[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".
**********************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
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
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