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Unformatted text preview: 1 CONCEPT CALCULUS
by Harvey M. Friedman
Ohio State University
July 27, 2006
PREFACE. We present a variety of basic theories involving
fundamental concepts of naive thinking, of the sort that
were common in "natural philosophy" before the dawn of
physical science. The most extreme forms of infinity ever
formulated are embodied in the branch of mathematics known
as abstract set theory, which forms the accepted foundation
for all of mathematics. Each of these theories embodies the
most extreme forms of infinity ever formulated, in the
following sense. ZFC, and even extensions of ZFC with the
so called large cardinal axioms, are mutually interpretable
with these theories. This is an extended abstract. Proofs
of the claims will appear elsewhere.
INTRODUCTION.
1. BETTER THAN.
1.1. Better Than, Much Better Than.
1.2. Better Than, Real.
1.3. Better Than, Real, Conceivable.
2. VARYING QUANTITIES.
2.1. Single Varying Quantity.
2.2. Two Varying Quantities, Three Separate Scales.
2.3. Varying Bit.
2.4. Persistently Varying Bit.
2.5. Naive Time.
3. BINARY RELATIONS.
3.1. Binary Relation, Single Scale.
3.2. Binary Relation, Two Separate Scales.
4. MULTIPLE AGENTS, TWO STATES.
5. POINT MASSES.
5.1. Discrete Point Masses in One Dimension.
5.2. Discrete Point Masses with End Expansion.
5.3. Discrete Point Masses with Inner Expansion.
5.4. Point Masses with Inner Expansion.
5.5. Discrete Point Masses with Inner Expansion
Revisited.
6. TOWARDS THE MEREOLOGICAL. 2 INTRODUCTION
We present a variety of basic theories involving
fundamental concepts of naive thinking, of the sort that
were common in "natural philosophy" before the dawn of
physical science.
The most extreme forms of infinity ever formulated are
embodied in the branch of mathematics known as abstract set
theory, which forms the accepted foundation for all of
mathematics.
Each of these theories embodies the most extreme forms of
infinity ever formulated, in the following sense. ZFC, and
even extensions of ZFC with the so called large cardinal
axioms, are mutually interpretable with these theories.
Physical science, as we know it today, is based on
measuring quantities via the usual real number system, and
counting objects via the usual natural number system.
Here we instead use abstract linear orderings (sometimes
without linearity), and formulate principles that can be
seen to be incompatible with the identification of these
linear orderings with the real numbers, or segments of the
real numbers. The principles have a clear conceptual
meaning to any naive thinker.
At this point, we are not suggesting the overthrow of the
real number orthodoxy in physical science. That would be
grossly premature, and may never be appropriate. Our aims
are quite different and more philosophical.
i. We show how a complex of very simple intuitive ideas
that can be absorbed and reasoned with by naive thinkers,
leads surprisingly quickly and naturally to a consistency
proof for mathematics (ZFC, even with large cardinals).
Thus these naive ideas are surprisingly powerful in light
of the
fact that mathematics itself is not sufficient to prove its
own consistency (in the sense of Gödel's second
incompleteness theorem).
ii. Armed with this array of very powerful naive
principles, we can search for entirely new contexts in
which principles of a similar nature arise with similar
results. We expect to find more richly philosophical, or 3
even theological, contexts in which these emerging naive
principles are particularly compelling, and relevant to
ordinary thinking and life experiences.
iii. In particular, there are contexts in which there are
orderings but where the very idea of quantitative
measurement (as presently construed) is inappropriate or
even absurd, and so there will not be or cannot be any
associated real number orthodoxy. For example, "x is more
beautiful than y".
Or "idea x is more interesting than idea y". Or "act x is
morally preferable to act y". Or "agent x is morally
superior to agent y". Or "outcome x is more just than
outcome y". Or "act x is more pleasurable than act y." Or
"activity x is preferable to activity y" or "state of
affairs x is preferable to sate of affairs y".
iv. What is emerging is a true calculus of conceptual
principles, whose "logical strengths" is being
"calculated".
There is already an entirely solid mathematical basis for
the comparison of theories formulated in first order
predicate calculus. This is through the fundamental notion
of
*interpretation* or *interpretability* due to Alfred
Tarski.
See http://en.wikipedia.org/wiki/Interpretability and
A.Tarski, A.Mostovski and R.M.Robinson, Undecidable
Theories. NorthHolland, Amsterdam, 1953.
v. We call this prospective calculus of conceptual
principles, the CONCEPT CALCULUS.
vi. What is the methodology of the Concept Calculus? We
identify fundamental informal notions from ordinary
thinking, particularly from philosophical subjects where
there is a rich literature of discussion going back for
long periods of time.
We then consider various combinations of fundamental
principles capable of clear and concise formulation.
We then experiment with appropriate combinations of such
principles. Some of these first order theories may of
course be incompatible with others. 4 We then "calculate" the logical strengths of these
theories.
Here logical strength has come to mean "interpretation
power". Calculating logical strength has come to mean
a) an identification of a theory among the robust linearly
ordered hierarchy of set theories, arising out of the
foundations of mathematics.
b) the verification that the theory in question is mutually
interpretable with the identified set theory.
Thus the "measuring tool" is the robust hierarchy of set
theories already developed in the foundations of
mathematics.
vii. This measuring tool is the appropriate tool needed to
compare the logical strengths of two interesting theories
arising in Concept Calculus. For, let S and T be two
theories arising in Concept Calculus. To compare S and T,
we first calculate the logical strengths of S,T in the
sense above through identifying the appropriate two levels of set
theory. Then we can merely note the comparison of these two
levels of set theory.
viii. This is a good analogy with the measurement of, say,
the height of buildings. First they are measured in, say,
meters  as, say, a base 10 rational. Then the two base 10
rationals are compared. Generally, one may only commit to
an interval and not a single number, where the intervals
are
both small enough that the comparison can be made.
ix. We are at the very beginning of the development of
Concept Calculus, and here great experience with set
theories and various techniques developed from mathematical
logic are needed to obtain the first significant results.
In particular, one has to be facile with how to interpret
set theories and interpret into set theories in a wide
variety of contexts.
x. However, at some point, basic tools should arise that
will make the development of Concept Calculus more amenable
to more scholars. In particular, what I want to do now is 5
to develop an array of most basic theories for which I can
calculate  or approximately calculate  their
logical strength. Then, scholars can work with these basic
theories – which should be more 'friendly' than set
theories  in order to make calculations, or at least the
interpretations needed for approximate calculations.
We will make a number of calculations in Concept Calculus.
For a while, we will not try to operate fully
systematically, as our main interest at the moment is in
showing how naturally one obtains theories that are at
least as strong, logically, as mathematics  as identified
with ZFC.
What is emerging, over and over again, is two fundamental
principles, each of which take somewhat different forms
depending on the context in which they are applied.
A. Completeness/Randomness/Creativity (anything that can
happen will).
B. Symmetry/Indiscernibility/Horizons (any two horizons are
indiscernible to observers on the basis of their extent).
The first principle is well established in the
Philosophical literature, as the Aristotelian Principle of
Plenitude.
http://en.wikipedia.org/wiki/Principle_of_plenitude
http://poznanstudies.swps.edu.pl/vols/ps51abs.html
http://www.philosophypages.com/dy/p5.htm
http://plato.stanford.edu/entries/aristotlenatphil/notes.html footnote 30
http://64.233.167.104/search?q=cache:GYFG3KOAiJwJ:ethesis.h
elsinki.fi/julkaisut/teo/syste/vk/kukkonen/studiesi.pdf+Ari
stotle+Plenitude&hl=en&gl=us&ct=clnk&cd=4
In
http://www.philosophyprofessor.com/philosophies/plenitudeprinciple.php 6
some different forms of the Principle of Plentitude are
discussed, including the relevant one of Aristotle.
