[FOM] 291: Independently Free Minds/Collectively Random Agents/more
Harvey Friedman
friedman at math.ohio-state.edu
Tue Jun 20 14:59:04 EDT 2006
This is the second installment. First we correct some typos from the first
installment http://www.cs.nyu.edu/pipermail/fom/2006-June/010611.html
1. In the "informal interpretation of the symbols", under paragraph b,
replace 'marks' with 'marking'.
2. Under paragraph c, replace 'is to vastly' with 'is so vastly'.
3. Under paragraph h, replace 'Under the interpretation f' with 'Under the
interpretation e'.
4. In the formal presentation of the axioms, we do not need to make the
definition of <=.
5. In the formal axiom 3, replace 'R' by 'A'.
We also remark that we can omit the second part of axiom 2 that asserts
alpha < beta. This can be derived from the remaining axioms.
I prefer to use "CONTINUAL CREATION" instead of "CONTINUAL BIRTH".
We continue our development from
http://www.cs.nyu.edu/pipermail/fom/2006-June/010611.html with section 2.
#####################################
INDEPENDENTLY FREE MINDS/COLLECTIVELY RANDOM AGENTS
continued
It is clear that the very strong version in section 3 below is particularly
elegant. Nevertheless, we still keep section 1 (from the first posting) and
the following section 2, for completeness of the development.
Let me say something I said earlier, but perhaps better.
Let us look at the usual comprehension axiom over a set D. This says that we
can define a subset of D according to any condition on elements of D.
If we stay in normal mathematics, with its immutable objects, we must create
the notion of set of elements of D, or properties of D, with a special axiom
of comprehension over D.
On the other hand, if we think of the elements of D are agents, which change
their state according to time, then if they are independently free or
collectively random, or whatever, then we can assert that there is a time at
which their states are as according to a previously given condition.
Thus comprehension now becomes something that corresponds well to basic
probability theory. That any pattern of states will occur at some time.
This is just a special case of what is gained by introducing time into
mathematics. Here when we introduce time into set theory, we get a theory
which makes sense in informal ordinary thinking. It no longer looks like
mathematics. We get rid of the most unintuitive aspect of set theory or a
theory of predication, namely the iterative nature of it. Once we have sets
of sets, sets of sets of sets, or predicates of predicates, predicates of
predicates of predicates, and the like, also with mixed types, we are in the
realm of technical mathematics. Here we stay within the realm of ordinary
intuition and ordinary thinking.
One of a huge variety of directions to take this work is to see what happens
when we take other more advanced concepts from probability theory and see
what they correspond to in the general theory of concepts.
Or more generally, see what happens when we take various concepts from
mathematics and see what they correspond to in the general theory of
concepts.
Another direction is to apply this to vague concepts such as "being red". We
are already treating some vague concepts here, in our Horizon Effect. I.e.,
the notion of epoch or transition.
In fact, note that in section 3 below, we rely on the notion of
"transition". This is really equivalent to a notion "being close together"
in the sense of lying within the same epoch, or "there is no transition from
t1 to t2". Of course, this is different than the more usual notion of being
close together.
Another important direction is to get away from "points of time" and instead
use "regions of time", as in mereology. Even if we keep points of time, then
we will explore what happens if we do NOT have sharp boundaries between
epochs.
It should be noted that our theories have a very strong built in direction
of time.
2. IFM/CRA WITH FINITELY MANY TRANSITION POINTS.
Here we give an obvious generalization to the system of section 1 with two
transition points. These systems climb up the ineffable cardinal hierarchy.
This is not the kind of profound jump in strength that we get in section 3
below.
Fix an integer n >= 2. Here is IFM/CRA with n transitions.
We use the following primitives.
1. Variables over times. t,t1,t2,... .
2. Variables over minds. M,M1,M2,... .
3. The binary relation symbol < on two times.
4. n constant symbols alpha_1,...,alpha_n representing n times.
5. The binary relation symbol A on a mind and a time.
6. Equality between times. No equality between minds.
We will first present the axioms semiformally.
AXIOM 1. TIME IS LINEAR. Time is linearly ordered by <.
AXIOM 2. n TRANSITIONS. There is a time before alpha_1, and the alpha's are
in strictly increasing order.
AXIOM 3. CONTINUAL CREATION. At every time there is a mind active at that
time, that was not active prior to that time.
AXIOM SCHEME 4. UNRESTRICTED ACTIVITY. There is a time t < alpha_2 where the
minds that have been active at some time < alpha_1, are active or not at t
according to any given condition. (Here t depends on the given condition).
AXIOM SCHEME 5. n HORIZON EFFECT. Any true statement about both a given time
t < alpha_1 and alpha_1,...,alpha_n-1 remains true as a corresponding
statement about both t and alpha_2,...,alpha_n.
A sharpening of axiom scheme 5 can be derived. Here we can use any two
strictly increasing subsequences (not necessarily blocks) of
alpha_1,...,alpha_n of the same lengths.
Now we present these axioms formally.
AXIOM 1. TIME IS LINEAR. not t < t, t1 < t2 or t2 < t1 or t1 = t2.
AXIOM 2. n TRANSITIONS. (therexists t)(t < alpha_1 < ... < alpha_n).
