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What is the largest large-cardinal hypothesis consistent with $ZFC + V=L$? The reason for the question is this: under the assumption that all of 'ordinary mathematics' (as reverse mathematics understands the term) can be interpreted in $ZFC+ V=L$ (see Simpson's paper "The Goedel Hierarchy and Reverse Mathematics", preprint, November 24, 2009 draft), this would be the largest cardinal allowed by current mathematical practice.

What then is the significance of this cardinal to ordinary mathematics?

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    $\begingroup$ In the paper ``A large cardinal in the constructible universe'' Silver shows that if $\kappa\to (\alpha)^{<\aleph_0}$ for all countable $\alpha,$ then the same is true for $\kappa$ in $L$. On the other hand, by a result of Rowbottom, $\kappa \to (\omega_1)^{<\aleph_0} $ contradicts $V=L.$ Silver concludes with the following: >It does not seem extravagant, then, to assert that, for all practical purposes, $\kappa \to (\omega)^{<\aleph_0}$ is the strongest strong axiom of infinity know to be consistent with $V=L,$ and therein lies its chief interest. $\endgroup$
    – Mohammad Golshani
    Commented Dec 9, 2016 at 5:56
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    $\begingroup$ See mathoverflow.net/questions/190614/… $\endgroup$
    – Mohammad Golshani
    Commented Dec 9, 2016 at 5:57
  • $\begingroup$ The question should be about the "the largest large-cardinal hypothesis consistent with $V=L$" rather than "the largest cardinal consistent with $V=L$", which would be a phrase with uncertain meaning. Fortunately, the answers have answered the former question, rather than the latter. $\endgroup$
    – Joel David Hamkins
    Commented Jan 2 at 1:35
  • $\begingroup$ Thank you, Prof. Hamkins. $\endgroup$
    – Thomas Benjamin
    Commented Jan 2 at 4:34

3 Answers 3

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Re: ordinary mathematics, I think your assumptions are incorrect (although the question is interesting on its own, since ZFC+V=L is a natural set theory). ZFC+V=L is (most, including Simpson, would argue) more than sufficient for ordinary mathematics. But then, so is ZFC, and indeed much less; there's nothing that singles ZFC+V=L out, here. Moreover, "stronger" theories (in the sense of consistency or interpretive strength, not outright implication) like ZFC+large cardinals are, even more so, sufficient for ordinary mathematics. So "the largest large cardinal axiom compatible with V=L" doesn't really have any special connection with ordinary mathematics.

In particular, if you want a "conservative" set theory - that is, one sufficient for ordinary mathematics, but (close to) minimally so - then you've already overshot the mark with ZFC. I would argue ZC (= ZFC without replacement - vastly weaker) is even overshooting this, although at this point we get into the fine details of what exactly "ordinary mathematics" is (does Borel determinacy count? because if it does, then ZC isn't enough).

And if you don't want a conservative set theory, then in what sense are the large cardinals compatible with V=L the only ones "allowed by current mathematical practice"?

A comment on motivating foundations:

I suspect that what's going on is a sort of "hybrid foundations", as follows:

  • First, we commit ourselves to ZFC, together with the rule that we won't unnecessarily bound the height of the ordinals which exist (e.g. we won't add "No inaccessibles" as an axiom without good reason); this is for one set of reasons, which lean on the maximality side. In particular, Replacement is justified as its negation provides an "artificially small" universe of sets.

  • We then - within this framework - declare a "mood change": further axioms will be justified by appeal to mathematical practice, this time leaning towards the minimality side - no unneeded sets are conjured into existence. Things like nonprojective sets of reals are already provably extant, from our previous commitments; but, since nonconstructible reals don't currently play a role in ordinary mathematics, we not only don't assume their existence, we actively assume that they don't exist. In particular, we adopt V=L since it is the "narrowest" axiom compatible with ZFC.

This is a reasonably consistent approach; however, I find it flawed for several reasons. The choice of mood change is unjustified - why have we exhausted the foundational value of the "maximality" approach right at ZFC? Moreover, I'm not swayed by the "minimality" approach at all: I see no reason to artificially narrow the universe, except perhaps for reasons of consistency strength - but these reasons militate against small large cardinals like inaccessibles, and not against failures of constructibility (since we can force V=L to fail, without higher consistency strength)! Finally, I don't see why the background set theory has to limit itself to ordinary mathematics at all. While the question of exactly how much set theory is necessary for ordinary mathematics is an interesting one (I am a reverse mathematician among other things, after all), it doesn't carry a value judgment, at least for me; indeed, one of the roles of foundations of mathematics in my view is to go well beyond current ordinary mathematics, and blaze new trails, motivated by the mathematical optimism that those trails will eventually be useful.