The second principle is probably as least implicit in the
Philosophical literature. But since I don’t yet have
references, let me try to say something intelligible about
this general principle.
Suppose you look up to the sky with different strong powers
of vision. You will then experience different horizons,
some much farther out than others. Ideally speaking, you
should not be able to distinguish what you see under the
different strong powers of vision, since what you see is,
in each case, unimaginably vast.
The idea is that this principle is meant to apply to
contexts far more general than cosmology. There should be
much more clarifying things to say about it, including
possibly a way to combine it with the Principle of
Plenitude. However, here we will be content, here, to mold
it to some specific contexts of naive thinking.
From looking at the specific contexts in which we apply
this Principles, we see the emergence of RIGOROUS theories
of
naive
naive
naive
naive
naive probability.
statistics.
geometry.
physics.
theory of agents (minds). More speculatively,
naive
naive
naive
...
naive differential equations?
biology?
psychology?
anything? For instance, we could try to exploit the naïve idea of the
instantaneous rate of change of a quantity varying
according to time.
For instance, naive probability and statistics already
suggested by the Principle of Plentitude: 7
*time is so vast, that any possibility will eventually
occur*
This basic principle corresponds very well with what we
know from standard mathematically formalized probability
theory.
Specifically, in
http://en.wikipedia.org/wiki/Law_of_large_numbers
we see the following formulation of the law of large
numbers (although this is not the only formulation):
"The phrase "law of large numbers" is also sometimes used
to refer to the principle that the probability of any
possible event (even an unlikely one) occurring at least
once in a series increases with the number of events in the
series. For example, the odds that you will win the lottery
are very
low; however, the odds that someone will win the lottery
are quite good, provided that a large enough number of
people purchased lottery tickets."
In fact, the usual comprehension axiom scheme or separation
axiom scheme in set theory can be viewed as a kind of
informal, intuitive, naive probability theory in the
following sense. A more explicit form is
**time is so vast, that any given possible behavior over
time intervals will be realized over some time interval**
We can also think of a binary relation on two separate
scales as an ensemble of data. I.e., we can plot a diagram
of pairs (height, weight) of persons. We can assert that
the two parts  height and weight  are completely
independent (which is of course not actually the case).
More abstractly, we can speak of
naive independence.
I.e., that the heights and weights have nothing to do with
each other. Of course, in reality they are correlated. But
when we handle binary relations, we use naïve independence.
An important challenge is to incorporate some dependence
(correlations) in this program. 8
In reading the numerous theories presented below in diverse
contexts, note that the issue is not (at least yet) one of
truth. Various principles can be objected to on various
grounds – even if the underlying concepts are taken to be
idealized conceptions residing in the mind.
We believe that the various theories can be adjusted in
order to meet many kinds of objections. The resulting
adjusted theories would also have very high logical
strengths which can be calculated. In particular, they can
also be used to give consistency proofs of mathematics (as
formalized by ZFC).
Examples pursued here of adjusting theories according to
reasonable objections, appear in sections 1.3, 2.2, 2.4,
the use of discrete point masses in chapter 5, and further
developments anticipated in chapter 6. 1. BETTER THAN
1.1. Better Than, Much Better Than.
We use a one sorted predicate calculus with equality, with
a binary relation symbol > for "better than", and a binary
relation symbol >> for “much better than”.
BASIC. Nothing is better than itself. x >> y Æ x > y. x >>
y Ÿ y > z Æ x >> z. x > y Ÿ y >> z Æ x >> z. If x >> y
then there exists z such that x >> z > y. Given anything,
there is something much better that it.
MINIMAL. There is nothing that is better than all minimal
things.
EXISTENCE. Let x be a thing better than a given range of
things. There is something that is better than the given
range of things and the things that they are better than,
and nothing else. Here we use L(>,>>) to present the range
of things.
HORIZON. Let x,y,z be given, as well as a true statement
about x,y, using “better than”, and “much better than x,y”.
The corresponding statement about x,y, using “better than”,
and “much better than x,y,z” is also true. 9
THEROEM 1.2.1. Basic + Minimal + Existence + Horizon is
mutually interpretable with ZFC. This is provable in EFA. 1.2. Better Than, Real.
We use a one sorted predicate calculus with equality, with
a binary relation symbol > for "better than", and a unary
predicate R for "being real".
The idea is that we are dividing the objects up into those
that are real and those that are imaginary.
We define
x is minimal if and only if x is not better than anything.
BASIC. Nothing is better than itself. (x > y Ÿ y > z) Æ x >
z. Something is real.
IMAGINARY. There is something that is better than all real
things, and nothing else.
MINIMAL. There is nothing that is better than all minimal
things.
"Even the lowest level things, collectively, have something
to offer." This has economic, political, and social
ramifications.
REAL EXISTENCE. Let something real be better than a given
range of things. There is something real that is better
than the given range of things and the things they are
better than, and nothing else. Here we use L(>) to present
the range of things.
REAL EXAMPLES. If two real things bear a certain relation
to something, then they bear that relation to something
real. Here we use L(>) to present the relation.
THEOREM 1.2.1. Basic + Imaginary + Minimal + Real Existence
+ Real Examples is mutually interpretable with ZFC. This is
provable in EFA.
In Real Existence, we can put a strong restriction on the
formula: Let x be a real thing that is better than a given
range of things that is defined where all quantifiers are
bounded to x (i.e., to the y < x), but with arbitrary 10
parameters. There is something real that is better than the
given range of things and the things they are better than,
and nothing else. Here we use L(>) to present the range of
things.
If we use this restricted form of Real Existence, then
Theorem 1.2.1 remains unchanged. 1.3. Better Than, Real, Conceivable.
From some points of view, one can criticize Real Existence
of section 1.2 on the grounds that one is asserting the
existence of a real object using a formula that involves
things that are not real. By the claim after Theorem 1.2.1,
we have met this objection to a considerable extent.
However, we still use arbitrary parameters.
We meet this criticism by adding the new notion
"conceivable". We replace Real Existence by Conceivable
Existence, which has heavy restrictions.
We use a one sorted predicate calculus with equality, with
a binary relation symbol > for "better than", and two unary
predicates R, C. Here R(x) means "x is real". C(x) means "x
is conceivable".
We define
x is minimal iff x is not better than anything.
BASIC. Nothing is better than itself. If a first thing is
better than a second thing, and the second thing is better
than a third thing, then the first thing is better than the
third thing. Everything that is real is conceivable. The
only things that a real thing can be better than are real.
REAL MINIMAL. There is nothing real that is better than all
real minimal things.
CONCEIVABLE EXISTENCE. Let a range of real things be given,
defined with reference to real things only. There is a
conceivable thing that is better than the given range of
things and the things that they are better than, and
nothing else.
TRANSFER. Let a true statement be given involving two
specific real things, "better than", and "being real". The 11
corresponding statement is true involving the two real
things, "better than", and "being conceivable". Here we use
L(>,R) to present the true statement.
In Conceivable Existence, the range is given by a formula j
all of those parameters are real things, and whose
quantifiers all range over real things.
THEOREM 1.3.1. Basic + Real Minimal + Conceivable Existence
+ Transfer interprets ZFC + "there exists a totally
indescribable cardinal" and is interpretable in ZFC +
"there exists a subtle cardinal". This is provable in EFA. 2. VARYING QUANTITIES
2.1. Single Varying Quantity.
The language L(<,F,T) is in first order predicate calculus
with equality, the binary relation symbol <, the unary
function symbol F, and the unary relation symbol T.
We think of < as the linear ordering of time. F(x) is the
value of the varying quantity at time x.
