FOM: 41:Strong Philosophical Iniscernibles

Wed May 19 15:27:00 EDT 1999

This is the 40th in a series of self contained postings to fom covering a
wide range of topics in f.o.m. Previous ones are:

1:Foundational Completeness   11/3/97, 10:13AM, 10:26AM.
2:Axioms  11/6/97.
3:Simplicity  11/14/97 10:10AM.
4:Simplicity  11/14/97  4:25PM
5:Constructions  11/15/97  5:24PM
6:Undefinability/Nonstandard Models   11/16/97  12:04AM
7.Undefinability/Nonstandard Models   11/17/97  12:31AM
8.Schemes 11/17/97    12:30AM
9:Nonstandard Arithmetic 11/18/97  11:53AM
10:Pathology   12/8/97   12:37AM
11:F.O.M. & Math Logic  12/14/97 5:47AM
12:Finite trees/large cardinals  3/11/98  11:36AM
13:Min recursion/Provably recursive functions  3/20/98  4:45AM
14:New characterizations of the provable ordinals  4/8/98  2:09AM
14':Errata  4/8/98  9:48AM
15:Structural Independence results and provable ordinals  4/16/98
10:53PM
16:Logical Equations, etc.  4/17/98  1:25PM
16':Errata  4/28/98  10:28AM
17:Very Strong Borel statements  4/26/98  8:06PM
18:Binary Functions and Large Cardinals  4/30/98  12:03PM
19:Long Sequences  7/31/98  9:42AM
20:Proof Theoretic Degrees  8/2/98  9:37PM
21:Long Sequences/Update  10/13/98  3:18AM
22:Finite Trees/Impredicativity  10/20/98  10:13AM
23:Q-Systems and Proof Theoretic Ordinals  11/6/98  3:01AM
24:Predicatively Unfeasible Integers  11/10/98  10:44PM
25:Long Walks  11/16/98  7:05AM
26:Optimized functions/Large Cardinals  1/13/99  12:53PM
27:Finite Trees/Impredicativity:Sketches  1/13/99  12:54PM
28:Optimized Functions/Large Cardinals:more  1/27/99  4:37AM
28':Restatement  1/28/99  5:49AM
29:Large Cardinals/where are we? I  2/22/99  6:11AM
30:Large Cardinals/where are we? II  2/23/99  6:15AM
31:First Free Sets/Large Cardinals  2/27/99  1:43AM
32:Greedy Constructions/Large Cardinals  3/2/99  11:21PM
33:A Variant  3/4/99  1:52PM
34:Walks in N^k  3/7/99  1:43PM
35:Special AE Sentences  3/18/99  4:56AM
35':Restatement  3/21/99  2:20PM
36:Adjacent Ramsey Theory  3/23/99  1:00AM
37:Adjacent Ramsey Theory/more  5:45AM  3/25/99
38:Existential Properties of Numerical Functions  3/26/99  2:21PM
39:Large Cardinals/synthesis  4/7/99  11:43AM
40:Enormous Integers in Algebraic Geometry  5/17/99  11:07AM

A complete archiving of fom, message by message, is available at
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Also, my series of self contained postings (only) is archived at
http://www.math.ohio-state.edu/foundations/manuscripts.html

FAVORITE SELF CONTAINED POSTINGS: 21, 25, 27, 31, 32, 34, 35', 37, 38, 39,
40, 41.

RE: #40. When I wrote J(R,4,4) I meant J(C,4,4). At that time I was
thinking about how to get the lower bounds for the reals instead of the
complexes. I have figured out how to do this. Note that for smaller fields,
the lower bound results are sharper since J(F,n,m) decreases (>=) as the
field F decreases.

In fact, I can now do the lower bounds over the field of rationals -
getting the same lower bound results. Moreover, this works even over the
ring of integers! In 4 dimensions, as you increase the degree of the
starting algebraic set, you exponentiate the lower bound. I want to revisit
this carefully, because I still hold out some hope that I can get something
dramatic in 3 dimensions.
********************

I considered the question of what sentences in predicate calculus with
equality are valid in all structures in which the domain is absolutely
everything. The basic results are:

1. This is the same as the sentences provable in predicate calculus with
the axioms asserting that there are infinitely many objects if and only if
there is a linear ordering of the universe of all things.

