[FOM] 245:Relational System Theory 1
Harvey Friedman
friedman at math.ohio-state.edu
Mon May 16 12:24:40 EDT 2005
RELATIONAL SYSTEM THEORY
ABSTRACT
by
Harvey M. Friedman
http://www.math.ohio-state.edu/%7Efriedman/
May 15, 2005
1. Introduction.
2. Ternary.
3. Systems, Subsystems, Reductions, Full Systems, Complete Systems.
4. Results.
5. Schematic Versions.
1. Introduction.
Here we present some formal systems concerning ternary relations which
relate to informal conceptual ideas that are arguably more fundamental than
those that drive modern set theory.
We establish mutual translations between these formal systems and various
systems of set theory, ranging from countable set theory, ZFC, to large
cardinal hypotheses such as the existence of a Ramsey cardinal, and the
existence of an elementary embeddings from V into V.
The primitives are identity, and x[y,z,w]. The latter is read "x is a
ternary relation which holds at the objects y,z,w."
The use of x[y,z,w] rather than the more usual y epsilon x has many
advantages for this work. One of them is that we have found a convenient way
to eliminate any need for axiom schemes. All axioms considered are single
sentences with clear meaning. (In one case only, the axiom is a conjunction
of a manageable finite number of sentences).
The theories are single sorted, and are based on the idea that some objects
appear as arguments of other objects (ternary relations), whereas some
objects do not so appear. The former objects are called arguments or
argumental, whereas the remaining objects are called nonarguments or
nonargumental. This is totally analogous to one sorted formalizations of the
theory of classes where sets are defined to be classes that are an element
of a class.
The basic axioms form a system called Ternary, which are, with one minor
exception, all easy consequences of the axiom scheme asserting the
following. We can form an object x which holds of any argumental objects
y,z,w if and only if a given first order formula holds of y,z,w, where x
does not appear and where parameters for objects are allowed. The minor
exception is the axiom that asserts that there is an argumental x which
holds at no arguments. As indicated above, here we avoid any use of axiom
schemes.
Although Ternary is not strong enough to derive the above mentioned axiom
scheme, it does derive instances sufficient for our purposes when combined
with the main axioms.
The main axioms are stated in terms of what we call 3-systems. A 3-system
consists of three objects x,y,z, where x and y and z all have at least one
argument (nonemptiness), and where every two distinct arguments of any of x
or y or z combined, are "related" by x or y or z, using some third argument.
We use the most appropriate notion of subsystem of 3-systems, essentially
borrowed from model theory.
We use two different notions of reduction of 3-systems. A reduction of a
3-system is always required to be a subsystem.
One is a-reduction, where argumental components remain the same and
nonargumental components become argumental.
The other is na-reduction, where argumental components remain the same and
nonargumental components remain nonargumental, but are cut back.
All of the axioms considered take the following two general forms:
***Every 3-system (of a specific kind) has an a-reduction (of the same
specific kind).***
***Every 3-system (of a specific kind) has an na-reduction (of the same
specific kind).***
The axioms of the first form correspond to roughly ZF and fragments thereof.
The axioms of the second form correspond to large cardinal hypotheses
ranging from roughly Ramsey cardinals (just below measurable cardinals) and
the existence of an elementary embedding from V into V (over NBG without
choice).
Now let me now go somewhat beyond what has been established at this point.
I believe that set theory is the canonical mathematical limit of informal
common sense thinking. Let me explain with an example you are all familiar
with.
People, using common sense, think about, say, a full head of hair. They
think that if you remove one strand of hair from a full head of hair, then
it remains a full head of hair.
Scientific thinking has a problem with this. After all, one can perform a
thought experiment whereby the number of strands of hair is counted, and
pulled out one by one, until there are no more. Clearly complete baldness is
not a full head of hair.
At this point, set theory enters the picture. The idea of a full head of
hair is associated with the precise set theoretic notion of: infinite set.
It is provable in set theory that if you take an element out of an infinite
set, then it remains infinite. It is provable in set theory that infinite
sets are not empty. It is provable in set theory that infinite sets cannot
be numerically counted - the count never terminates.
There is a more sophisticated idea of this sort. There is the common sense
idea of a large system. Not just an inert clump like a head of hair, but
rather a large system with a number of interlocking components with
complicated internal connections. Like the physical universe, or like the
human body, or like the world of living organisms, or like the world
economic system.
There is the idea that in any large system, we can take something away
without the system falling apart.
In the case of the head of hair, it doesn't make any difference which strand
of hair you take away. Here, we are merely asserting that something can be
taken away - not that anything can be taken away. And I am not asserting
that you can get away with taking only one item away. It may be a
significant amount of stuff.
Furthermore, in any large system, we can take something away without the
system falling apart, and where the system remains "similar".
So what are the missing parts of this analogy?
1. Any large clump stays large after some (any) point removal
(infinite sets).
2. Any large system remains a large system after some portion removal (XXX).
3. Any large system remains a large system, unaffected, after some portion
removal (YYY).
4. Any large system remains unaffected after some expansion (ZZZ).
XXX ~ Jonsson cardinals ~ Ramsey cardinals.