Note that the values of the varying quantity are from the
same scale as time. This can rather easily be objected to,
and section 2.2 is in response to this objection.
T(x) is read "x is a transition". Thus if T(x) then x
represents the beginning of a new epoch. Thus the
transitions are spaced unimaginably apart. One can also
think of x as representing an “horizon”. If y < x and T(x),
then x is unimaginably far beyond y.
Bounded intervals are intervals of the form
(x,y), [x,y], (x,y], [x,y)
where x,y are points. We allow x ≥ y when using interval
notation.
LINEARITY. < is a linear ordering.
TRANSITIONS. Every time is < some transition. 12
ARBITRARY BOUNDED RANGES. Every bounded range of values is
the range of values over some bounded interval. Here we use
L(<,F,T) to present the bounded range of values.
TRANSITION SIMILARITY. Any true statement involving x,y,
and a given transition z > x,y, remains true if z is
replaced by any transition w > z. Here we use L(<,F) to
present the true statement.
THEOREM 2.1.1. Linearity + Transitions + Arbitrary Bounded
Ranges + Transition Similarity interprets ZFC + "there
exists a subtle cardinal" and is interpretable in ZFC +
"there exists an almost ineffable cardinal". This is
provable in EFA.
TAIL TRANSITION SIMILARITY. Let x,y,z be specific times,
and let a true statement involving the times x,y, the
relation <, and “being a transition > x,y” be given. The
statement remains true about the times x,y, the relation <,
and “being a transition > x,y,z”. Here we use L(<,F) to
present the true statement.
THEOREM 2.1.2. Linearity + Transitions + Arbitrary Bounded
Ranges + Tail Transition Similarity interprets ZFC + "for
all x Õ w, x# exists" and is interpretable in ZFC + "there
exists a measurable cardinal". This is provable in EFA.
TRANSITION ACCUMULATION. There is a point which is the
limit of earlier transitions.
THEOREM 2.1.3. Linearity + Transitions + Arbitrary Bounded
Ranges + Tail Transition Similarity + Transition
Accumulation interprets ZFC + "there exists a measurable
cardinal k with kappa many measurable cardinals below k" and
is interpretable in ZFC + "there exists a measurable
cardinal k with normal measure 1 measurable cardinals below
k". This is provable in EFA.
STRONG TRANSITION ACCUMULATION. There is a transition which
is the limit of earlier transitions.
THEOREM 2.1.4. Linearity + Transitions + Arbitrary Bounded
Ranges + Tail Transition Similarity + Strong Transition
Accumulation interprets ZFC + "there exists a measurable
cardinal k of order ≥ k”, and is interpretable in ZFC +
“there exists a measurable cardinal above infinitely many
Woodin cardinals”. This is provable in EFA. 13 2.2. Two Varying Quantities, Three Separate Scales.
In most contexts from naive thinking, a quantity varies
over time, where the scale for that quantity is not
appropriately identified with the time scale. In
mathematical science, there is usually just one scale, as
the real numbers are normally used for all scales.
We can carry out a development that is analogous to that of
section 2.1, except that
i. We use two varying quantities rather than one.
ii. We use (and need) a deeper formulation of Arbitrary
Bounded Ranges.
We introduce axioms formulated in the three sorted
predicate calculus with equality on each sort.
The three sorts are: the time scale, the first quantity
scale, and the second quantity scale.
We use t,t1,t2,... for variables over the time scale.
We use x,x1,x2,... for variables over the first quantity
scale.
We use y,y1,y2,... for variables over the second quantity
scale.
We use the binary relation symbol <1 on the time scale.
We use the binary relation symbol <2 on the first quantity
scale.
We use the binary relation symbol <3 on the second quantity
scale.
We use the unary function symbol F from the time scale to
the first quantity scale.
We use the unary function symbol G from the time scale to
the second quantity scale.
We also use the unary relation symbol T on the time scale
for "transitions" (as in section 2.1).
We refer to this language as L[3,T]. If we do not use T,
then we write L[3]. 14
It will also be convenient to refer to the time scale as
"the first scale", the first quantity scale as "the second
scale", and the second quantity scale as "the third scale".
LINEARITY. <1,<2,<3 are strict linear orderings.
BOUNDEDNESS. The values of F on any bounded interval of
time are included in some bounded interval in the second
scale. The values of G on any bounded interval of time are
included in some bounded interval in the third scale.
TRANSITIONS. Every time is earlier than some transition.
By a "suitable pair" we mean a pair x,y, where x is from
the first scale and y is from the second scale.
ARBITRARY BOUNDED RANGES. Every bounded range of suitable
pairs is the range of values of F,G over some bounded time
interval. Here we use L[3,T] to present the bounded range
of suitable pairs.
We now present a more subtle form of Arbitrary Bounded
Ranges. This is a key idea needed to obtain high logical
strength in this section.
Let us digress into an informal discussion. Recall the
informal idea from the Introduction,
*time is so vast, that any possibility will eventually
occur*
A more refined form for our purposes is
**time is so vast, that any given possible behavior over
time intervals will be realized over some time interval**
Clearly Arbitrary Bounded Ranges is a special case of this
general principle, where the behavior over a time interval
focuses on the range of pairs of quantities.
However, there is a more subtle kind of behavior over a
time interval, that we can consider.
Let J be a time interval and x lie in the second scale (the
first quantity scale). We can consider the range of values
in the third scale (the second quantity scale) that occur
at some time in J where the second scale value is x. This 15
gives what we call a range of cross sections (in the third
scale)
over J. One cross section for each x.
We want to formulate a principle concerning the realization
of a (suitably bounded) arbitrary range of cross sections
over some time interval.
How are we going to specify "ranges of cross sections" in
order to formulate this principle? How are we going to
present "ranges of cross sections"?
The idea is very simple. Suppose we are presented an nary
relation R, n ≥ 2, where the last argument lies within the
third scale, and the earlier arguments are all from various
scales. It is clear what we mean by the range of cross
sections of R (obtained by fixing the first n1 arguments
in any way).
ARBITRARY BOUNDED CROSS SECTIONS. For any given bounded
relation R with sorted arguments, the last sort being the
third scale, there is a bounded time interval J such that
the range of cross sections of R is the same as the range
of cross sections over J. Furthermore, we can reverse the
role of the second and third scales in this principle. We
use L[3,T] to present R.
We say that a transition t is later than a point in any of
the three scales if and only if
i. If the point is a time t1 then t1 < t; and
ii. if the point is a quantity then that quantity appears
before time t.
TRANSITION SIMILARITY. Any true statement involving points
z,y,z in the various three scales and a transition later
than these points remains true if the transition is
replaced by any later transition. Here we use L[3] to
present the statement.
THEOREM 2.2.1. Linearity + Boundedness + Transitions +
Arbitrary Bounded Ranges + Arbitrary Bounded Cross Sections
+ Transition Similarity interprets ZFC + "there exists a
subtle cardinal" and is interpretable in ZFC + "there
exists an almost ineffable cardinal". This is provable in
EFA. 16
TAIL TRANSITION SIMILARITY. Let x,y,z,w be points in the
various three scales, and let a true statement involving
x,y,z, the relations <1,<2,<3, and “being a transition later
than x,y,z”. Then the statement remains true about x,y,z,
the relations <1,<2,<3, and “being a transition later than
x,y,z,w”.
THEOREM 2.2.2. Linearity + Boundedness + Transitions +
Arbitrary Bounded Ranges + Arbitrary Bounded Cross Sections
+ Tail Transition Similarity interprets ZFC + "for all x Õ
w, x# exists" and is interpretable in ZFC + "there exists a
measurable cardinal". This is provable in ZFC.
TRANSITION ACCUMULATION. There is a time which is the limit
of earlier transitions.
