# [FOM] 283: A theory of indiscernibles

Harvey Friedman friedman at math.ohio-state.edu
Sun May 7 18:42:41 EDT 2006

We give a simple theory of indiscernibles IND, and show that it interprets
an extension of ZFC with a large cardinal axiom, and is interpretable in an
extension of ZFC with a somewhat stronger large cardinal xiom.

IND could be subject to interesting philosophical interpretations.

IND is a two sorted system with the following language. One sort is for
objects, a second sort is for binary relations on objects.

Here is the language of IND.

i. variables over objects. We use x,x1,x2,... .
ii. variables over binary relations on objects. We use R,R1,R2,... .
iii. = between objects.
iv. the ternary relation symbol HD (holds of) between a binary relation on
objects, an object, and an object.
v. the binary function symbol CH (choose) from binary relations on objects
and objects, into objects.
vi. the binary relation symbol < on objects.

The axioms of IND are as follows.

1. Relation extensionality. (forall x,y)(HD(R1,x,y) iff HD(R2,x,y)) implies
R1 = R2.

2. Relation comprehension. (therexists R)(forall x1,x2)(HD(R,x1,x2) iff
phi), where phi is a formula in the language of IND in which R is not free.

3. Choice. HD(R,x,y) implies HD(R,x,CH(R,x)).

4. Linearity. not x < x, (x < y and y < z) implies x < z, ((x < z or z < x)
and (y < w or w < y)) implies (x < y or y < x or x = y).

5. Indiscernibility. Let phi be a formula in the language of IND with at
most the free variable x3, in which x1,x2 do not appear. x1 < x2 < x3
implies (phi[\x1] iff phi[\x2]).

We need to explain the notation phi[\xi], where xi does not appear in phi.
Informally, phi[\xi] results from phi by deleting xi in <. Formally, we
replace every subformula of phi of the form

xj < xk

by

the formula xj < xk and xj,xk not= xi.

THEOREM 1. IND is interpretable in ZFC + "there exists a measurable
cardinal" and interprets ZFC + "there exists a Ramsey cardinal".

**********************************

manuscripts. This is the 283rd 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 #1-249 can be found at
http://www.cs.nyu.edu/pipermail/fom/2005-June/008999.html in the FOM
archives, 6/15/05, 9:18PM. NOTE: The title of #269 has been corrected from
the original.

250. Extreme Cardinals/Pi01  7/31/05  8:34PM
251. Embedding Axioms  8/1/05  10:40AM
252. Pi01 Revisited  10/25/05  10:35PM
253. Pi01 Progress  10/26/05  6:32AM
254. Pi01 Progress/more  11/10/05  4:37AM
255. Controlling Pi01  11/12  5:10PM
256. NAME:finite inclusion theory  11/21/05  2:34AM
257. FIT/more  11/22/05  5:34AM
258. Pi01/Simplification/Restatement  11/27/05  2:12AM
259. Pi01 pointer  11/30/05  10:36AM
260. Pi01/simplification  12/3/05  3:11PM
261. Pi01/nicer  12/5/05  2:26AM
262. Correction/Restatement  12/9/05  10:13AM
263. Pi01/digraphs 1  1/13/06  1:11AM
264. Pi01/digraphs 2  1/27/06  11:34AM
265. Pi01/digraphs 2/more  1/28/06  2:46PM
266. Pi01/digraphs/unifying 2/4/06  5:27AM
267. Pi01/digraphs/progress  2/8/06  2:44AM
268. Finite to Infinite 1  2/22/06  9:01AM
269. Pi01,Pi00/digraphs  2/25/06  3:09AM
270. Finite to Infinite/Restatement  2/25/06  8:25PM
271. Clarification of Smith Article  3/22/06  5:58PM
272. Sigma01/optimal  3/24/06  1:45PM
273: Sigma01/optimal/size  3/28/06  12:57PM
274: Subcubic Graph Numbers  4/1/06  11:23AM
275: Kruskal Theorem/Impredicativity  4/2/06  12:16PM
276: Higman/Kruskal/impredicativity  4/4/06  6:31AM
277: Strict Predicativity  4/5/06  1:58PM
278: Ultra/Strict/Predicativity/Higman  4/8/06  1:33AM
279: Subcubic graph numbers/restated  4/8/06  3:14AN
280: Generating large caridnals/self embedding axioms  5/2/06  4:55AM
281: Linear Self Embedding Axioms  5/5/06  2:32AM
282: Adventures in Pi01 Independence  5/7/06

Harvey Friedman