YYY ~ ZZZ ~ elementary embedding axioms roughly around a rank into itself.
We have already backed up these statements with precise theorems in set
theory involving set theoretic structures of sufficiently large cardinality.
We call this subject, the theory of large algebras. Initial developments
along these lines have been published in [Fr04]. Some further developments
are implicit in the unpublished notes [Fr03].
The break point (how large is large?) depends on the notion of "unaffected"
one uses. In [Fr04}, the break point is the first measurable cardinal. The
break points discussed in [Fr03] are much higher. Some relevant notions are
language oriented, whereas others are more directly mathematical.
But the bold new idea here goes well beyond any theory of large algebras
within set theory. Here is what we anticipate:
A. There is a nonmathematical common sense oriented theory of systems and
components which corresponds to various well studied levels of set theory
with/without large cardinals.
B. More generally, one can formulate transparent principles of a plausible
nature about ANY common sense notions, which correspond to various well
studied levels of set theory with/without large cardinals.
A word of caution here. It may well be the case that in any region of common
sense thinking, if one goes far enough, one reaches outright contradictions.
That is to be expected. What we are anticipating is the designation of
formal systems closely associated with commonsense thinking, that are in
some ways extrapolations of commonsense thinking, and in other ways are
restrictions of commonsense thinking, which correspond exactly to various
levels of abstract set theory.
As we shall see, A is arguably already implicit in the development below,
where we focus entirely on the notion of a ternary relation. However, we
look forward to reworking the development here based on richer notions that
are an integral part of everyday thinking.
In fact, it can be argued that common sense thinking is incredibly richer,
logically, than mathematics or science. Of course, common sense thinking is
not subject to the same kind of deep and subtle constraints that mathematics
and science operate under. In a separate longstanding development, I
struggle, with increasing success, to find normal mathematics that requires
large cardinals. But I speculate that large cardinals are everywhere
implicit in common sense thinking.
We will hopefully get to the point where set theory with large cardinals
emerges as the one mathematical area which applies to just about everything
outside of science - across the board.
In fact, set theory with large cardinals may be to common sense thinking as
the Newton/Leibniz calculus is to science.
The "calculus" aspect of set theory with large cardinals is as follows.
There will be a proliferation of natural formal systems involving various
groups of common sense notions. One will want to know how these systems
compare under interpretability. One will see that, in fact, there is a quasi
linear ordering under interpretability. One will want to "calculate", for
any pair of such systems, how they compare in this quasi linear ordering.
The only way to make such comparisons will be to have a manageable set of
representatives for each level that arises, and first identify where each of
the two systems to be compared fits in.
The manageable set of representatives is, of course, just various well
studied levels of set theory with large cardinals.
So set theory with large cardinals may be the appropriate measuring tool for
the comparison of systems based on common sense notions. Perhaps it can then
emerge as the most generally and transparently useful area of modern
mathematics.
REFERENCES
[Fr03] `Restrictions and extensions', February 17, 2003, 3 pages, draft,
http://www.math.ohio-state.edu/%7Efriedman/manuscripts.html
[Fr04] Working with Nonstandard Models, in: Nonstandard Models of Arithmetic
and Set Theory, American Mathematical Society, ed. Enayat and Kossak, 71-86,
2004.
[Mi79] W. Mitchell, Ramsey Cardinals and Constructibility, JSL, Vol. 44, No.
2 (June 1979), 260-266.
*************************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 245th 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 #1-149 can be found at
http://www.cs.nyu.edu/pipermail/fom/2003-May/006563.html in the FOM
archives, 5/8/03 8:46AM. Previous ones counting from #150 are:
150:Finite obstruction/statistics 8:55AM 6/1/02
151:Finite forms by bounding 4:35AM 6/5/02
152:sin 10:35PM 6/8/02
153:Large cardinals as general algebra 1:21PM 6/17/02