THEOREM 2.2.3. Linearity + Boundedness + Transitions +
Arbitrary Bounded Ranges + Arbitrary Bounded Cross Sections
+ Tail Transition Similarity + Transition Accumulation
interprets ZFC + "there exists a measurable cardinal
k with k many measurable cardinals below k" and is
interpretable in ZFC + "there exists a measurable cardinal k
with normal measure 1 measurable cardinals below k". This is
provable in EFA.
STRONG TRANSITION ACCUMULATION. There is a transition which
is the limit of earlier transitions.
THEOREM 2.2.4. Linearity + Boundedness + Transitions +
Arbitrary Bounded Ranges + Arbitrary Bounded Cross Sections
+ Tail Transition Similarity + Strong Transition
Accumulation interprets interpret ZFC + "there exists a
measurable cardinal k of order ≥ k" and is interpretable in
ZFC + "there exists a measurable cardinal above infinitely
many Woodin cardinal”. This is provable in EFA. 2.3. Varying Bit.
In this section, we work with a bit varying over time. This
is obviously far more elemental than the varying quantities
considered in sections 2.1, 2.2.
In order to get logical strength out of this very elemental
situation, we need to use forward translations of time.
This is most conveniently introduced through an addition
function. 17
Forward time translation is based on the idea that for any
times a,b, time going forward from a looks the same as time
going forward from b.
Conceptually, a+b from the point of view of a looking
forward, corresponds exactly to b from the point of view of
infinity looking forward. I.e., the "amount of time from a
to a+b is measured by b".
We also say that a+b is the result of translating a by b.
Let A be a set of points. The translation of A by b is the
set of all a+b such that a lies in A.
We work in the usual predicate calculus with equality with
the binary relation symbol <, the unary predicates T,P, and
the binary operation +.
Here, as before, T(t) means that "t is a transition".
P corresponds to the varying bit. P(t) means "the varying
bit at time t is 1".
We can give several informal interpretations of this setup.
a. A light is flashing on and off, forever. P(t) means that
the light is on at time t.
b. An agent is active or inactive, at any given time. P(t)
means that the agent is active at time t.
c. Instead of a time scale, we think of a physical ray, or
one dimensional space with a direction. There are
pointmasses at some positions. P(t) means that there is a
pointmass at position t. Or P(t) means that the mass
density at time t is 1; otherwise 0.
All three of these interpretations suggest some natural and
fundamental restrictions on behavior, that are not
considered in this section. There are considered in section
2.4 and chapter 5.
LINEARITY. < is a linear ordering.
TRANSITIONS. Every time is earlier than some transition.
BOUNDED TIME TRANSLATION. For every given range of times
before a given time b, there exists a translation time c 18
such that a time before b lies in the range of times if and
only if the bit at time b+c is 1. Here we use L(<,T,P,+) to
present the range of times.
The idea of Bounded Time Translation is our usual one: that
the behavior of P over bounded intervals is arbitrary, up
to translation. As mentioned above, various reasonable
objections can be made to this idea in the full generality
used here, some of which are met in the next section on
Flashing Lights.
TRANSITION SIMILARITY. Any true statement involving x,y and
a transition z > x,y, remains true if z is replaced by any
transition w > z. Here we use L(<,P,+) to present the
statement.
TAIL TRANSITION SIMILARITY. Let x,y,z be specific times,
and let a true statement involving x,y, the notions <,P,+,
and “being a transition > x,y” be given. The statement
remains true about x,y, the notions <,P,+, and “being a
transition > x,y,z”.
TRANSITION ACCUMULATION. There is a point which is the
limit of earlier transitions.
STRONG TRANSITION ACCUMULATION. There is a transition which
is the limit of earlier transitions.
Note that these axioms correspond to those given in section
2.1, except that we have incorporated addition in Bounded
Time Translation (instead of the previous Arbitrary Bounded
Ranges).
Note also that we do not need any fact whatsoever about
addition, except that which is embodied in Bounded Time
Translation.
THEOREM 2.3.1. Linearity + Transitions + Bounded Time
Translation + Transition Similarity interprets ZFC + "there
exists a subtle cardinal" and is interpretable in ZFC +
"there exists an almost ineffable cardinal". This is
provable in EFA.
THEOREM 2.3.2. Linearity + Transitions + Bounded Time
Translation + Tail Transition Similarity interprets ZFC +
"for all x Õ w, x# exists" and is interpretable in ZFC + 19
"there exists a measurable cardinal". This is provable in
EFA.
THEOREM 2.3.3. Linearity + Transitions + Bounded Time
Translation + Tail Transition Similarity + Transition
Accumulation interprets ZFC + "there exists a measurable
cardinal k with k many measurable cardinals below k" and is
interpretable in ZFC + "there exists a measurable cardinal k
with normal measure 1 measurable cardinals below k". This is
provable in EFA.
THEOREM 2.3.4. Linearity + Transitions + Bounded Time
Translation + Tail Transition Similarity + Strong
Transition Accumulation interprets ZFC + "there exists a
measurable cardinal k of order ≥ k" and is interpretable in
ZFC + "there exists a measurable cardinal above infinitely
many Woodin cardinals”. This is provable in EFA.
We would like to know that these theories are compatible
with substantial principles concerning time and addition.
This turns out to be the case, as we see in the next two
sections. 2.4. Persistently Varying Bit.
In this section, we weaken Bounded Time Translation
considerably to meet objections concerning what is viewed
as 'unrealistic' bit variation.
The idea is to reflect the view that a varying bit must be
persistent – or alternatively, that one is only interested
in persistently varying bits.
Specifically, the value of the bit at any time persists up
to some later time. I.e., if a bit is on then it remains on
for a while, and if a bit is off then it remains off for a
while.
In the case of discrete time, where every time has an
immediate successor, persistence is not a restriction.
However, for continuous time, or even dense time (where
between any two times there is a third), this is a major
restriction.
More specifically, we say that a range of times is
persistent if and only if for any time t there exists t1 > t
such that i) if t lies in the range of times then every 20
time in the interval [t,t1) lies in the range of times; if t
does not lie in the range of times then no time in the
interval [t,t1) lies in the range of times.
PERSISTENT TIME TRANSLATION. For any time b and persistent
range of times before b, there exists a translation time c
such that any time before b lies in the range of times if
and only if the bit at time b+c is 1. Here we use
L(<,T,P,+) to present the range of times.
The idea can be given in two closely related alternative
forms:
a. Any possible behavior of a varying bit over a bounded
time interval will occur + the behavior of a varying bit
must be persistent.
b. Any possible behavior of a persistently varying bit over
a bounded time interval will occur.
We cannot carry out the development in section 2.3 with
this fundamentally weakened Time Translation axiom scheme.
However, we can carry it out if we add
ADDITION. y < z implies x+y < x+z.
ORDER COMPLETENESS. Every nonempty range of times with an
upper bound has a least upper bound. Here we use L(<,T,P,+)
to present the nonempty range of times.
THEOREM 2.4.1. Linearity + Addition + Order Completeness +
Transitions + Persistent Time Translation + Transition
Similarity interprets ZFC + "there exists a subtle
cardinal" and is interpretable in ZFC + "there exists an
almost ineffable cardinal". This is provable in EFA.
THEOREM 2.4.2. Linearity + Addition + Order Completeness +
Transitions + Persistent Time Translation + Tail Transition
Similarity interprets ZFC + "for all x Õ w, x# exists" and
is interpretable in ZFC + "there exists a measurable
cardinal". This is provable in EFA.
THEOREM 2.4.3. Linearity + Addition + Order Completeness +
Transitions + Persistent Time Translation + Tail Transition
Similarity + Transition Accumulation interprets ZFC +
"there exists a measurable cardinal k with k many measurable
cardinals below k" and is interpretable in ZFC + "there 21
exists a measurable cardinal k with normal measure 1
measurable cardinals below k". This is provable in EFA.