2. There is a simple principle going in the opposite direction which
directly contradicts that there is a linear ordeirng of the universe, and
which completely solves this problem in a different way. This is the
principle of symmetric arguments, which asserts that for any k-ary
predicate on the universe  there are k distinct objects such that the
predicate holds or fails of all permutations of these k objects.

An extended abstract of this work, "A Complete Theory of Everything," is on
my website. There will be a major revised version within a week.

THe principle of symmetric objects can be viewed as an indiscernibility
principle and is cited by Kit Fine as having a philosophically plausible
justification according to his theory of arbitrary objects.

This suggests the following program. Find strong principles of
indiscernibility which are enough to demonstrate the consistency of ZFC,
possibly together with large cardinal axioms, which are sufficiently
natural and fundamental from a philosophical point of view that they might
be subjected to philosophical analysis - as Fine did with the principle of
symmetric arguments. This posting reports steps in this direction.

The first version of strong indiscernibility, we use =, epsilon, and the
monadic predicate IN (for being an indiscernible). The axioms are

i) extensionality, pairing, union, and separation with respect to all
formulas in the expanded language;
ii) for all x there exists y such that x in y and IN(y);
iii) for all phi(x) with only the free variable shown and no occurrence of IN:
for all x,y in the extension of IN and of the same finite cardinality,
phi(x) iff phi(y).

[We could add a primitive for "x,y are of the same finite cardinality,"
add obvious axioms for it, and insist that this primitive can be used in
the separation scheme. This is an alternative to defining "x,y are of the
>ame finite cardinality" using axioms i)].

This proves the consistency of ZFC + a cardinal just a little bit lower
than a Ramsey cardinal (i.e., kappa arrows kappa in the partition
calculus).

If we strengthen iii) to:

iv) for all phi(x) with only the free variable shown and no occurrence of
IN: for all x,y in the extension of IN and of the same at most
countable cardinality, phi(x) iff phi(y),

then we derive the consistency of ZFC + much larger cardinals; e.g.,
>measurable cardinals of high order.

There is  a way of eliminating a condition like ii). We can assert that
indiscernibles like the extension of IN can be obtained as a proper
subclass of any proper class. In this purely set theoretic setup,
we give this alternative formulation as follows. For each formula phi(x)
with only the free variable shown and using at most =,epsilon, we
introduce a unary predicate symbol IN_phi.

i) extensionality, pairing, union, and separation with respect to all
formulas in the expanded language;
ii) for all phi(x) with only the free variable shown, if there is no set
which includes the extension of phi then the extension of IN_phi is
included in the extension of phi and also there is no set which includes
the extension of IN_phi;
iii) for all psi(x) with only the free variable shown and no occurrences
of unary predicates: for all x,y included in the extension of IN_phi of the
same finite cardinality, psi(x) iff psi(y).

We can also strengthen iii) to:

iv) for all psi(x) with only the free variable shown and no occurrences of
unary predicates: for all x,y included in the extension of IN_phi and of
the same at most countable cardinality, psi(x) iff psi(y),

The results are the same as for the first two versions.

The mathematically cleanest of all formulations is perhaps in class
theory. We have variables over classes, and a unary predicate symbol M(x)
for "x is a set." We have extensionality for classes, pairing for sets,
union for sets, every element of a class is a set, the intersection of a
class and a set is a set, and separation for all formulas in the language
that do not have any bound class variables. A proper class is a class
which is not a set.

The additional axiom is: Let F:A into x, where A is a proper class and x
is a set. There exists a proper class B contained in A such that for all n
>= 2, F is constant on the n element subsets of B.

We can strengthen this to:

Let F:A into x, where A is a proper class and x is a set. There exists a
proper class B contained in A such that for F agrees at any two at most
countable subsets of B of the same cardinality.

And again we get the exact same results.