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    $\begingroup$ Indeed, essentially number-theoretic systems suffice for virtually all "ordinary" mathematics --- predicativists have known this for a long time, and the reverse mathematics school has precisely calibrated just how much second order arithmetic is needed for a large body of results from mainstream mathematics. ZFC + V = L is vastly more than you need. $\endgroup$
    – Nik Weaver
    Commented Dec 9, 2016 at 4:55
  • $\begingroup$ @NikWeaver: Would it be safe to say that there are no nonconstructible sets in 'ordinary mathematics'? $\endgroup$
    – Thomas Benjamin
    Commented Dec 9, 2016 at 11:50
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    $\begingroup$ If "nonconstructible" means "not in L", then sure. You don't even really need the real line, or the power set of the natural numbers, to be sets --- you can treat them as proper classes and still do essentially all of (what I consider to be) ordinary mathematics. $\endgroup$
    – Nik Weaver
    Commented Dec 9, 2016 at 15:18
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    $\begingroup$ @ThomasBenjamin There's a difference between "There are no nonconstructible sets in ordinary mathematics" and "Ordinary mathematics demands that there be no nonconstructible sets"; similarly, there are no non-projective sets of reals in ordinary mathematics, but nobody would claim (I think) that the right set theory is one which proves "Every set of reals is projective," based on this. $\endgroup$
    – Noah Schweber
    Commented Dec 9, 2016 at 16:11
  • $\begingroup$ Does KP^P+AC prove Borel determinacy? $\endgroup$
    – user21820
    Commented Jun 17, 2022 at 13:35
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For your first question: If $\alpha$ is countable, then $\alpha$-Erdos cardinals are consistent with $V=L$ (if they are consistent at all). On the other hand, the existence of an $\omega_1$-Erdos cardinal implies the existence of $0^\sharp$ (which imples $V \neq L$).

I don't really know how to answer your second question.

Let me also say that I think your assertion "[these $\alpha$-Erdos cardinals] would be the largest cardinal[s] allowed by current mathematical practice" is a bit hasty. Supposing "ordinary mathematics" can be interpreted in $L$, it does not follow that $\omega_1$-Erdos cardinals (or bigger, stranger beasts) are not "allowed" -- only that they aren't necessary from a certain point of view.

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    $\begingroup$ I think I've figured out an answer to my own question: for any function, if we have homogenous sequences of arbitrary countable ordinal length, taking a union of ones having a single color gives a homogenous sequence of length $\omega_1$. $\endgroup$
    – Wojowu
    Commented Dec 8, 2016 at 18:30
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    $\begingroup$ @ThomasBenjamin Obviously $\omega_1$-Erdos cardinals aren't allowed when ZFC+V=L holds. The question is, why on earth does ordinary mathematics commit us to V=L? If the answer is because nonconstructible sets aren't (currently) relevant to ordinary mathematics, then ordinary mathematics should commit us to ZC+ the negation of Replacement, since sets not in $V_{\omega+\omega}$ aren't relevant to ordinary mathematics. You seem to conflate ordinary mathematics not demanding nonconstructible sets, with ordinary mathematics militating against nonconstructible sets. (cont'd) $\endgroup$
    – Noah Schweber
    Commented Dec 9, 2016 at 16:09
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    $\begingroup$ Similarly, non-projective sets of reals have (currently) no role in ordinary mathematics; does that commit us to a set theory stating that all sets of reals are projective? $\endgroup$
    – Noah Schweber
    Commented Dec 9, 2016 at 16:10
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    $\begingroup$ @ThomasBenjamin: I see that Noah has beet me to the comments -- I second what he wrote. I am not saying that we are allowed to assume the existence of $\omega_1$-Erdos cardinals when $V=L$ holds. What I am saying is that knowing we can interpret some bit of mathematics in $ZFC+V=L$ does not mean that we must work under the assumption $V=L$. $\endgroup$
    – Will Brian
    Commented Dec 9, 2016 at 16:17
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    $\begingroup$ @Wojowu A union of homogeneous sequence of the same color does not have to be homogeneous, and in fact $\sup\{\kappa(\alpha):\alpha\lt\omega_1\}\lt\kappa(\omega_1)$. $\endgroup$
    – bof
    Commented Jan 2 at 0:53
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In the paper "Generalized Erdős cardinals and $0^\sharp$", Annals of Mathemtical Logic 15 (1978) 289–313, Baumgartner and Galvin define some large cardinals which are smaller than the Erdős cardinal $\kappa(\omega_1)$ but greater than all the Erdős cardinals $\kappa(\alpha)$ where $\alpha$ is a countable ordinal. I won't try to summarize their work here as it's over my head, but it seems that some of those "generalized Erdős cardinals" are consistent with V=L and some are not.

I think the definition of "generalized Erdős cardinal" is something like this: Given a function $a:\{0,1\}^\omega\to\omega_1$, the "generalized Erdős cardinal" $\kappa(a)$ is the least $\kappa$ such that, for any $\{0,1\}$-coloring of the finite subsets of $\kappa$, there exist a function $f\in\{0,1\}^\omega$ and a set $A\subseteq\kappa$ of order type $a(f)$ such that, for each $n\in\omega$, all $n$-element subsets of $A$ have color $f(n)$. (So it's like the off-diagonal Ramsey number $R(m,n)$ where the size of the monochromatic clique depends on the color.)

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