[FOM] Re:Independence without forcing

Harvey Friedman friedman at math.ohio-state.edu
Fri Jul 11 12:09:28 EDT 2003

Reply to Eisworth 7/10/03 3:28PM.

>Todd Eisworth has asked about independence results in set theory that do
>not use forcing. Harvey Friedman has found numerous examples, mostly of a
>combinatorial nature, of propositions that are provable from the existence
>of large (but not very large) cardinals together with ZFC.

A large percentage of my postings concern this matter. I think well
over 100 such postings.

We are far beyond just quoting that Con(ZFC + V = L + there exists an
inaccessible cardinal) and Con(ZFC + V = L + there is no ianccessible
cardinal) are consistent.

The most stable of these results is the following. Let phi be the
most basic example of Boolean relation theory that I have discussed
many times on the FOM. The statement is entirely mathematical, and
involves only functions of several variables from the integers into
the integers.

Then phi is equivalent to the 1-consistency of ZFC + (there exists an
n-Mahlo cardinal)_n. Hence as a Corollary,

Con(ZFC + V = L + phi) and Con(ZFC + V = L + notphi) are both consistent.

I have also discussed many other related results. For example, the
6561 classification results, which also have the same
metamathematical status of being independent of ZFC. And also various
explicit forms and finite forms of phi, also with the same
metamathematical status. I have also discussed in detail the issue of
how Boolean relation theory fits into the general mathematical

Previous to this, there is the following paper that contains a
mathematical theorem involving only functions of several variables
from the integers into the integers, and also only finite functions
of this kind. However, the level of mathematical naturalness and the
connections with the general mathematical culture are not of the same
level as the more recent work involving Boolean relation theory.

H. Friedman, Finite functions and the necessary use of large
cardinals, Annals of Mathematics, 148 (1998), 803-893.

Also see

H. Friedman, Equational Boolean Relation Theory, 

which is a polished preprint.

For relevant FOM postings, see, for example, the following of my
numbered postings. In most cases, these contain examples of the kind
you are talking about.