THEOREM 2.4.4. Linearity + Addition + Order Completeness +
Transitions + Persistent Time Translation + Tail Transition
Similarity + Strong Transition Accumulation interprets ZFC
+ "there exists a measurable cardinal k of order ≥ k" and is
interpretable in ZFC + "there exists a measurable cardinal
above infinitely many Woodin cardinals”. This is provable
in EFA. 2.5. Naive Time.
In sections 2.3 and 2.4, we use an addition function. In
section 2.4, we used some properties of <,+. These
properties are fragments of a natural theory of Naive Time,
which is of interest in its own right.
In sections 2.3 and 2.4, we can add Naive Time to all of
the theories considered, and obtain the same results.
Naive Time uses <,+ (with identity). When we apply it, we
will generally have a larger language, and possibly sorts
in addition to the time scale sort.
NAIVE LINEARITY. < is a linear ordering with left endpoint
and no right endpoint.
NAIVE COMPLETENESS. Every range of times with an upper
bound has a least upper bound. Here we use L(<,+), or the
underlying language of the theory being axiomatized, to
present the nonempty range of times.
NAIVE ADDITION. For every x, the function x+y of y is
strictly increasing from all points onto the points ≥ x. We
call this the translation function at x. 0+x = x. x+(y+z) =
(x+y)+z. Here 0 is the left endpoint.
This completes the presentation of the axioms of Naive
Time.
Here are some consequences of Naive Time.
THEOREM 2.5.1. Every nonempty range of times has a greatest
lower bound. Here we use L(<,+) to present the nonempty
range of times, or whatever the underlying language of the
theory being considered. 22 THEOREM 2.5.2. If x £ y then there is a unique z such that
x+z = y.
THEOREM 2.5.3. 0+x = x+0 = x. y < z ´ x+y < x+z. y = z ´
x+y = x+z. For y > 0, x+y is the least strict upper bound
of the x+z, z < y. x £ y Æ x+z £ y+z.
Naive Time splits into two branches.
DISCRETENESS. There is an immediate successor of the left
endpoint.
DENSITY. There is no immediate successor of the left
endpoint.
THEOREM 2.5.4. Naive Time + Discreteness proves every point
has an immediate successor. Every nonempty range of values
has a least element. Here we use L(<,+), or the underlying
language of the theory being axiomatized, to present the
nonempty range of times.
THEOREM 2.5.5. Naive Time + Density proves that between any
two points there is a third.
In sections 2.3, 2.4, we can add Naive Time + Discreteness
to the various axiom systems considered and obtain the same
results. The same is true of Naïve Time + Density. 3. BINARY RELATIONS
3.1. Binary Relation, Single Scale.
Here we consider a time scale and a binary relation on
points in the time scale.
We can think of this as a naive Cartesian plane, with a
naively random pointset. Or we can think of it as naive two
dimensional space with a naively random distribution of
matter, where the point densities are 0 or 1.
Under such naive physical interpretations, it is important
to reflect the symmetry between first and second
coordinates.
Or we can think of the scale as the time scale, and a
single mind is reflecting on multiple past events as time 23
proceeds. Then R(t,t1) has the interpretation: the mind at
time t is reflecting on the past at time t1.
This immediately suggests a variant: that instead of a
binary relation on a scale, we look at a function f on the
scale, obeying f(x) < x. E.g., the mind, at any time, is
reflecting on exactly one past event.
Yet another variant: f(x) < x when defined. So we then use
a partial function. This corresponds to the mind, at any
time, reflecting on at most one past event.
Recall that a function f on a scale was already discussed
in detail in sections 1 and 2, and simplified in section 3.
We will discuss such variants in the future. This goes
under the heading: restricted (partial) functions on a
scale.
The axioms are formulated in the usual first order
predicate calculus with equality, based on the following:
1. The binary relation symbol <.
2. The binary relation R.
3. The unary relation T.
Here, as usual, T(x) means "x is a transition". We write
this language as L(<,R,T).
If we are thinking of the naive plane, or naive two
dimensional space, then transitions are like the
demarcations of physical or spatial horizons. For example,
consider
a.
b.
c.
d.
e.
f.
g. Looking out over the ocean.
Looking out over the solar system.
Looking out over the galaxy.
Looking out over the local group.
Looking out over the visible universe.
"Looking" out over the universe.
"Looking" out over the multiverse. Similarity phenomena have been noted by everyone. A
particularly pure form of it is the kind of similarity
embodied in our Transition Similarity axioms.
LINEARITY. < is a linear ordering with no right endpoint. 24
TRANSITIONS. Every point is earlier than some transition.
ARBITRARY BOUNDED RANGES (asymmetric). Every given range
of points £ x is precisely the points £ x that some y is
related to. Here we use L(<,R,T) to present the range of
points.
ARBITRARY BOUNDED RANGES (symmetric). Every given range of
points £ x is precisely the points £ x that some y is
related to, and precisely the points £ x that is related to
some z. Here we use L(<,R,T) to present the range of
points.
Note how appropriate Arbitrary Bounded Ranges (symmetric)
is under the "naive random" interpretation.
Note that there is a sharper form of Arbitrary Bounded
Ranges (asymmetric) that is obtained by removing the second
occurrence of "£ x" in Arbitrary Bounded Ranges. This
strengthening is not appropriate if we think of R as
naively random. However, it makes sense under some mind
interpretations.
TRANSITION SIMILARITY. Any true statement involving x,y and
a transition z > x,y, remains true if z is replaced by any
transition w > z. Here we use L(<,R) to present the
statement.
TAIL TRANSITION SIMILARITY. Let x,y,z be specific points,
and let a true statement involving x,y, the relation <, and
“being a transition > x,y” be given. The statement remains
true about x,y, the relation <, and “being a transition >
x,y,z”.
TRANSITION ACCUMULATION. There is a point which is the
limit of earlier transitions.
STRONG TRANSITION ACCUMULATION. There is a transition which
is the limit of earlier transitions.
THEOREM 3.1.1. Linearity + Transitions + Arbitrary Bounded
Ranges (symmetric) + Transition Similarity interprets ZFC +
"there exists a subtle cardinal" and is interpretable in
ZFC + "there exists an almost ineffable cardinal". This is
provable in EFA. 25
THEOREM 3.1.2. Linearity + Transitions + Arbitrary Bounded
Ranges (symmetric) + Tail Transition Similarity interprets
ZFC + "for all x Õ w, x# exists" and is interpretable in
ZFC + "there exists a measurable cardinal". This is
provable in ZFC.
THEOREM 3.1.3. Linearity + Transitions + Arbitrary Bounded
Ranges (symmetric) + Tail Transition Similarity +
Transition Accumulation interprets ZFC + "there exists a
measurable cardinal k with k many measurable cardinals below
k" and is interpretable in ZFC + "there exists a measurable
cardinal k with normal measure 1 measurable cardinals below
k". This is provable in EFA.
THEOREM 3.1.4. Linearity + Transitions + Arbitrary Bounded
Ranges (symmetric) + Tail Transition Similarity + Strong
Transition Accumulation interprets ZFC + "there exists a
measurable cardinal k of order ≥ k” and is interpretable in
ZFC + "there exists a measurable cardinal above infinitely
Woodin cardinals”. This is provable in EFA.
We now consider
SYMMETRY. R is symmetric. I.e., R(x,y) ´ R(y,x).
This makes good sense if we are thinking of two
communicating agents  i.e., two agents communicating over
time.
THEOREM 3.1.5. Theorems 3.1.1  3.1.4 hold if we add
Symmetry to all of the theories. We can also use Arbitrary
Bounded Ranges (asymmetric). 3.2. Binary Relation, Two Separate Scales.
