[FOM] 290: Independently Free Minds/Collectively Random Agents

Harvey Friedman friedman at math.ohio-state.edu
Mon Jun 12 11:01:11 EDT 2006

Harvey M. Friedman
June 12, 2006


We develop a theory of independently free minds, or collectively random
agents. At this point, it is not clear which is the preferable description,
as the axioms that we present have good motivating interpretations of either

Time plays an essential role in the theory. We readily obtain fundamental
theories that are interpretable in set theory with large cardinals and vice

We can look at this ongoing development as the introduction of temporal and
other intuitive conceptions into set theory. When we do this, certain
characteristic ideas of set theory that seem to have at most rather weak
counterparts in ordinary reasoning and in normal mathematics, get replaced
by intuitive concepts from ordinary thinking. E.g., the iterative structure,
so foreign outside set theory, becomes flat.

In fact, we can step back and put this clearly into the general context of
mathematical science. For centuries, we have modeled moving bodies as
*mathematical functions* whose domains represent (the set of points of) time
and whose values represent positions in space. We thus model a moving body
as something that not only doesn't move, but also never changes - namely,
the immutable mathematical objects of modern mathematics. The function
itself is immutable, as are all mathematical objects.

This lead to various problems with the function concept, which then lead to
sets of ordered pairs. This in turn led to the iterative concept of set as a
foundation for mathematics, since ordered pairs themselves need to be
ultimately treated (or not treated, at some related cost).

This whole development, one thing leading to another, is responsible for the
great rigorous power of our mathematics and its current foundations, but
also pushes our mathematics away from "normal ordinary thinking".

I now believe that we have had enough experience with this way of modeling
things in mathematics, that we may be ready to really see what happens when
we do things differently.

Namely, I am calling for an introduction - or reintroduction - of the kind
of informal concepts that mathematics has successfully taken great pains to
rid itself of - in its (wholly necessary and successful) quest for absolute

Am I saying that the move to absolute rigor via set theory was misguided??

Absolutely not!!!

On the contrary, this move to absolute rigor via (the iterative concept of)
sets was entirely necessary - even entirely necessary in order for us to
begin now to **rigorously** move away from it.

The lesson to be learned is: we had to first gain the great experience and
power of doing things crudely, before we can now properly do them with
greater finesse, clarity, power, beauty, and imagination.

We can view what I am doing now, in this posting, as rigorously introducing
time directly into the foundations of mathematics at the foundational level.

This begs for the rigorous introduction of other informal notions from
everyday ordinary thinking, at the foundational level.

This also begs for the rigorous introduction of various informal notions
from everyday ordinary thinking, at the level of mathematical practice.

These enterprises should join. I.e.,

present day mathematics and present day foundations of mathematics

are currently quite far apart, and I have spent a lot of time moving them
closer together - in particular, through BRT = Boolean Relation Theory.


this new mathematics and this new foundations of new mathematics

should be naturally quite close together!!

Of course, I expect that the new foundations of new mathematics will be
mutually interpretable with the present foundations of present mathematics
via abstract set theory. But a lot of transformation can happen under mutual

>From what you see below, it appears that when I do this - introduce time and
minds - I come to a theory which is mutually interpretable with set theory
with large cardinals, but which has many conceptual advantages. In

i. The setup is much more closely related to everyday intuitive thinking of
ordinary people than set theory.

ii. In particular, there is no directly iterative concept that is so
characteristic of set theory or theories of properties.


Here IFM = independently free minds, and CRA = collectively random agents.
As I said in the Introduction, both titles seem appropriate at this point.

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. Two constant symbols alpha,beta representing times.
5. The binary relation symbol A on a mind and a time.
6. Equality between times. No equality between minds.

Here is the informal interpretation of the symbols.

a. Times (points of time) are as usually conceived, with < as the strict
linear ordering on times. We will not make any explicit assumption other
than strict linearity.

b. The time alpha is a transition marking the end of the "first epoch" and
the beginning of the "second epoch". The time beta is a transition marks the
end of the "second epoch" and the beginning of the "third epoch". We make no
assumption about further "epochs", if any.

c. The time alpha is so vastly far into the future, and the time beta is to
vastly far into the future compared to alpha, and "all of time up to alpha"
forms such a perfect whole, forming a kind of "first eternity", and "all of
time up to beta" forms such a perfect whole, also forming a kind of "second
eternity", that we are led to the axiom of Horizon Effect, below. Of course,
these "little eternities" are not to be compared at all with "absolute
eternity" which is much further out still, and also perfect.

d. A(M,t) is read "the mind M at time t is active".

e. We view the minds as operating independently and freely. Thus each mind
can become active or inactive at will, independently of other minds. We also
assume that a mind, or the minds under discussion, will not, at some point,
remain forever inactive. (E.g., reincarnation). This leads to the axiom of
Unrestricted Activity below.

f. There is an alternative interpretation of minds that also leads to the
axiom of Unrestricted Activity, which goes like this. Although the various
minds may first become active at arbitrary times, once they do, they act
randomly - even randomly with respect to each other.

