[FOM] Justifying SRP?
Rupert McCallum
rupertmccallum at yahoo.com
Mon Sep 15 05:57:51 EDT 2014
Let's think about reflection with second-order parameters and what happens if you try to allow third-order parameters. If we let X^(3) be a third-order parameter and let phi be the assertion that every element of X^(3) is a bounded subset of On, then we get easily see that for any cardinal kappa we could come up with a value of the parameter X^(3) such that kappa does not satisfy reflection for that formula and that parameter. Thus full reflection with third-order parameters is inconsistent. So if we are going to try to generalise reflection principles to higher-order parameters then we need some principled grounds for distinguishing the principles that are consistent from the ones that are inconsistent.
William Tait suggested the following idea Suppose that A is a parameter and we are reflecting a formula down from some cardinal kappa to some smaller cardinal beta. So we want to reflect down from the structure (V_kappa, V_(kappa+1), epsilon, A) to the structure (V_beta, V_(beta+1), epsilon, A^beta). But in the case where A is third or higher order, the inclusion map of the smaller structure into the bigger-structure is no longer single-valued. X \neq Y does not imply X^beta \neq Y^beta, and for A third-order and X \in V_(kappa+1), X \neq A does not imply X^beta \neq A^beta. So William Tait used this to motivate the idea that the formula to be reflected should only have "positive" components, it shouldn't contain the relation X=Y between second-order objects or the relation X \neq Y between second-order and third-order objects. So he used this to motivate the following set of reflection principles.
A higher-order formula is positive if it is built up by means of the operations or, and, \forall and \exists from atoms of the form x=y, x \neq y, x \in y, x \neq y, x \in Y x \notin Y, X = Y, and (X,...,Y)\in Z. Let \Gamma_n denote the set of all positive formulas of the form \forall X_1 \exists Y_1 ... \forall X_n \exists Y_n \psi, where \psi is first-order, the X_i are all second-order, and for i=1, ... n, \exists Y_j is a sequence \exists Z_j,1 ... \exists Z_j,m_j and the Z_j,i are variables of arbitrary types. A subclass B of the cardinal \kappa is n-reflective in \kappa if \forall X ... \forall Y [\phi(X,...,Y) implies \exists beta \in B \phi^beta(X^\beta, ... Y^\beta)] for all \phi(X,...,Y) in \Gamma_n with X, .... Y of any order at most one. Then the reflection principle is that there exists a cardinal kappa which is n-reflective in kappa. This reflection principle is provably consistent relative to a measurable (Peter Koellner later improved
this to kappa(omega)) and implies SRP.
But the trouble was that Tait also suggested that it should be okay to allow the variables following the universal quantifiers to be of order higher than second, and Peter Koellner showed that this leads to inconsistency. So then the difficulty becomes to find principled reasons why the reflection principle should be accepted in the first case but not the second.
My idea for how to explain this was to think of the formula to be reflected as a formula involving universal quantifiers only and also parameters which are Skolem functions. When you think about the formula in this way you once again get the issue that when you pass to the smaller substructure the Skolem functions are no longer single-valued. So I formulated a reflection principle which made the requirement that there should be a mapping j from the smaller substructure into the bigger substructure which guides the reflection, and that this mapping j should be used to choose what the argument of the Skolem functions should be. I was able to show that this was provably consistent relative to kappa(omega) and could be used to justify those of Tait's reflection principles that were consistent and some stronger ones as well. I suggested (tentatively) that this could be used as a motivation for why Tait's reflection principles should be regarded as justified
whereas the ones which are inconsistent should not be. The difficulty appears to be whether there are sound philosophical reasons for still regarding the reflection principle as justified when we use the mapping j to guide the reflection. Examining this would probably require a deeper look at why we regard ordinary reflection with second-order parameters as justified in the first place. Perhaps Tait and others would be interested in having a discussion about this.
I'll try to be a bit more complete about references to the literature.
William Tait, "Foundations of Set Theory", in the anthology "Truth in Mathematics". (But he later expressed dissatisfaction with some of the philosophical ideas in this essay and said that it should be regarded as superseded by the later essay).
William Tait, "Constructing Cardinals from Below", in the anthology "Provenance of Pure Reason".
Peter Koellner, "On Reflection Principles":
http://logic.harvard.edu/koellner/ORP_final.pdf
I also have an early draft of an essay about some stronger reflection principles as well as an attempt to discuss the underlying philosophical issues, but it is still in an early draft stage:
http://rupertmccallum.com/large_cardinals.pdf
On Monday, 15 September 2014, 0:57, Harvey Friedman <hmflogic at gmail.com> wrote:
http://www.cs.nyu.edu/pipermail/fom/2014-September/018168.html Rupert
McCallum wrote:
William Tait wrote an essay that appeared in "The Provenance of Pure
Reason" called "Constructing Cardinals from Below" which discussed a
set of reflection principles that justify SRP. Unfortunately Peter
Koellner later observed that some of the reflection principles he
considered were inconsistent. I wrote down my own thoughts in a recent
Mathematical Logic Quarterly article about how one might find
principled grounds for distinguishing the consistent ones from the
inconsistent ones.
http://onlinelibrary.wiley.com/doi/10.1002/malq.201200015/abstract
******************************
I'm sure that the FOM readers would be most interested if you could
give a simple brief account of the ideas behind some of the reflection
principles that work - at least in the sense that they can be
obtained from standard large cardinal hypotheses. Of course, subtle
cardinals themselves are based on a very simple idea - but that idea
would not normally be characterized as reflection.
For just subtle, we have
kappa is essentially subtle if and only if kappa is a cardinal such
that for all binary relations R on kappa, there exists infinite alpha
< beta < kappa such that the sections of R at alpha,beta agree below
alpha.
Note that essentially subtle is closed upward, so it is not quite the
same as being subtle. HOWEVER, the first subtle cardinal is exactly
the first essentially subtle cardinal. ALSO "there exists a subtle
cardinal" is equivalent to "there exists an essentially subtle
cardinal".
If FOM readers relate to your simple brief account, they can of course
delve into publications. FOM readers can also get a chance to interact
online starting from what you write.
Harvey
PS: Maybe I see how to do this using some arguable reflection using
multiple universes. Let's consider two universes V and V', where V' is
longer than V. Let's not worry about the most philosophically honest
way to formalize this just yet.
Let R be a binary relation on V and let phi be a sentence that holds
in (V',V,R). "Reflection" says that there exists kappa in V such that
phi holds in (V',V(kappa),R|V(kappa)). This seems to prove Con(ZFC +
"there exists a subtle cardinal"). I think that if you use
V,V',V'',V''',... then you will get Con(SRP).
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