AXIOM 3. CONTINUAL CREATION. (forall t)(therexists M)(A(M,t) and (forall t1
< t)(not R(M,t1))).
AXIOM SCHEME 4. UNRESTRICTED ACTIVITY. (therexists t < alpha_2)(forall M, t1
< alpha_1)(A(M,t1) implies (A(M,t) iff phi)), where phi is any formula in
the language of IFM/CRA with n transitions, in which t is not free.
AXIOM SCHEME 5. n HORIZON EFFECT. t < alpha_1 implies (phi implies
phi[alpha_1/alpha_2,alpha_2/alpha_3,...,alpha_n-1,alpha_n]), where phi is
any formula in the language of IFM/CRA with n transitions, which doesn't
mention alpha_n and has at most the free variable t. Here
phi[alpha_1/alpha_2,alpha_2/alpha_3,...,alpha_n-1,alpha_n] is the result of
replacing all occurrences of alpha_1,...,alpha_n-1, by alpha_2,...,alpha_n,
respectively, in phi.
We now come to the main Theorems.
THEOREM 2.1. Let n >= 2. ZFC + On is an n-1 subtle cardinal (scheme), and
IFM/CRA with n transitions are mutually interpretable and equiconsistent.
This is provable in EFA = exponential function arithmetic.
THEOREM 2.2. Let n >= 3. There is an interpretation of the finitely
axiomatizable NBG + Global Choice + "there exists an n-2 subtle cardinal" in
IFM/CRA with n transitions. We can prove the consistency of IFM/CGA with n
transitions in ZFC + "there exists an n-1 subtle cardinal".
We remark that we can omit the second part of axiom 2, which asserts that
the alpha's are strictly increasing.
3. IFM/CRA WITH ETERNAL TRANSITIONS.
We now make a profound jump in strength through the addition of a primitive
for "being a transition".
We now formulate IFM/CRA with eternal transitions.
We use the following primitives.
1. Variables over times. t,t1,t2,... .
2. Variables over minds. M,M1,M2,... .
3. The binary relation symbol < on two times.
4. The unary relation symbol T on times.
5. The binary relation symbol A on a mind and a time.
6. Equality between times. No equality between minds.
The informal interpretation of the symbols is as in the first posting,
except for the following.
T(t) is read "t is a transition". I.e., t marks the end of an old epoch and
the beginning of a new epoch. The transitions mark out vast regions of time.
We first present the axioms semiformally.
AXIOM 1. TIME IS LINEAR. Time is linearly ordered by <.
AXIOM 2. ETERNAL TRANSITIONS. There is a transition after every time.
AXIOM 3. CONTINUAL CREATION. At every time there is a mind active at that
time, that was not active prior to that time.
AXIOM SCHEME 4. UNRESTRICTED ACTIVITY. For any given time t there is a time
t1 such that the minds that have been active at some time prior to t, are
active or not at t1 according to any given condition.
AXIOM SCHEME 5. TRANSITION HORIZON EFFECT. Any true statement about any
given time remains true if we hypothetically restrict the transitions to
those that are at least any given transition after the given time.
Here are the formal axioms. We use the abbreviation <=.
AXIOM 1. TIME IS LINEAR. not t < t, t1 < t2 or t2 < t1 or t1 = t2.
AXIOM 2. TWO TRANSITIONS. (therexists t)(t < alpha < beta).
AXIOM 3. CONTINUAL CREATION. (forall t)(therexists M)(A(M,t) and (forall t1
< t)(not A(M,t1))).
AXIOM SCHEME 4. UNRESTRICTED ACTIVITY. (therexists t1)(forall M,t)(A(M,t)
implies (A(M,t1) iff phi)), where phi is any formula in the language of
IFM/ERA with eternal transitions, in which t is not free.
AXIOM SCHEME 5. TRANSITION HORIZON EFFECT. t1 < t implies (phi implies
phi[>=t]), where phi is any formula in the language of IFM/CRA with eternal
transitions, with at most the free variable t1, in which t does not appear.
Here phi[>=t] is the result of replacing each T[ti] by (T[ti] and t <= ti).
We don't have a precise calculation of the strength of this system. It is
somewhere above the existence of sharps, and the existence of a measurable
cardinal. So it lies well above
there exists a cardinal that arrows every countable ordinal
and breaks into the large cardinal assumptions that are incompatible with V
= L.
In order to get past measurable cardinals, we add the following.
AXIOM 6. TRANSITION ACCUMULATION. There is a time which occurs later than a
segment of transitions without end.
AXIOM 6. TRANSITION ACCUMULATION. (therexists t)((therexists t1 < t)(T(t1)
and (forall t1 < t)(T(t1) implies (therexists t2)(t1 < t2 < t and T(t2)))).
This bumps us well past ZFC + "there exists a measurable cardinal", and also
well past ZFC + "there exists a measurable cardinal kappa with kappa many
measurable cardinals below kappa". However, it is weaker than ZFC + "there
exists a measurable cardinal kappa with normal measure 1 measurable
cardinals below kappa".
We get into the concentrator measurable hierarchy if we strengthen axiom 6
to assert that t is itself a transition.
######################################
In the next posting we will discuss strong extensions of IFM/CRA which
correspond to yet higher levels of set theory.
**********************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 291st 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 6/1/06 11:28AM
Harvey Friedman
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