TITLE: Arguments against large cardinals
QUESTION [60 upvotes]: I started to learn about large cardinals a while ago, and I read that the existence, and even the consistency of the existence of an inaccessible cardinal, i.e. a limit cardinal which is additionally regular, is unprovable in ZFC. Nevertheless large cardinals were studied extensively in the last century and (apart from attempts that went too far as the Reinhardt-Cardinals) nobody ever found a contradiction to ZFC. As a consequence it seems to me that set theorists today don't consider the possibility of the non-existence of a large cardinal. Therefore my questions:
Why is it so unreasonable to think that the existence of large cardinals contradicts ZFC? Are there any mathematicians that do believe that large cardinals don't exist? And what are their arguments?
EDIT: I want to say thank you for all your very interesting answers and comments. It will take some time for me to fully understand them though, however I feel like I have already learned a lot due to this discussion. Thank you!

REPLY [11 votes]: Large cardinals offer a detailed coherent picture — with a single principle, that of symmetry, reaching even (essentially) the strongest large cardinals.  They continually offer new results — without inconsistency.  For weaker large cardinals (up through Woodin cardinals) we have canonical well-understood fine-structural models, with a variety of intuitively true (or otherwise well-analyzed) seemingly unrelated combinatorial principles about real numbers and other sets implying their consistency.  And while we are not there yet, we expect many more such results at the level of supercompact cardinals.
However, per the question title, the rest of the answer will be arguments against large cardinals, which are important both because of genuine doubts (especially about certain things) and for the insight the arguments offer.  Many of the arguments also apply to ZFC, but become stronger as the large cardinal strength increases.
The arguments separate into:
• Arguments for inconsistency of large cardinals.
• Arguments for non-existence of large cardinals (even if they are consistent).
• Arguments for not accepting large cardinals as axioms.
Moreover regarding truth/consistency, we can approximately separate mere doubts from affirmative arguments for falsehood/inconsistency, with most arguments being the former.
Arguments against acceptance as axioms:
• Doubts about large cardinal axioms (see the other paragraphs).
• The importance for the foundations of mathematics to be free of unnecessary doubt.
• The unfortunate general disinterest in foundations.  Most mathematicians do not even know exactly what ZFC is. (This is relevant here because 'acceptance' has a sociological component.)
• Related to the above, the limited need to go beyond ZFC for results most mathematicians care about.
• Knowing that something is provable in a weak theory (or just ZFC) often gives us important information that goes beyond knowing that it is true.
• One can still use large cardinal axioms by including them as assumptions.
Doubts about consistency:
• Doubt about metaphysical existence of infinite sets, or if they exist, about intuitive reasoning about such sets.  Many believe that the observable physical universe is finite and/or that the operation of physical laws is computable, and that humans do not have privileged access to truth.  Also, some facts that are obviously true about finite sets fail for the infinite.
• Lack of inner model and core model theory beyond Woodin cardinals.  While (as set theorist John Steel put) "all roads lead to projective determinacy", we do not yet know whether all roads lead to analogs of supercompact cardinals.
  - Core model induction has not yet reached a Woodin limit of Woodin cardinals (but there is progress).  Core models (including core model induction) are our main method of showing that natural plausible combinatorial propositions at low levels of $V$ imply consistency of large cardinals.
  - Existence of canonical inner models for a stationary set of Woodin cardinals is still open as of 2018.
  - While under plausible conjectures, current definitions of fine-structural models reach subcompact cardinals (and slightly further), supercompact cardinals show a qualitively different and anti-core-model behavior, such as consistency of indestructibility.
• Lack of sufficient heuristic 'proofs' of consistency of ZFC:
  - It is hard to find compelling arguments for consistency of ZFC that make no mention of infinity (other than the argument that consequences of ZFC have been well-understood with no inconsistency found; also, "hard" need not mean "impossible").
  - Lack of enough natural true arithmetic statements that are known to imply consistency of ZFC.
  - Lack of reasonable ordinal notation systems that have been proved to reach ZFC (or even just $\mathrm{Z}_2$).  Ordinal notation systems (and to a lesser extent, canonical inner models) give us a well-understood picture of how a theory operates, which (among many other benefits) counteracts doubts about consistency.
Affirmative historical arguments for inconsistency:
• Historically, paradoxes (including Russell paradox) were used to argue against infinity (including consistency of infinity) but these arguments have receded, as ZFC has no shown sign of being inconsistent through such a paradox.
• Kunen's inconsistency was used to argue against superficially similar axioms:  Since existence of nontrivial elementary $j:V_{λ+2}→V_{λ+2}$ is inconsistent, why should using $V_{λ+1}$ be different?  However, the current understanding is that there is no inconsistency without the axiom of choice (but see below), and unlike $V_{λ+2}$, choice does not cause problems for $V_{λ+1}$ and (starting from a stronger axiom for ZF) we can get choice in a generic extension.
