Thank you for your interest in my views on the set-theoretic
multiverse.
Yes, indeed, the well-foundedness mirage axiom you mention is
probably the most controversial of my multiverse axioms, and so
allow me to explain a little about it.
The axiom expresses in a strong way the idea that we don't actually
have a foundationally robust absolute concept of the finite in
mathematics. Specifically, the axiom asserts that every universe of
set theory is ill-founded even in its natural numbers from the
perspective of another, better universe. Thus, every set-theoretic
background in which we might seek to undertake our mathematical
activity is nonstandard with respect to another universe.
My intention in posing the axiom so provocatively was to point out
what I believe is the unsatisfactory nature of our philosophical
account of the finite.
You might be interested in the brief essay I wrote on the topic, A
question for the mathematics
oracle,
published in the proceedings of the Singapore workshop on Infinity
and Truth. For an interesting and entertaining interlude, the
workshop organizers had requested that everyone at the workshop
pose a specific question that might be asked of an all-knowing
mathematical oracle, who would truthfully answer. My question was
whether in mathematics we really do have a absolute concept of the
finite.
To explain a bit more, the naive view of the natural numbers in
mathematics is that they are the numbers, $0$, $1$, $2$, and so
on. The natural numbers, with all the usual arithmetic structure,
are taken by many to have a definite absolute nature; arithmetic
truth assertions are taken to have a definite absolute nature, in
comparison for example with the comparatively less sure footing of
set-theoretic truth assertions.
To be sure, many mathematicians and philosophers have proposed a
demarcation between arithmetic and analysis, where the claims of
number theory and arithmetic are said to have a definite absolute
nature, while the assertions of higher levels of set theory,
beginning with claims about the set of sets of natural numbers, are
less definite. Nik Weaver, for example, has suggested that
classical logic is appropriate for the arithmetic realm and
intuitionistic logic for the latter realm, and a similar position
is advocated by Solomon Feferman and others.
But what exactly does this phrase, "and so on" really mean in the
naive account of the finite? It seems truly to be doing all the
work, and I find it basically inadequate to the task. The situation
is more subtle and problematic than seems to me to be typically
acknowledged. Why do people find their conception of the finite to
be so clear and absolute? It seems hopelessly vague to me.
Of course, within the axiomatic system of ZFC or other systems, we
have a clear definition of what it means to be finite. The issue is
not that, but rather the extent to which these internal accounts of
finiteness agree with the naive pre-reflective accounts of the
finite as used in the meta-theory.
Some mathematicians point to the various categoricity arguments as
an explanation of why it is meaningful to speak of the natural
numbers as a definite mathematical structure. Dedekind proved,
after all, that there is up to isomorphism only one model
$\langle\mathbb{N},S,0\rangle$ of the second-order Peano axioms,
where $0$ is not a successor, the successor function $S$ is
one-to-one, and $\mathbb{N}$ is the unique subset of $\mathbb{N}$
containing $0$ and closed under successor.
But to my way of thinking, this categoricity argument merely pushes
off the problem from arithmetic to set theory, basing the
absoluteness of arithmetic on the absoluteness of the concept of an
arbitrary set of natural numbers. But how does that give one any
confidence?
We already know very well, after all, about failures of
absoluteness in set theory. Different models of set theory can
disagree about whether the continuum hypothesis holds, whether the
axiom of choice holds, and so with innumerable examples of
non-absoluteness. Different models of set theory can disagree on
their natural number structures, and even when they agree on their
natural numbers, they can still disagree on their theories of
arithmetic truth (see Satisfaction is not
absolute).
So we know all about how mathematical truth assertions can seem to
be non-absolute in set theory.
Skolem pointed out that there are models of set theory $M_1$, $M_2$
and $M_3$ with a set $A$ in common, such that $M_1$ thinks $A$ is
finite; $M_2$ thinks $A$ is countably infinite and $M_3$ thinks $A$
is uncountable. For example, let $M_3$ be any countable model of
set theory, and let $M_1$ be an ultrapower by a ultrafilter on
$\mathbb{N}$ in $M_3$, and let $A$ be a nonstandard natural number
of $M_1$. So $M_1$ thinks $A$ is finite, but $M_3$ thinks $A$ has
size continuum. If $M_2$ is a forcing extension of $M_3$, we can
arrange that $A$ is countably infinite in $M_2$.