g. In connection with f, it may be useful to make analogies with standard
situations in probability theory. Consider the following. Suppose that we
have a finite number n of coins, and we flip them for a very long (finite)
amount of discrete time. Assume that each coin has a nonzero probability of
coming up heads and a nonzero probability of coming up tails. Then almost
certainly there will be a time at which they will come up with any assigned
pattern of heads/tails. (This requires that the amount of discrete time is
large relative to the reciprocals of these probabilities). And in an
infinite amount of discrete time, we can say that 'certainly' this will
happen. This can be studied by probabilistic methods with infinitely many
coins, or a continuous ensemble of coins, operating under either discrete or
continuous time. Of course, ordinary probability theory cannot be literally
invoked here - only intuitive 'probability' theory: Because of our Horizon
Effect axiom, it is easy to see that we cannot be talking about a time
ordering that is anything like (an interval, possibly infinite, of) the real
numbers, at least in the sense that explicitly in the theory we have a
decent halving function on time. A decent halving function would immediate
violate the Horizon Effect axiom, by taking alpha/2 and comparing it to
alpha and beta. (We would have alpha/2 < alpha < beta, and twice alpha/2 is
alpha, yet twice alpha/2 is NOT beta, contradicting Horizon Effect).

h. Under the interpretation f, we call this IFM = Independently Free Minds.
Under the interpretation g, we call this CRA = Collectively Random Agents.

We first present the axioms semiformally.

AXIOM 1. TIME IS LINEAR. Time is linearly ordered by <.

AXIOM 2. TWO TRANSITIONS. There is a time before alpha, and alpha is before

AXIOM 3. CONTINUAL BIRTH. 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 < beta where the
minds that have been active at some time < alpha, are active or not at t
according to any given condition. (Here t depends on the given condition).

AXIOM SCHEME 5. HORIZON EFFECT. Any true statement about both a given time t
< alpha and alpha, remains true as a corresponding statement about both t
and beta.  

Regarding 3 This says that at every time, at least one mind is "born". This
might sound rather extravagant. However, consider this.

a. How often, somewhere on earth, is a living thing born? This may include
bacteria and viruses. Probably sufficiently often that "normal measuring
tools" cannot locate a time where this does not happen. And consider all
forms of life everywhere in the universe. It makes sense, informally, to say
that "living things are born all the time".

b. Another argument for consideration of axiom 2 is that God creates life,
and why would God waste any time doing so?

Regarding 4. The condition is allowed to mention any particular minds and
any particular times.

Regarding 5. By taking negations, we can derive the converse of Horizon
Effect. I.e., the iff form.

Regarding 5. The statement can only mention objects t and alpha, and not any
other particular times and not any other particular minds.

Now we present these axioms formally. We define t1 <= t2 as t1 < t2 or t1 =

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 BIRTH. (forall t)(therexists M)(A(M,t) and (forall t1 <
t)(not R(M,t1))).

AXIOM SCHEME 4. UNRESTRICTED ACTIVITY. (therexists t < beta)(forall M, t1 <
alpha)(A(M,t1) implies (A(M,t) iff phi)), where phi is any formula in the
language of IFM/CRA in which t is not free.

AXIOM SCHEME 5. HORIZON EFFECT. t < alpha implies (phi implies
phi[alpha/beta]), where phi is any formula in the language of IFM/CRA which
doesn't mention beta and has at most the free variable t. Here
phi[alpha/beta] is the result of replacing all occurrences of alpha by beta,
in phi. 

We now come to the main Theorems.

THEOREM 1. ZFC + On is a subtle cardinal (scheme), and IFM/CRA are mutually
interpretable and equiconsistent. This is provable in EFA = exponential
function arithmetic.

THEOREM 2. We can exhibit an interpretation of the finitely axiomatizable
NBG + Global Choice + "there exists a totally indescribable cardinal" in
IFM/CRA. We can prove the consistency of IFM/CGA in ZFC + "there exists a
subtle cardinal". 

There are a number of natural extensions of IFM/CGA for which these results
still hold. 

a. We can add axioms that pin down more about the linear structure of time.
E.g., we can say that time is dense, and prescribe the endpoint situation
(four possibilities). Or, alternatively, we can assert that every time has
an immediate successor, and again prescribe the endpoint situation in any of
four ways. However, we cannot assert that every point with something below
it, has an immediate predecessor, as this would run afoul of Horizon Effect
when applied to alpha.

b. We can add the axiom scheme of completeness (lubs, glbs) of the ordering
of time (without committing to density).

c. We can strengthen Continual Birth in several ways. Firstly, we can assert
that every mind is active at some time. We can even assert that every mind
is born at some time. We can also assert something about the number of minds
that are born at every time. For instance, we can assert that exactly one
mind is born at any given time. However, in order to assert this, we need to
add equality between minds. We can alternatively assert that at least two
minds are born at any given time.

d. We can add some additional principles of Unrestricted Activity. We can

(therexists t > beta)(forall M, t1 <= t2)(A(M,t1) implies (A(M,t) iff phi)).

(therexists t < alpha)(forall M, t1 <= t2 < alpha)(A(M,t1) implies (A(M,t)
iff phi)).

e. In the Horizon Effect, we can use any finite number t1,...,tn <= alpha.


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 290th 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

Harvey Friedman 

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