compatible with V = L, and some large cardinals compatible with V = L are interpretable in MBT + STAR."
For PP and IP to be true, in a sense that can prove mathematics, they need to be stated precisely like the axioms in MBT. That formulation is much more complex than PP and IP as it must be to interpret ZF.
It is important to keep in mind that consistency does not imply truth. The statement that a formal system is consistent is equivalent to a statement of the form $\forall_{n\in\omega} r(n)$ where $r$ is a recursive relationship. This is equivalent to the halting problem for a particular Turing machine.
The following quote from Friedman is, I suspect, a big part of his and others interest in this work:
STARTLING OBSERVATION. Any two natural theories S,T, known to interpret PA, are known (with small numbers of exceptions) to have: S is interpretable in T or T is interpretable in S. The exceptions are believed to also have comparability.
It is an interesting and even startling observation, but it is worth keeping in mind that that rigorous theories are, among other things, recursive processes for enumerating theorems. To say that one theory is interpretable in another is to say a subset of one processes outputs are, in a specific well defined sense, isomorphic to the outputs of the process defined from the theory being interpreted. Whatever other significance it may have, this is a statement about unbounded recursive processes.
My personal view (see what is Mathematics About?) is that the only mathematics that can be interpreted as a properties of recursive processes is objectively true or false. This is based on the old idea that infinite is a potential that can never be realized. In this view Cantor's proof that the reals are not countable is an incompleteness theorem. The cardinal hierarchy is a hierarchy of the ways the real numbers provably definable in a formal system can always be expanded. Because of the Lowheheim Skolem theorem, we know such an interpretation exists. Interpretations that assume the absolutely uncountable are inevitably ambiguous at least as far as they can be expressed formally in the always finite universe that we seem to inhabit.
Best Answer
I heard from Kai Hauser that the letter $L$ comes from "law", and it is because the model is constructed using some laws.