No amount of set-theoretic information in our set-theoretic
background could ever establish that our current conception of the
natural numbers, whatever it is, is the truly standard one, since
whatever we assert to be true is also true in some nonstandard
models, whose natural numbers are not standard.
The well-foundedness mirage axiom asserts that this phenomenon is
universal: all universes are wrong about well-foundedness.
In defense of the mirage axiom, let me point out that whatever attitude toward it one might harbor, nevertheless the axiom cannot be seen as incoherent or inconsistent, because Victoria Gitman and I have proved that all of my multiverse axioms are true in the multiverse consisting of the countable computably saturated models of ZFC. So the axiom is neither contradictory nor incoherent. See A natural model of the multiverse axioms.
I have discussed my multiverse views in several papers.
Hamkins, Joel David, The set-theoretic multiverse, Rev. Symb. Log. 5, No. 3, 416-449 (2012). Doi:10.1017/S1755020311000359, ZBL1260.03103.
Hamkins, Joel David, A multiverse perspective on the axiom of constructibility, Chong, Chitat (ed.) et al., Infinity and truth. Based on talks given at the workshop, Singapore, July 25--29, 2011. Hackensack, NJ: World Scientific (ISBN 978-981-4571-03-6/hbk; 978-981-4571-05-0/ebook). Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore 25, 25-45 (2014). DOI:10.1142/9789814571043_0002,
ZBL1321.03061.
Gitman, Victoria; Hamkins, Joel David, A natural model of the multiverse axioms, Notre Dame J. Formal Logic 51, No. 4, 475-484 (2010). DOI:10.1215/00294527-2010-030, ZBL1214.03035.
Hamkins, Joel David; Yang, Ruizhi, Satisfaction is not absolute, to appear in the Review of Symbolic Logic.
But finally, to address your specific question. Of course, there
are specific finite numbers that will be finite with respect to any
alternative set-theoretic background. As Michael Greinecker points
out in the comments, the number 35253586543 has that value
regardless of your meta-mathematical position. So of course, there
are many proofs that are standard finite with respect to any of the
alternative foundations.
Meanwhile, I find it very interesting to consider the situation
where different foundational systems disagree on what is provable.
In very recent work of mine, for example, we are looking at the
theory of set-theoretic and arithmetic potentialism, where
different foundational systems disagree on what is true or
provable.
For example, recently with Hugh Woodin, I have proved that there is
a universal finite set $\{x\mid\varphi(x)\}$, a set that ZFC proves
is finite, and which is empty in any transitive model of set
theory, but if the set is $y$ in some countable model of set theory
$M$ and $z$ is any finite set in $M$ with $y\subset z$, then there
is a top-extension of $M$ to a model $N$ inside of which the set is
exactly $z$. The key to the proof is playing with the non-absolute
nature of truth between $M$ and its various top-extensions.
I've noticed that recently you have asked a few questions about my work, and so let me thank you; you are kind to take an interest.
This particular question can be seen as part of the subject of set-theoretic potentialism, which my co-author Øystein Linnebo and I recently investigated in our paper:
In that paper, we consider several different notions of set-theoretic accessibility, as below, from forcing accessibility to Grothendieck-Zermelo potentialism to rank extensions or transitive extensions or submodel potentialism and others.
In each case, for each concept of accessibility we determine the corresponding modal logic of that concept of potentialism. In other related work, W. Hugh Woodin and I recently looked at the case of top-extensional set-theoretic potentialism, proving that the modal logic is S4.
(You may also be interested in my more philosophical remarks on this topic at the end of my paper, The modal logic of arithmetic potentialism and the universal algorithm.)
Nevertheless, despite all that, I am very sorry to say that none of these cases are exactly the case you asked about, which is outer-model accessibility. Outer-model potentialism is certainly a natural case of potentialism, and so let me try to tell you what I know about it. Like forcing potentialism, this would be a case of width potentialism and height actualism, since outer models increase only the width of the universe and not the height.