154:Orderings on theories 5:28AM 6/25/02
155:A way out 8/13/02 6:56PM
156:Societies 8/13/02 6:56PM
157:Finite Societies 8/13/02 6:56PM
158:Sentential Reflection 3/31/03 12:17AM
159.Elemental Sentential Reflection 3/31/03 12:17AM
160.Similar Subclasses 3/31/03 12:17AM
161:Restrictions and Extensions 3/31/03 12:18AM
162:Two Quantifier Blocks 3/31/03 12:28PM
163:Ouch! 4/20/03 3:08AM
164:Foundations with (almost) no axioms 4/22/03 5:31PM
165:Incompleteness Reformulated 4/29/03 1:42PM
166:Clean Godel Incompleteness 5/6/03 11:06AM
167:Incompleteness Reformulated/More 5/6/03 11:57AM
168:Incompleteness Reformulated/Again 5/8/03 12:30PM
169:New PA Independence 5:11PM 8:35PM
170:New Borel Independence 5/18/03 11:53PM
171:Coordinate Free Borel Statements 5/22/03 2:27PM
172:Ordered Fields/Countable DST/PD/Large Cardinals 5/34/03 1:55AM
173:Borel/DST/PD 5/25/03 2:11AM
174:Directly Honest Second Incompleteness 6/3/03 1:39PM
175:Maximal Principle/Hilbert's Program 6/8/03 11:59PM
176:Count Arithmetic 6/10/03 8:54AM
177:Strict Reverse Mathematics 1 6/10/03 8:27PM
178:Diophantine Shift Sequences 6/14/03 6:34PM
179:Polynomial Shift Sequences/Correction 6/15/03 2:24PM
180:Provable Functions of PA 6/16/03 12:42AM
181:Strict Reverse Mathematics 2:06/19/03 2:06AM
182:Ideas in Proof Checking 1 6/21/03 10:50PM
183:Ideas in Proof Checking 2 6/22/03 5:48PM
184:Ideas in Proof Checking 3 6/23/03 5:58PM
185:Ideas in Proof Checking 4 6/25/03 3:25AM
186:Grand Unification 1 7/2/03 10:39AM
187:Grand Unification 2 - saving human lives 7/2/03 10:39AM
188:Applications of Hilbert's 10-th 7/6/03 4:43AM
189:Some Model theoretic Pi-0-1 statements 9/25/03 11:04AM
190:Diagrammatic BRT 10/6/03 8:36PM
191:Boolean Roots 10/7/03 11:03 AM
192:Order Invariant Statement 10/27/03 10:05AM
193:Piecewise Linear Statement 11/2/03 4:42PM
194:PL Statement/clarification 11/2/03 8:10PM
195:The axiom of choice 11/3/03 1:11PM
196:Quantifier complexity in set theory 11/6/03 3:18AM
197:PL and primes 11/12/03 7:46AM
198:Strong Thematic Propositions 12/18/03 10:54AM
199:Radical Polynomial Behavior Theorems
200:Advances in Sentential Reflection 12/22/03 11:17PM
201:Algebraic Treatment of First Order Notions 1/11/04 11:26PM
202:Proof(?) of Church's Thesis 1/12/04 2:41PM
203:Proof(?) of Church's Thesis - Restatement 1/13/04 12:23AM
204:Finite Extrapolation 1/18/04 8:18AM
205:First Order Extremal Clauses 1/18/04 2:25PM
206:On foundations of special relativistic kinematics 1 1/21/04 5:50PM
207:On foundations of special relativistic kinematics 2 1/26/04 12:18AM
208:On foundations of special relativistic kinematics 3 1/26/04 12:19AAM
209:Faithful Representation in Set Theory with Atoms 1/31/04 7:18AM
210:Coding in Reverse Mathematics 1 2/2/04 12:47AM
211:Coding in Reverse Mathematics 2 2/4/04 10:52AM
212:On foundations of special relativistic kinematics 4 2/7/04 6:28PM
213:On foundations of special relativistic kinematics 5 2/8/04 9:33PM
214:On foundations of special relativistic kinematics 6 2/14/04 9:43AM
215:Special Relativity Corrections 2/24/04 8:13PM
216:New Pi01 statements 6/6/04 6:33PM
217:New new Pi01 statements 6/13/04 9:59PM
218:Unexpected Pi01 statements 6/13/04 9:40PM
219:Typos in Unexpected Pi01 statements 6/15/04 1:38AM
220:Brand New Corrected Pi01 Statements 9/18/04 4:32AM
221:Pi01 Statements/getting it right 10/7/04 5:56PM
222:Statements/getting it right again 10/9/04 1:32AM
223:Better Pi01 Independence 11/2/04 11:15AM
224:Prettier Pi01 Independence 11/7/04 8:11PM
225:Better Pi01 Independence 11/9/04 10:47AM
226:Nicer Pi01 Independence 11/10/04 10:43AM
227:Progress in Pi01 Independence 11/11/04 11:22PM
228:Further Progress in Pi01 Independence 11/12/04 2:49AM
229:More Progress in Pi01 Independence 11/13/04 10:41PM
230:Piecewise Linear Pi01 Independence 11/14/04 9:38PM
231:More Piecewise Linear Pi01 Independence 11/15/04 11:18PM
232:More Piecewise Linear Pi01 Independence/correction 11/16/04 8:57AM
233:Neatening Piecewise Linear Pi01 Independence 11/17/04 12:22AM
234:Affine Pi01 Independence 11/20/04 9:54PM
235:Neatening Affine Pi01 Independence 11/28/04 6:08PM
236:Pi01 Independence/Huge Cardinals 12/2/04 3:49PM
237:More Neatening Pi01 Affine Independence 12/6/04 12:56AM
238:Pi01 Independence/Large Large Cardinals/Correction 12/7/04 10:31PM
239:Pi01 Update 12/11/04 1:12PM
240:2nd Pi01 Update 12/13/04 2:49AM
241:3rd Pi01 Update 12/13/04 4:08AM
242:4th Pi01 Update 12/18/04 9:47PM
243:Inexplicit Pi01/Ordinals of Set Theories 12/19/04 11:48PM
244:LUB Systems 2/26/05 8PM
Harvey Friedman
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