[FOM] 280: Generating large cardinals/self embedding axioms

Harvey Friedman friedman at math.ohio-state.edu
Tue May 2 04:55:55 EDT 2006

```During recent discussions with set theorists concerning the possibility of
simple axioms generating large cardinals, I think that some work by me may
have been overlooked. Also see at the end, a particularly nice looking self
embedding axiom.

I first turn to the rather dense abstract

Restrictions and Extensions, February 17, 2003, 3 pages, draft.

My website is at

http://www.math.ohio-state.edu/%7Efriedman/

The only results that have been published concern:

THEOREM 1.2. The following are equivalent.
i) for all sufficiently large kappa, E(kappa,fg);
ii) for all sufficiently large k, E(kappa,omega);
iii) there exists a measurable cardinal.

There is a corresponding result concerning the ³turning
point².

THEOREM 1.3. The following are equivalent.
i) kappa is least such that for all lambda >= kappa, E(lambda,fg);
ii) kappa is least such that for all lambda >= kappa, E(lambda,omega);
iii) kappa is the least measurable cardinal.

The work appears in

Working with Nonstandard Models, in: Nonstandard Models of Arithmetic and
Set Theory, American Mathematical Society, ed. Enayat and Kossak, 71-86,
2004.

But I think that the work appearing there only treats ii),iii), and not the
fg = finitely generated case, if I recall (I have to go look back at it).

Other results there concern such large cardinal hypotheses as: weakly
compact cardinals, extendible cardinals, and elementary embeddings from
V(kappa+1) into itself. These have not been published.

So my approach reported there is very systematic and produces a diverse
level of large cardinals, including some of the most extreme such.

Note that for the large cardinals going far beyond measurables, in this
approach we use the language of second order logic.

I had an earlier approach that does not use languages but gets to the
extreme cardinals, beyond the huge cardinal hierarchy. See

`Combinatorial set theoretic statements of great logical strength', 1995, 4
pages, abstract.

also at

http://www.math.ohio-state.edu/%7Efriedman/

###A NEW SELF EMBEDDING AXIOM IN CLASS THEORY AND SET THEORY###

An ordered binary function is a triple (A,R,f), where A is a class, R is a
transitive relation on A, and f:A2 into A.

An initial segment of (A,R,f) is a (B,R|B,f|B), where B is a subclass of A
and f:B2 into B.

We say that (A,R,f) is self embeddable if and only if there is a class
bijection f:A into A which preserves R and f, and which is not surjective.

We say that (A,R,f) is class like if A is a proper class, and the R
predecessors of any element of A forms a set.

We say that (A,R,f) is kappa like if A has cardinality kappa, and the R
predecessors of every element of A has cardinality < kappa.

SELF EMBEDDING AXIOM (classes). Every class like surjective ordered binary
function has a self embeddable surjective initial segment.

SELF EMBEDDING AXIOM (sets). There is a cardinal kappa such that the
following holds. Every kappa like ordered binary function has a self
embeddable initial segment.

These axioms imply ranks into themselves, and follow from j:V into M with
V(lambda) contained in M.

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

manuscripts. This is the 278th 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

Harvey Friedman

```