[FOM] 308:Large large cardinals
joeshipman@aol.com
joeshipman at aol.com
Thu Jul 5 16:42:16 EDT 2007
How do you express
"the existence of a nontrivial elementary embedding j:V into M, where
V(lambda)
containedin M."
as a statement or scheme in ZFC, that refers only to sets and not
classes?
-- JS
-----Original Message-----
From: Harvey Friedman <friedman at math.ohio-state.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>
Sent: Thu, 5 Jul 2007 5:01 am
Subject: [FOM] 308:Large large cardinals
We present a new statement whose strength is between the existence of a
nontrivial elementary embedding of some V(lambda) into V(lambda), and
the
existence of a nontrivial elementary embedding j:V into M, where
V(lambda)
containedin M.
A partial linearly ordered product is a triple (A,<,dot), where (A,<)
is a
linear ordering and dot is a partial function from A^2 into A.
A subset of A is bounded if and only if there exists x in A which is >=
all
elements of A. A subset of A is unbounded if and only if it is not
bounded.
Let B be a set of tuples from A. We write fld(B) for the set of all
coordinates of elements of B. We say that B is bounded (unbounded) if
and
only if fld(B) is bounded (unbounded).
Let f:A^2 into A be partial. We treat f as a subset of A^3.
An initial self embedding of f is a partial function g:A into A such
that
i. dom(g) is a bounded subset of fld(f) which is downward closed.
ii. rng(g) containedin dom(g).
iii. g is strictly increasing.
iv. For all x,y,z in A, f(x,y) = z iff f(gx,gy) = gz.
We say that partial h:A into A is cross sectional if and only if there
exists x in A such that for all y,z in A,
h(y) = z iff f(x,y) = z.
PROPOSITION A. There exists a partial linearly ordered product in which
every unbounded definable partial binary function has a cross sectional
initial self embedding with a least fixed point and a least non fixed
point.
It is obvious by the completeness theorem for first order predicate
calculus
with equality that Proposition A is provably equivalent to a Pi01
sentence,
over WKL_0.
**********************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 308th 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
283: A theory of indiscernibles 5/7/06 6:42PM
284: Godel's Second 5/9/06 10:02AM
285: Godel's Second/more 5/10/06 5:55PM
286: Godel's Second/still more 5/11/06 2:05PM
287: More Pi01 adventures 5/18/06 9:19AM
288: Discrete ordered rings and large cardinals 6/1/06 11:28AM
289: Integer Thresholds in FFF 6/6/06 10:23PM
290: Independently Free Minds/Collectively Random Agents 6/12/06
11:01AM
291: Independently Free Minds/Collectively Random Agents (more) 6/13/06
5:01PM
292: Concept Calculus 1 6/17/06 5:26PM
293: Concept Calculus 2 6/20/06 6:27PM
294: Concept Calculus 3 6/25/06 5:15PM
295: Concept Calculus 4 7/3/06 2:34AM
296: Order Calculus 7/7/06 12:13PM
297: Order Calculus/restatement 7/11/06 12:16PM
298: Concept Calculus 5 7/14/06 5:40AM
299: Order Calculus/simplification 7/23/06 7:38PM
300: Exotic Prefix Theory 9/14/06 7:11AM
301: Exotic Prefix Theory (correction) 9/14/06 6:09PM
302: PA Completeness 10/29/06 2:38AM
303: PA Completeness (restatement) 10/30/06 11:53AM
304: PA Completeness/strategy 11/4/06 10:57AM
305: Proofs of Godel's Second 12/21/06 11:31AM
306: Godel's Second/more 12/23/06 7:39PM
307: Formalized Consistency Problem Solved 1/14/07 6:24PM
Harvey Friedman
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