Here we do not assume that the scale for one axis is the
same as the scale for the other.
We can also think of a binary relation on two separate
scales as an ensemble of data. I.e., we can plot a diagram
of pairs (height, weight) of persons. We can assert that
the two parts  height and weight  are completely
independent (which is of course not actually the case).
More abstractly, we can speak of
naive independence. 26
This leads of course to the idea that we can rework the
whole of probability and statistics as
naive probability theory.
naive statistics.
where the study will of course require that we go well
beyond the usual axioms for mathematics (ZFC), and even use
large cardinals.
We work with a two sorted predicate calculus, with equality
on each sort. These correspond to the two separate scales.
We use x,x1,x2,... for variables over the first scale.
We use y,y1,y2,... for variables over the second scale.
We use the binary relation symbol <1 on the first scale.
We use the binary relation symbol <2 on the second scale.
We use the binary relation symbol R whose first arguments
are from the first scale, and whose second arguments are
from the second scale.
We use the unary relation symbol T1 over the first scale.
We use the unary relation symbol T2 over the second scale.
T1(x) means that x is a transition in the first scale.
T2(y) means that y is a transition in the second scale.
The idea is that transitions demarcate horizons.
Transitions are unimaginably far apart. E.g., people say
they both play chess, but George is a true chess
professional. There is a transition from amateur to
professional chess player.
There are rating systems that demarcate different leagues
of chess players. There are also different leagues in
baseball.
But as we shall see, we are going to rely on the idea that
at the upper reaches of the two independent scales, there
are "more" fine gradations.
In scales used for measuring finite populations, this is
not going to be the case. As one moves up the scales, there
are fewer and fewer examples. 27 However, the situation is arguably quite different for
"abstract populations", which consider all possibilities.
LINEARITY. <1,<2 are linear orderings on the first and
second scales, respectively, with no right endpoints.
TRANSITIONS. Every point in the first scale is less than
some first scale transition. Every point in the second
scale is less than some second scale transition.
ARBITRARY BOUNDED RANGES. Every bounded range of points
from the second scale is precisely the points that some
point from the first scale is related to. Every bounded
range of points from the first scale is precisely the
points that are related to some point from the first scale.
Here we use L(<1,<2,T1,T2,R) to present bounded ranges of
points.
TRANSITION SIMILARITY. Any true statement involving points
x1,x2 in the first scale and a transition x > x1,x2 remains
true if x is replaced by any transition > x. Any true
statement involving points y1,y2 in the second scale and a
transition y > y1,y2 remains true if y is replaced by any
transition > y. Here we use L(<1,<2,R) to present the true
statements.
TAIL TRANSITION SIMILARITY. Let x,y,z be points in the
first scale, and let a true statement involving x,y, the
relations <1,<2,R, and “being a transition > x,y in the
first scale” be given. The statement remains true about
x,y, the relations <1,<2,R, and “being a transition > x,y,z
in the first scale”. Let x,y,z be points in the second
scale, and let a true statement involving x,y, the
relations <1,<2,R, and “being a transition > x,y in the
second scale” be given. The statement remains true about
x,y, the relations <1,<2,R, and “being a transition > x,y,z
in the second scale”.
TRANSITION ACCUMULATION. There is a point in the first
scale which is the limit of earlier transitions. There is a
point in the second scale which is the limit of earlier
transitions.
STRONG TRANSITION ACCUMULATION. There is a transition in
the first scale which is the limit of earlier transitions. 28
There is a transition in the second scale which is the
limit of earlier transitions.
THEOREM 3.2.1. Linearity + Transitions + Arbitrary Bounded
Ranges + Transition Similarity interprets ZFC + "there
exists a subtle cardinal" and is interpretable in ZFC +
"there exists an almost ineffable cardinal". This is
provable in EFA.
THEOREM 3.2.2. Linearity + Transitions + Arbitrary Bounded
Ranges + Tail Transition Similarity interprets ZFC + "for
all x Õ w, x# exists" and is interpretable in ZFC + "there
exists a measurable cardinal". This is provable in EFA.
THEOREM 3.2.3. Linearity + Transitions + Arbitrary Bounded
Ranges + Tail Transition Similarity + Transition
Accumulation interprets ZFC + "there exists a measurable
cardinal k with k many measurable cardinals below k" and is
interpretable in ZFC + "there exists a measurable cardinal k
with normal measure 1 measurable cardinals below k". This is
provable in EFA.
THEOREM 3.2.4. Linearity + Transitions + Arbitrary Bounded
Ranges + Tail Transition Similarity + Strong Transition
Accumulation interprets ZFC + "there exists a measurable
cardinal k of order ≥ k”, and is interpretable in ZFC +
"there exists a measurable cardinal above infinitely many
Woodin cardinals”. This is provable in EFA. 4. MULTIPLE AGENTS, TWO STATES.
We use a two sorted predicate calculus, with equality on
each sort. The two sorts are the time sort, and the agent
(mind) sort.
We use
i. Variables t,t1,t2,... over times.
ii. Variables x,x1,x2,... over agents (minds).
iii. Binary relation < on the time sort.
iv. Binary relation A relating agents and times.
v. Unary relation T on the time sort.
Here A(x,t) means "mind x is active at time t". T(t) means
"t is a transition". 29
The idea is that an agent will be active at some times and
not at other times. The first time an agent is active is
regarded as its date of its creation. Although it is
natural to do so, we will not assume that every agent has a
birthdate (date of creation). However, the axiom of
Continual Creation asserts that every time is the birthdate
of some agent.
LINEARITY. < is a linear ordering.
TRANSITIONS. Every time is < some transition.
CONTINUAL CREATION. At any time there is some agent which
is active at that time but not previously.
UNRESTRICTED ACTIVITY. Let a time t be given. As time
varies, the then active agents that have been active
previous to t, are arbitrary. Here we use L(<,A,T) to
present the arbitrary condition.
TRANSITION SIMILARITY. Any true statement involving times
t1,t2 and a later transition t remains true t is replaced by
any later transition. Here we use L(<,A) to present the
true statement.
TAIL TRANSITION SIMILARITY. Lte times t1,t2,t3 be given, and
let a true statement involving times t1,t2, the relations
<,A, and “being a transition > t1,t2” be given. The
statement remains true about t1,t2, the relations <,A, and
“being a transition > t1,t2,t3”.
TRANSITION ACCUMULATION. There is a point which is the
limit of earlier transitions.
STRONG TRANSITION ACCUMULATION. There is a transition which
is the limit of earlier transitions.
THEOREM 4.1. Linearity + Transitions + Continual Creation +
Unrestricted Activity + Transition Similarity interprets
ZFC + "there exists a subtle cardinal" and is interpretable
in ZFC + "there exists an almost ineffable cardinal". This
is provable in EFA.
THEOREM 4.2. Linearity + Transitions + Continual Creation +
Unrestricted Activity + Tail Transition Similarity
interprets ZFC + "for all x Õ w, x# exists" and is 30
interpretable in ZFC + "there exists a measurable
cardinal". This is provable in ZFC.
THEOREM 4.3. Linearity + Transitions + Continual Creation +
Unrestricted Activity + Tail Transition Similarity +
Transition Accumulation interprets ZFC + "there exists a
measurable cardinal k with k many measurable cardinals below
k" and is interpretable in ZFC + "there exists a measurable
cardinal k with normal measure 1 measurable cardinals below
k". This is provable in EFA.
THEOREM 4.4. Linearity + Transitions + Continual Creation +
Unrestricted Activity + Tail Transition Similarity + Strong
Transition Accumulation interprets ZFC + "there exists a
measurable cardinal k of order ≥ k”, and is interpretable in
ZFC + “there is a measurable cardinal above infinitely many
Woodin cardinals”. This is provable in EFA. 5. POINT MASSES
5.1. Discrete Point Masses In One Dimensional
Space.