• More recently, Hugh Woodin has argued (see his work on the HOD dichotomy) that Reinhardt cardinals for ZF are inconsistent.  Specifically, for cardinals such as measurable (or even strong), there are canonical ordinal definable inner models that use the restrictions of the extenders from the embeddings on $V$ (and get the same large cardinals from these extenders), but he was able to show that modulo consistency of a strengthening of ZF + Reinhardt cardinal (that is weaker than Berkeley cardinals), this need not hold for supercompact cardinals.  Personally I think that this failure is why supercompacts have been so useful for consistency of combinatorial propositions — unlike weaker cardinals, the embeddings (in a sense) capture a large part of $V$, and thus cannot be generally included into canonical inner models as is — but there is still much that we do not understand.
In between consistency and existence, we have various degrees of soundness.  Properties inconsistent with the axiom of choice can still hold at those levels.  Thus, for example, near the top of the known hierarchy (and thus, with the highest doubts about consistency), we have the $Π^V_2$ statement "For every cardinal $κ$, there is a model of 'ZF + Berkeley cardinal' closed under $κ$ sequences."  The truthfulness of such statements is generally regarded as closely related to consistency.  However, this belief assumes a fundamental soundness of the large cardinal hierarchy, which has strong analytical and empirical support, but can also be argued against (for example, see the arguments for $V=L$).
Doubts on existence of large cardinals:
• Formalists and adherents to various related philosophies believe that infinite (or failing that, uncountable) sets do not actually exist.  What may exist are finite quasimodels (or countable models) of various set axioms, but they hold that there is no single preferred truth predicate for $(V,∈)$.  The lack of consensus on CH is often cited to support this.  Various large cardinal axioms may hold in some but not other theories/quasimodels/models, and the use of ZFC is a useful common convention.  For them, existence of large cardinals morphs into goodness of using ZFC + A (for various A) for mathematical work.
• Consistency of large cardinals is fundamentally different from existence.  To fully accept inaccessible cardinals, a set platonist would need to know that they metaphysically exist, or for a symmetry platonist (a type of truth-value platonist), that we have enough symmetry that truth values of statements at the level of inaccessible cardinals are unambiguous.
• The replacement axiom schema (and choice) may intuitively follow from "For every ordinal $α$, a process that can be always be continued can be iterated $α$ times", but this reasoning is insufficient to reach inaccessible cardinals.
• By Kunen's inconsistency, some otherwise natural large cardinal axioms are inconsistent with choice.  Perhaps we will discover a stronger natural true combinatorial principle that refutes more large cardinal axioms.
• A canonical theory for a given expressiveness level may require consistency strength that on its face corresponds to a higher level.  For example, a true reasonably complete axiomatization of second order arithmetic requires projective determinacy, and projective determinacy can be ascertained by just studying real numbers (disclaimer: not everyone agrees with this), which in turn gives consistency of Woodin cardinals.  In this manner, doubt about consistency of Woodin cardinals may translate into doubt about existence of the set or class of all real numbers (and quantification over it).  We do not know how much strength is required for third order arithmetic and higher expressiveness levels, and a breakdown would argue for non-existence of higher levels.
• Large cardinals can be destroyed by forcing.  For example, we can destroy all measurables by Prikry forcing (making them cofinality $ω$ without changing other cofinalities), and it is arguably unclear why the ground model rather than the extension is a better analog to $V$.  Also, by adding clubs, starting with a model of GCH, I think we can destroy all Mahlo cardinals while preserving GCH and cofinalities; and no forcing can resurrect Mahloness.
Affirmative arguments for non-existence:
• Predicativist argument for $V=L$.  Predicativists hold that certain objects are (essentially) constructed in stages, and that quantification is fully permitted only over the previously constructed stages.  Most predicativists start with only natural numbers, but if one also accepts arbitrary ordinals, the result is the constructible universe $L$ — each stage of $L$ is added predicatively.  One would then object to zero sharp analogously to the traditional predicativist objection to zero hyperjump (Kleene's $\mathcal{O}$).  Also, some mathematicians simply like $L$ as a canonical tidy universe that satisfies ZFC (and is $Σ^1_2$-correct).
• Against measurable cardinals:
  - The nonrigidity (nontrivial elementary embedding $V→M$) arguably qualitatively alters the behavior of $V$ and is contrary to how $V$ behaves at lower levels.
  - Measurable cardinals cause a few counterexamples in topology.
  - Under $HOD(ℝ)⊨\mathrm{AD}$, $ω_1^V$ is the least measurable in HOD (assuming the proof of lack of lower measurables applies here).  The absence of lower measurable cardinals, combined with the fact that every ordinal definable subset of $ω_1$ is constructible from a real, and that the measure on $ω_1$ is 'caused' by an external structure (the club filter on $ω_1$) (along with the general largeness of measurables and existence of Prikry forcing) suggest a possibility that measurable cardinals do not occur in the cumulative hierarchy on their own, but reflect a restriction on $\mathcal{P}(κ)$.
(However, on the balance, I find that evidence supports existence of measurables.)