FOM: 153:Large cardinals as general algebra

Sat Jun 15 23:21:43 EDT 2002

By an algebra we will mean a nonempty set (the domain) together with a finite 
list of functions from the domain into itself, of various finite arities >= 0. 
An (proper) extension of an algebra A is an algebra B such that A is a (proper) 
subalgebra of B. 

MY FAVORITE IS PROPOSITION 6.

UNDERLYING PHILOSOPHICAL PRINCIPLE: every sufficiently large/complex system can 
be enlarged/complexified without changing its local character.

I have already worked with something like this principle to give axioms for two 
communicating minds, or the latest version - the expanding mind, which 
correspond to large cardinals. The latest informal account is in my recent 
Goedel lecture given in Las Vegas, June 2, 2002, by Ted Slaman. 

This general algebraic work here is in the context of ordinary set theory, and 
so it can be presented in the simplest possible general algebraic terms, 
without resorting to predicate calculus schemes, etcetera. 

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THEOREM 1. Every algebra with infinite domain has a proper extension with the 
same finite subalgebras up to isomorphism. In fact, it begins an infinite 
sequence of successive proper extensions all with the same finite subalgebras 
up to isomorphism.

Proposition 1 follows from the compactness theorem. 

PROPOSITION 2. There is a domain such that every algebra on that domain has a 
proper extension with the same countable (finitely generated, finitely 
presented) subalgebras up to isomorphism. 

Having the same countable, finitely generated, finitely presented, finite, 
subalgebras up to isomorphism is hereby called countable, local, presentable, 
finite, equivalence. 

PROPOSITION 3. There is a domain such that every algebra on that domain has a 
proper extension, with the same subalgebras up to countable (local, 
presentable, finite) equivalence. 

PROPOSITION 4. There is a domain such that every algebra on that domain begins 
an infinite chain of proper extensions, where all algebras in the chain have 
the same subalgebras up to countable (local, presentable, finite) equivalence. 

THEOREM 5. All three forms of Proposition 2 follow from a weakly compact 
cardinal and prove "there is a weakly compact cardinal in L", over ZFC. Thus 
ZFC + Proposition 2 is mutually interpretable and equiconsistent with ZFC + 
"there is a weakly compact cardinal". All four forms of Proposition 3 and all 
four forms of Proposition 4 follow from a subtle cardinal and prove "there are 
lots of weakly compact cardinals in L", over ZFC. They are strictly weaker than 
a subtle cardinal. 

PROPOSITION 6. There is a domain such that every algebra on at least that 
domain has a proper extension with the same countable (finitely generated, 
finitely presented) subalgebras up to isomorphism. 

PROPOSITION 7. There is a domain such that every algebra on at least that 
domain has a proper extension with the same subalgebras up to countable (local, 
presentable, finite) equivalence. 

PROPOSITION 8. Let n >= 1. There is a domain such that every algebra on at 
least that domain begins a chain of proper extensions of length n, where all 
algebras in the chain have the same subalgebras up to countable (local, 
presentable, finite) equivalence. 

PROPOSITION 9. There is a domain such that every algebra on at least that 
domain begins an infinite chain of proper extensions, where all algebras in the 
chain have the same subalgebras up to countable (local, presentable, finite) 
equivalence. 

THEOREM 10. All three forms of Proposition 6 are equivalent to "there exists a 
measurable cardinal" over ZFC. All four forms of Propositioni 7 are equivalent 
to "there exists an extendible cardinal" over ZFC. All four forms of 
Proposition 8 are equivalent to "for all n, there is an n-huge cardinal" over 
ZFC. All four forms of Proposition 9 prove the existence of arbitrarily large 
ranks that can be properly elementarily embedded in themselves, and follow from 
j:V into M, where M contains the rank on the first fixed point above the 
critical point. 

We can weaken the base theory ZFC in the discussion to a critical fragment of 
Zermelo set theory with the axiom of choice.

POSSIBLE APPLICATION OF THESE IDEAS: speculative cosmology. the universe is 
expanding without local changes. 

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I use http://www.mathpreprints.com/math/Preprint/show/ for manuscripts with
proofs. Type Harvey Friedman in the window.

This is the 153rd in a series of self contained postings to FOM covering
a wide range of topics in f.o.m. Previous ones counting from #100 are:

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
103:Hilbert's Program for Consistency Proofs/1 4/11/01  11:10AM
104:Turing Degrees/3   4/12/01  3:19PM
105:Turing Degrees/4   4/26/01  7:44PM
106.Degenerative Cloning  5/4/01  10:57AM
107:Automated Proof Checking  5/25/01  4:32AM
108:Finite Boolean Relation Theory   9/18/01  12:20PM
109:Natural Nonrecursive Sets  9/26/01  4:41PM
110:Communicating Minds I  12/19/01  1:27PM
111:Communicating Minds II  12/22/01  8:28AM
112:Communicating MInds III   12/23/01  8:11PM
113:Coloring Integers  12/31/01  12:42PM
114:Borel Functions on HC  1/1/02  1:38PM
115:Aspects of Coloring Integers  1/3/02  10:02PM
116:Communicating Minds IV  1/4/02  2:02AM
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
122:Communicating Minds IV-revised  1/31/02  2:48PM
123:Divisibility  2/2/02  10:57PM
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/statisics  8:55AM  6/1/02
151:Finite forms by bounding  4:35AM  6/5/02  
152:sin  10:35PM  6/8/02