We will treat one dimensional space as we did time  a
linearly ordered ray, with no right endpoint. Here one
dimensional space is made up of points. At some points
there lies a point mass. So the point masses form a
subclass of
the points.
Throughout this chapter, our treatment of point masses is
very primitive – we do not assign masses to point masses.
Nor do these point masses move.
Consider the case of point masses whose distribution is
unrestricted. This is exactly the case of a Varying Bit
treated in section 2.3.
We want to consider the case of point masses which are
discrete (discretely arranged). I.e., they are isolated
from each other.
This situation has much in common with section 2.4, but is
a little bit different. So we restate the results. 31
The language is again predicate calculus with equality in
the language <,T,P,+. T(x) means "x is a transition
(horizon) point". P(x) means "there is a point mass at
position x".
LINEARITY. < is a linear ordering.
ADDITION. y < z implies x+y < x+z.
ORDER COMPLETENESS. Every nonempty range of points with an
upper bound has a least upper bound. Here we use L(<,T,P,+)
to present the nonempty range of points.
TRANSITIONS. Every point is earlier than some transition.
We say that a range of points is discrete if and only if
for any x in the range, there exists y,z such that y < x <
z and x is the only point in the range in the open interval
(y,z).
DISCRETE POINT MASS TRANSLATION. For any point b and
discrete range of points before b, there exists a
translation distance c such that any point x lies in the
range of points if and only if there is a point mass at
position x+c. Here we use L(<,T,P,+) to present the
discrete range of points.
TRANSITION SIMILARITY. Any true statement involving x,y and
a transition z > x,y, remains true if z is replaced by any
transition w > z. Here we use L(<,P,+) to present the
statement.
TAIL TRANSITION SIMILARITY. Let x,y,z be points, and let a
true statement involving x,y, the notions <,P,+, and “being
a transition > x,y” be given. The statement remains true
about x,y, the notions <,P,+, and “being a transition >
x,y,z”.
TRANSITION ACCUMULATION. There is a point which is the
limit of earlier transitions.
STRONG TRANSITION ACCUMULATION. There is a transition which
is the limit of earlier transitions.
THEOREM 5.1.1. Linearity + Addition + Order Completeness +
Transitions + Discrete Point Mass Translation + Transition
Similarity interprets ZFC + "there exists a subtle 32
cardinal" and is interpretable in ZFC + "there exists an
almost ineffable cardinal". This is provable in EFA.
THEOREM 5.1.2. Linearity + Addition + Order Completeness +
Transitions + Discrete Point Mass Translation + Tail
Transition Similarity interprets ZFC + "for all x Õ w, x#
exists" and is interpretable in ZFC + "there exists a
measurable cardinal". This is provable in EFA.
THEOREM 5.1.3. Linearity + Addition + Order Completeness +
Transitions + Discrete Point Mass Translation + Tail
Transition Similarity + Transition Accumulation interprets
ZFC + "there exists a measurable cardinal k with
k many measurable cardinals below k" and is interpretable in
ZFC + "there exists a measurable cardinal k with normal
measure 1 measurable cardinals below k". This is provable in
EFA.
THEOREM 5.1.4. Linearity + Addition + Order Completeness +
Transitions + Discrete Point Mass Translation + Tail
Transition Similarity + Strong Transition Accumulation
interprets ZFC + "there exists a measurable cardinal
k of order ≥ k" and is interpretable in ZFC + "there exists
a measurable cardinal above infinitely many Woodin
cardinals". This is provable in EFA.
As in section 2.4, we can add the naive time principles
reformulated as naive one dimensional space principles, in
the sense of section 2.5, without changing any of the above
results. We can choose between Discreteness and Density. We
can also add the axiom "the point masses are discrete"
without changing the results.
We now come to bodies. We take this to be non overlapping
closed intervals, where the endpoints form a discrete set.
This is very similar to what we have just done, although
not quite the same. The same results hold. 5.2. Discrete Point Masses With End Expansion.
Here we will consider exactly two snapshots of one
dimensional space. The first snapshot we will call "the
present". The second snapshot we will call "the future".
Every present point is a future point, but not vice versa.
The variables range over the future points. 33
We will also assume that we have a set of point masses.
Every point mass that exists at present also exists at the
future, in the same position. However, new point masses may
be created, at new points.
We do not take into account motion of point masses.
We use predicate calculus with equality, with the following
additional symbols.
1. The binary relation symbol < on all points; i.e., points
of one dimensional space at the future.
2. The unary relation symbol P where P(x) means "there is a
point mass at position x in one dimensional space".
3. The unary relation symbol R where R(x) means "x is a
point in one dimensional space at the present".
4. Addition, +.
Note that we do not use transition points.
LINEARITY. < is a linear ordering.
ADDITION. y < z implies x+y < x+z.
ORDER COMPLETENESS (present). Every nonempty range of
points in the present, with an upper bound in the present,
has a least upper bound in the sense of the present. Here
we use L(<,P,R,+) to present the nonempty range of points.
ORDER COMPLETENESS (future). Every nonempty range of
points, with an upper bound, has a least upper bound. Here
we use L(<,P,R,+) to present the nonempty range of points.
END EXPANSION. If a point is before some point that exists
in the present, then that point also exists in the present.
There is a point that is not in the present.
PROPERTY PRESERVATION. Any true statement stated in terms
of the points existing at the present, and involving a
given point existing at the present, remains true when
stated in terms of all points, and the given point. Here we
use L(<,P,+) to present the true statement.
The idea of Property Preservation is that the expansion of
space does not effect any property of points. 34
DISCRETE POINT MASS TRANSLATION (present). For any point b
in the present, and discrete range of points before b in
the present, there exists a translation distance c in the
present such that any present point x lies in the range of
points if and only if there is a point mass at position
x+c. Here
we use L(<,P,R,+) to present the discrete range of points.
DISCRETE POINT MASS TRANSLATION (future). For any point b
and discrete range of points before b, there exists a
translation distance c such that any point x before b lies
in the range of points if and only if there is a point mass
at position x+c. Here we use L(<,P,R,+) to present the
discrete range of points.
It can be seen that Discrete Point Mass Translation
(future) and Order Completeness (present) follow from the
preceding axioms.
THEOREM 5.2.1. Linearity + Addition + Order Completeness
(present,future) + End Expansion + Property Preservation +
Discrete Point Mass Translation (present,future) is
mutually interpretable with ZFC. This is provable in EFA.
We can add the naive time principles reformulated as naive
one dimensional space principles, in the sense of section
2.5, without changing any of the above results. We can
choose between Discreteness and Density. We can also add
the axiom "the point masses are discrete" without changing
this result. 5.3. Discrete Point Masses With Inner Expansion.
We carry out the development in section 5.2, but with the
idea that there exists a point not in the present which is
earlier than some point in the present.
We use the same language as in section 5.2.
NAIVE LINEARITY. < is a linear ordering with left endpoint
and no right endpoint.
NAIVE ADDITION. For every x,
strictly increasing from all
We call this the translation
x+(y+z) = (x+y)+z. Here 0 is the function x+y of y is
points onto the points >= x.
function at x. 0+x = x.
the left endpoint. 35
ORDER COMPLETENESS (present). Every nonempty range of
points in the present, with an upper bound in the present,
has a least upper bound in the sense of the present. Here
we use L(<,P,R,+) to present the nonempty range of points.
ORDER COMPLETENESS (future). Every nonempty range of
points, with an upper bound, has a least upper bound. Here
we use L(<,P,R,+) to present the nonempty range of points.