First, let us fix a countable model of ZFC plus V=L, say, and consider it in the context of all its outer models. Since we can force so as to destroy any stationary set in this model, while preserving others, we see that there is an infinite family of independent buttons, "the $n^{th}$ stationary set in the $L$-least partition of $\omega_1^L$ is no longer stationary." This can be made true in an outer model (by forcing) and once true, remains true in all further outer models; and the statements can be controlled independently. And since we also have a family of independent switches arising from the GCH patterns, it follows by the main modal logic analysis (as in Structural connections...), it follows that the modal logic of outer-models is contained within S4.2. And it certainly contains S4, because it is reflexive and transitive.
So the answer to your question this a modal logic between S4 and S4.2.
I don't actually know which side it will end up on, or if it will end up strictly between, and I think this is a good question.
If you hope to prove that the model logic is S4.2, then usually one does this by proving an directedness or amalgamation theorem. The problem here, however, is that we already know that outer model possibility is not directed, since you can move from a model to outer models in incompatible ways that cannot be amalgamated (this is due to Mostowski). So that avenue of showing (.2) is valid is closed off. I don't know if (.2) is valid for this notion of possibility or not, but I am inclined against it. I think there will be fundamentally incompatible possibily necessary statements, which is to say, railway switches, and this will be incompatible with S4.2.
If you hope to prove that only S4 is valid, then you could follow some of the recent work on the universal algorithm and universal finite sets. The general consequence of the existence of these finite sequences with the universal extension property is that they cause the existence of railyard labelings, which then cause the modal logic to be contained in S4. I don't know if there is any universal finite sequence phenomenon for outer models, and this also is an interesting question.
So I believe that the exact modal logic of outer-model potentialism is an open question.
Best Answer
How kind of you to take an interest in my paper. Please see also my blog post about the dream solution and the arxiv entry for the paper.
First, I shall make a quibble, and then I'll address your question at the end.
The quibble is that your quotation from the paper is not accurate. The full paragraph from the paper reads:
The difference is that it should say "extensive experience of the contrary" rather than "extensive evidence of the contrary", a difference that affects the meaning, since the point is that we have experience in both the CH and in the $\neg$CH worlds. In particular, there is a symmetry here, and I hope it was clear that implicitly include your variation as part of my intended meaning.
Now, let me consider your final question, which is very good.
I take the answer to be yes, these views are made coherent by what I have called the universe view in my paper The set-theoretic multiverse, from which the dream solution paper is adapted. The universe view is the view I am arguing against, and although I have attacked the universe view for being mistaken, I do not attack it as incoherent. The question is whether the alternative set-theoretic universes that we seem to have discovered via forcing and other methods exist as legitimate concepts of set or not. I have argued at length that they do. But the opposing universe view is that no, there is just one absolute background concept of set, and the purpose of set theory is to discover what is true there. This seems to be a perfectly coherent view. It is a view advanced explicitly by Daniel Isaacson, who I quote extensively in my dream solution paper, and also by Donald Martin, in his paper "Multiple universes of sets and indeterminism in set theory", Topoi 20, 5--16, 2001, among others.
Criticizing my argument, Peter Koellner has emphasized that one can view my account of the naturalist account of forcing, rather than providing evidence that forcing extensions are real, instead as the desired explanation of the illusion of forcing extensions of $V$. And perhaps this criticism is the detailed answer to your question. That is, Koellner argues that the details of the proof of the naturalist account of forcing is how one explains away the illusion of forcing. So that would seem to be a coherent view. My reply to that argument, in my multiverse paper, is that such an account of forcing seems fundamentally crippling to our mathematical intuition, if we must regard all talk of actual forcing extensions of $V$ as ever-more-fantastical simulations of the extensions inside $V$, something like the writings of the exotic-travelogue writer who never actually ventures west of sixth avenue, or the absurdity of the mathematician who insists that yes, the real numbers exist with a full Platonic existence, but the complex numbers do not; they must be simulated inside the reals, such as with ordered pairs. The multiverse perspective makes a philosophically simple position, taking the existence of the forcing extensions at face value, while nurturing a robust use of forcing that will ultimately aid our set-theoretical understanding.
Finally, let me say that I agree completely with Andrej's point about geometry, and I discuss this analogy in section 4 of my multiverse paper.