[FOM] 545: Set Theoretic Consistency/SRM/SRP

Sun Sep 14 22:06:59 EDT 2014

I am participating in a small email group concerning higher set
theory, focusing originally on whether the continuum hypothesis is a
"genuine" problem. The discussion has partly gone into issues
surrounding the foundations of higher set theory. The ideas in this
posting were inspired by the interchange there.

CAUTION: I am not an active expert in this kind of higher set theory,
and so what I say may be either known, partly true, or even false.

As a modification of existing ideas concerning "maximality" I offer
the following axiom over MK class theory with the global axiom of
choice.

DEFINITION 1. Let phi be a sentence of set theory. We say that phi is
set theoretically consistent if and only if ZFC + phi is consistent
with the usual axioms and rules of infinitary logic. At the minimum,
the standard axioms and rules of L(V,omega), with quantifiers ranging
over V, with epsilon,=, but one may consider the much stronger
language L(V,On) where set length blocks of quantifiers are used. I
think that for what I am going to do, it makes no difference, so that
there is stability here.

Then we can formulate in class theory,

POSTULATE A. If a sentence is set theoretically consistent then it
holds in some transitive model of ZFC containing all ordinals. It
follows that it holds in some L[x], x a real.

Now what happens if we relativize this in an obvious way?

DEFINITION 2. Let alpha be an infinite ordinal. Let phi be a sentence
of set theory. We say that phi is set theoretically consistent
relative to POW(alpha) if and only if ZFC + phi is consistent with the
usual axioms and rules of infinitary logic together with "all subsets
of alpha are among the actual subsets of alpha", where the latter is
formulated in the obvious way using infinitary logic.

POSTULATE B(alpha). If a sentence is set theoretically consistent
relative to POW(alpha) then it holds in some transitive model of ZFC
containing all ordinals.

POSTULATE C. Postulate B(alpha) holds for all infinite alpha.

To prove consistency, or consistency of this for some alpha, we seem to need

1. An extension of Jensen's coding the universe where we add a subset
of alpha+ that codes the universe without adding any subsets of alpha.
This has probably been done.
2. Cone determinacy for subsets of alpha+.E.g., using the equivalence
relation x == y if and only if x,y are interdefinable over V(alpha+).
I don't know if this has been done.

Turning now to SRM once again from
http://www.cs.nyu.edu/pipermail/fom/2014-September/018171.html, let's
look at

1. Linearly ordered integral domain axioms.
2. Finite interval. [x,y] exists.
3. Boolean difference. A\B exists.
4. Set addition. A + B = {x+y: x in A and y in B} exists.
5. Set multiplication. AxB = {xy: x in A and y in B} exists.
6. Least element. Every nonempty set has a least element.
7. n^0 = 1. m >= 0 implies n^(m+1) = n^m x n. n^m defined implies m >= 0.
8. {n^0,...,n^m} exists.
9. {0+n^0,...,m+n^m}.

I originally said that 1-8 is a conservative extension of EFA, which
is wrong. I corrected this by saying that 1-7,9 is a conservative
extension of EFA. There are some subtle points, and it is possible
that I might need some standard laws of exponentiation, maybe just
n^(m+r) = n^m dot n^r, or even less, BUT: much easier is to use the
full 1-9. 1-9 is a conservative extension of EFA. I really like the
Foundational Traction here, as I unravel the subtleties of the
situation properly.

Now to justifications of SRP.

On Sep 13, 2014, at 2:18 AM, Rupert McCallum <rupertmccallum at yahoo.com> 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

Bill Tait wrote http://www.cs.nyu.edu/pipermail/fom/2014-September/018169.html

"Thanks for the announcement, Rupert; I look forward to reading the paper.

In the interests of immodesty, let me mention that here is a bit of
unclarity: I considered reflection principles G^m_n for m, n < omega
and used the G^m_2 to derive the existence of m-ineffable cardinals. I
also proved the G^m_2 consistent relative to a measurable. Peter
showed that they are consistent relative to kappa(omega).

What was unfortunate was certainly not that Peter found the G^m_n
inconsistent for n>2, rather it was my proposing them."

This illustrates a systematic problem with attempts at philosophical
foundations of higher set theory. One formulates a reasonable looking
idea, makes some associated philosophy, and obtains some partial
information. Then perhaps somebody shows there is an inconsistency.
Then one adjusts the philosophy to explain why the stuff that seems
fine was a good idea, and the stuff that turned sour was, in
retrospect, a bad idea.

Of course, so far, we have seen that proposals that survive a lot of
attention and work, especially detailed structure (not so clear what
this means for proposals compatible with V = L), invariably have not,
so far, led to inconsistencies. Kunen with Reinhardt's j:V into V
wasn't around too long, and didn't have any structure theory. However,
I1,I2,I3 have been around a long time, but not too much structure
theory, still a fairly substantial amount of research. What confidence
should we have that they are consistent?

I don't have any problem with this process as research - there is
little or no alternative.  BUT, in order to respond to inquiries like
"why should I believe subtle cardinals exist, or that they are
consistent with ZFC?" from top core mathematicians, it is not going to
be well received in anything like its present form. Probably also my
approach with V,V',V'',V''',... is still not innovative enough.

Having said this, it still appears that there is a kind of comfort
zone with ZFC. But I explain this as follows. ZF is a kind of unique
transfer from the finite world, and I have reported on this some time
ago on the FOM.

I have just seen http://rupertmccallum.com/large_cardinals.pdf which
FOM readers may find of interest. It also contains a good list of
references. A quick glance reminds me of the SRP characterization in
terms of countable transitive models with outside elementary
embeddings.

My feeling at the moment is that this area of justifying cardinals
relatively low down (SRP hierarchy = subtle cardinal hierarchy =
ineffable cardinal hierarchy) needs a really striking simple new idea.

Harvey Friedman

****************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 545th in a series of self contained numbered
postings to FOM covering a wide range of topics in f.o.m. The list of
previous numbered postings #1-527 can be found at the FOM posting
http://www.cs.nyu.edu/pipermail/fom/2014-August/018092.html

528: More Perfect Pi01  8/16/14  5:19AM
529: Yet more Perfect Pi01 8/18/14  5:50AM
530: Friendlier Perfect Pi01
531: General Theory/Perfect Pi01  8/22/14  5:16PM
532: More General Theory/Perfect Pi01  8/23/14  7:32AM
533: Progress - General Theory/Perfect Pi01 8/25/14  1:17AM
534: Perfect Explicitly Pi01  8/27/14  10:40AM
535: Updated Perfect Explicitly Pi01  8/30/14  2:39PM
536: Pi01 Progress  9/1/14 11:31AM
537: Pi01/Flat Pics/Testing  9/6/14  12:49AM
538: Progress Pi01 9/6/14  11:31PM
539: Absolute Perfect Naturalness 9/7/14  9:00PM
540: SRM/Comparability  9/8/14  12:03AM
541: Master Templates  9/9/14  12:41AM
542: Templates/LC shadow  9/10/14  12:44AM
543: New Explicitly Pi01  9/10/14  11:17PM
544: Initial Maximality/HUGE  9/12/14  8:07PM

Harvey Friedman