   12:Finite trees/large cardinals  3/11/98  11:36AM
   13:Min recursion/Provably recursive functions  3/20/98  4:45AM
   16:Logical Equations, etc.  4/17/98  1:25PM
   16':Errata  4/28/98  10:28AM
   17:Very Strong Borel statements  4/26/98  8:06PM
   18:Binary Functions and Large Cardinals  4/30/98  12:03PM
   26:Optimized functions/Large Cardinals  1/13/99  12:53PM
   28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM
   28':Restatement  1/28/99  5:49AM
   29:Large Cardinals/where are we? I  2/22/99  6:11AM
   30:Large Cardinals/where are we? II  2/23/99  6:15AM
   31:First Free Sets/Large Cardinals  2/27/99  1:43AM
   32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM
   33:A Variant  3/4/99  1:52PM
   35:Special AE Sentences  3/18/99  4:56AM
   35':Restatement  3/21/99  2:20PM
   36:Adjacent Ramsey Theory  3/23/99  1:00AM
   37:Adjacent Ramsey Theory/more  5:45AM  3/25/99
   39:Large Cardinals/synthesis  4/7/99  11:43AM
   45:Result #1/Program A  7/14/99  11:07AM
   52:Cardinals and Cones  7/18/99  3:33PM
   57:Fixpoints/Summation/Large Cardinals  9/10/99  3:47AM
   57':Restatement  9/11/99  7:06AM
   58:Program A/Conjectures  9/12/99  1:03AM
   59:Restricted summation:Pi-0-1 sentences  9/17/99  10:41AM
   60:Program A/Results  9/17/99  1:32PM
   62:Approximate fixed points revisited  10/11/99  1:35AM
   63:Disjoint Covers/Large Cardinals  10/11/99  1:36AM
   64:Finite Posets/Large Cardinals  10/11/99  1:37AM
   67:Nested Min Recursion/Large Cardinals  10/25/99  8:00AM
   68:Bad to Worse/Conjectures  10/28/99  10:00PM
   72:New finite forms/large cardinals  12/12/99  6:11AM
   80.Boolean Relation Theory  3/10/00  9:41AM
   82:Simplified Boolean Relation Theory  3/15/00  9:23AM
   83:Tame Boolean Relation Theory  3/20/00  2:19AM
   84:BRT/First Major Classification  3/27/00  4:04AM
   85:General Framework/BRT   3/29/00  12:58AM
   88:Boolean Relation Theory  6/8/00  10:40AM
   89:Model Theoretic Interpretations of Set Theory  6/14/00 10:28AM
   95:Boolean Relation Theory III  12/19/00  7:29PM
   96:Comments on BRT  12/20/00  9:20AM
   98:Model Theoretic Interpretation of Large Cardinals  3/5/01  3:08PM
   99:Boolean Relation Theory IV  3/8/01  6:08PM
100:Boolean Relation Theory IV corrected  3/21/01  11:29AM
101:Turing Degrees/1  4/2/01  3:32AM
102: Turing Degrees/2  4/8/01  5:20PM
104:Turing Degrees/3   4/12/01  3:19PM
105:Turing Degrees/4   4/26/01  7:44PM
108:Finite Boolean Relation Theory   9/18/01  12:20PM
114:Borel Functions on HC  1/1/02  1:38PM
117:Discrepancy Theory   1/6/02  12:53AM
118:Discrepancy Theory/2   1/20/02  1:31PM
119:Discrepancy Theory/3  1/22.02  5:27PM
120:Discrepancy Theory/4  1/26/02  1:33PM
121:Discrepancy Theory/4-revised  1/31/02  11:34AM
124:Disjoint Unions  2/18/02  7:51AM
125:Disjoint Unions/First Classifications  3/1/02  6:19AM
126:Correction  3/9/02  2:10AM
127:Combinatorial conditions/BRT  3/11/02  3:34AM
128:Finite BRT/Collapsing Triples  3/11/02  3:34AM
129:Finite BRT/Improvements  3/20/02  12:48AM
130:Finite BRT/More  3/21/02  4:32AM
131:Finite BRT/More/Correction  3/21/02  5:39PM
132: Finite BRT/cleaner  3/25/02  12:08AM
133:BRT/polynomials/affine maps  3/25/02  12:08AM
134:BRT/summation/polynomials  3/26/02  7:26PM
135:BRT/A Delta fA/A U. fA  3/27/02  5:45PM
136:BRT/A Delta fA/A U. fA/nicer  3/28/02  1:47AM
137:BRT/A Delta fA/A U. fA/beautification  3/28/02  4:30PM
138:BRT/A Delta fA/A U. fA/more beautification  3/28/02  5:35PM
139:BRT/A Delta fA/A U. fA/better  3/28/02  10:07PM
140:BRT/A Delta fA/A U. fA/yet better  3/29/02  10:12PM
141:BRT/A Delta fA/A U. fA/grammatical improvement  3/29/02  10:43PM
142:BRT/A Delta fA/A U. fA/progress  3/30/02  8:47PM
143:BRT/A Delta fA/A U. fA/major overhaul  5/2/02  2:22PM
144: BRT/A Delta fA/A U. fA/finesse  4/3/02  4:29AM
145:BRT/A U. B U. TB/simplification/new chapter  4/4/02  4:01AM
146:Large large cardinals  4/18/02  4:30AM
147:Another Way  7:21AM  4/22/02
148:Finite forms by relativization  2:55AM  5/15/02
149:Bad Typo  1:59PM  5/15/02
150:Finite obstruction/statistics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02
153:Large cardinals as general algebra  1:21PM  6/17/02
156:Societies  8/13/02  6:56PM
157:Finite Societies  8/13/02  6:56PM
161:Restrictions and Extensions  3/31/03  12:18AM
170:New Borel Independence  5/18/03  11:53PM
171:Coordinate Free Borel Statements  5/22/03  2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals  5/34/03  1:55AM
173:Borel/DST/PD  5/25/03  2:11AM
186:Grand Unification 1
187:Grand Unification 2

>I've kept up with a tiny amount of Harvey Friedman's work in this area
>(mostly things about subtle cardinals and linear orderings). I guess the
>easiest answer to my question would be that ZFC + V=L neither proves nor
>refutes the existence of an inaccessible cardinal assuming that inaccessible
>cardinals are consistent with ZFC.

I would like to know how you got this entirely false impression.

>I am more interested in the hard version of the question --- assuming only
>CON(ZFC), how might one go about finding a "mathematical" statement P for
>which both CON(ZFC + V=L + P) and CON(ZFC + V=L + not P) hold? Does anyone
>have any speculations on a plausible scenario for establishing such results?
>Any guesses on what such a P should look like? Is it a waste of time to be
>concerned with such matters?

This can also be done, but is not generally worth the effort at this
time. One can obtain serious control over the strength of the
statements involved, although this is not trivial, and is certainly
not a high priority. The highest priority is to deepen the
connections with general mathematical culture.

Technically, what I do has Con(ZFC + V = L + notP) equivalent to
Con(ZFC), but Con(ZFC + V = L + P) equivalent to some form of the
consistency of some large cardinals. Most recently, in #173, the
statements involved reach "infinitely many Woodin cardinals" in

>I suspect that what I am asking for is (given the current state of our
>knowledge) the mathematical equivalent of science fiction, but I like to
>think science fiction has a hand in the development of science.

Science fiction is here!

>My interest in this stems from conversations with Shelah while I was in
>Jerusalem.  Some of his speculations are recorded in Section 9 of "On what I
>do not understand..." in Fund. Math. (166) 1 1-82. I thought it might be
>interesting if others who have thought out such matters shared their

It might be of interest to the FOM for you to explain what Shelah was
talking about.

Harvey Friedman

More information about the FOM mailing list