INNER EXPANSION. There are points x < y such that [x,y]
contains no points in the present.
PROPERTY PRESERVATION. Any true statement stated in terms
of the points existing at the present, and involving a
given point existing at the present, remains true when
stated in terms of all points, and the given point. Here we
use L(<,P,+) to present the true statement.
POINT MASS TRANSLATION (present). For any point b in the
present, and range of points before b in the present, there
exists a translation distance c in the present such that
any present point x lies in the range of points if and only
if there is a point mass at position x+c. Here we use
L(<,P,R,+) to present the range of points.
POINT MASS TRANSLATION (future). For any point b and
discrete range of points before b, there exists a
translation distance c such that any point x lies in the
range of points if and only if there is a point mass at
position x+c. Here we use L(<,P,R,+) to present the range
of points.
THEOREM 5.3.1. Naive Linearity + Naive Addition + Order
Completeness (present,future) + Inner Expansion + Property
Preservation + Point Mass Translation (present,future)
interprets ZFC + "there exists a Ramsey cardinal" and is
interpretable in ZFC + "there exists a measurable
cardinal".
We have a number of choices of additional axioms, without
changing the results.
i. Discreteness of points.
ii. Density of points.
iii. Discreteness of point masses.
iv. Every point is earlier than some present point.
v. There is a point later than all present points. 36 Of course, if we add ii then we cannot add i. If we add iv
then we cannot add v. 5.4. Point Masses With Inner Expansion.
Here we carry out the development of section 5.3 without
restricting to discreteness. We aim for extremely high
logical strength.
We will use "transition points of the present". It makes
sense to also have transition points associated with the
future, but we will not need these.
We use predicate calculus with equality, with the following
additional symbols.
1. The binary relation symbol < on all points; i.e., points
of one dimensional space at the future.
2. The unary relation symbol P where P(x) means "there is a
point mass at position x in one dimensional space".
3. The unary relation symbol R where R(x) means "x is a
point in one dimensional space at the present".
4. Addition, +.
5. The unary relation symbol T, where T(x) means "x is a
transition point of the present".
NAIVE LINEARITY. < is a linear ordering with left endpoint
and no right
endpoint.
NAIVE ADDITION. For every x,
strictly increasing from all
We call this the translation
x+(y+z) = (x+y)+z. Here 0 is the function x+y of y is
points onto the points >= x.
function at x. 0+x = x.
the left endpoint. ORDER COMPLETENESS (present). Every nonempty range of
points in the present, with an upper bound in the present,
has a least upper bound in the sense of the present. Here
we use L(<,P,R,T,+) to present the nonempty range of
points.
ORDER COMPLETENESS (future). Every nonempty range of
points, with an upper bound, has a least upper bound. Here
we use L(<,P,R,T,+) to present the nonempty range of
points. 37
INNER EXPANSION. There are points x < y such that [x,y]
contains no points in the present.
PROPERTY PRESERVATION. Any true statement stated in terms
of the points existing at the present, and involving a
given point existing at the present, remains true when
stated in terms of all points, and the given point. Here we
use L(<,P,T,+) to present the true statement.
The idea of Property Preservation is that the expansion of
space does not effect the relationship between present
points and present point masses.
POINT MASS TRANSLATION (present). For any point b in the
present, and range of points in the present before b, there
exists a translation distance c in the present such that
any present point x lies in the range of points if and only
if there is a point mass at position x+c. Here we use
L(<,P,R,T,+) to present the range of points.
POINT MASS TRANSLATION (future). For any point b and
discrete range of points before b, there exists a
translation distance c such that any point x lies in the
range of points if and only if there is a point mass at
position x+c. Here we use L(<,P,R,T,+) to present the range
of points.
TRANSITIONS. Every transition is at the present. Every
present point is earlier than some transition point.
We now come to Transition Similarity. We need a natural
strengthening of this principle.
EXTENDED TRANSITION SIMILARITY. Any true statement
involving a range of points before a point before a
transition, and that transition, remains true if we use any
later transition. Here we use L(<,P,R,T,+) and any points,
to present the range of points, and L(<,P,R,+) to present
the true statement.
THEOREM 5.4.1. Naive Linearity + Naïve Addition + Order
Completeness (present,future) + Inner Expansion + Property
Preservation + Point Mass Translation (present,future) +
Transitions + Extended Transition Similarity interprets ZF
+ “there exists a nontrivial elementary embedding from some
V(k) into V(k), where V(k) is an elementary substructure of
V (scheme)”. Hugh Woodin has shown some time ago that ZFC + 38
“there exists a nontrivial elementary embedding from some
successor rank into itself” is interpretable in this
theory, and hence into Naive Linearity + Naive Addition +
Order Completeness (present,future) + Inner Expansion +
Property Preservation + Point Mass Translation
(present,future) + Transitions + Extended Transition
Similarity.
We have a number of choices of additional axioms, without
changing the results.
i. Discreteness of points.
ii. Density of points.
iii. Every point is earlier than some present point.
iv. There is a point later than all present points.
Of course, if we add iv then we cannot add v. 5.5. Discrete Point Masses With Inner Expansion
Revisited.
We carry out the development in section 5.4, but with the
idea that the point masses are discrete. In order to carry
this off, we need to consider three snapshots of one
dimensional space. We call these, respectively, the present
points, the intermediate points, and the future points.
Every present point is an intermediate point, and every
intermediate point is a future point. The variables range
over the future points.
We use the language
1. The binary relation symbol < on all points; i.e., points
of one dimensional space at the future.
2. The unary relation symbol P where P(x) means "there is a
point mass at position x in one dimensional space".
3. The unary relation symbol R where R(x) means "x is a
point in one dimensional space at the present".
4. The unary relation symbol S where S(x) means "x is a
point in one dimensional space at the intermediate".
5. Addition, +.
6. The unary relation symbol T, where T(x) means "x is a
transition point of the present".
NAIVE LINEARITY. < is a linear ordering with left endpoint
and no right endpoint. 39
NAIVE ADDITION. For every x, the function x+y of y is
strictly increasing from all points onto the points ≥ x. We
call this the translation function at x. 0+x = x. x+(y+z) =
(x+y)+z. Here 0 is the left endpoint.
ORDER COMPLETENESS (present). Every nonempty range of
points in the present, with an upper bound in the present,
has a least upper bound in the sense of the present. Here
we use L(<,P,R,S,T,+) to present the nonempty range of
points.
ORDER COMPLETENESS (intermediate). Every nonempty range of
points in the intermediate, with an upper bound in the
intermediate, has a least upper bound in the sense of the
intermediate. Here we use L(<,P,R,S,T,+) to present the
nonempty range of points.
ORDER COMPLETENESS (future). Every nonempty range of
points, with an upper bound, has a least upper bound. Here
we use L(<,P,R,S,T,+) to present the nonempty range of
points.
INNER EXPANSION. There are points x < y such that [x,y]
contains no points in the present.
DOUBLE PROPERTY PRESERVATION. Any true statement stated in
terms of the points existing at the present, the points
existing at the intermediate, and involving a given point
existing at the present, remains true when stated in terms
of the points existing at the intermediate, the points
existing at the future (i.e., all points), and the given
point. Here we use L(<,P,T,+) to present the true
statement.
DISCRETE POINT MASS TRANSLATION (present). For any point b
in the present, and discrete range of points in the present
before b, there exists a translation distance c in the
present such that any present point x lies in the range of
points if and only if there is a point mass at position
x+c. Here we use L(<,P,R,S,T,+) to present the discrete
range of points.
DISCRETE POINT MASS TRANSLATION (intermediate). For any
point b in the intermediate, and discrete range of points
in the intermediate before b, there exists a translation
distance c in the intermediate such that any intermediate 40
point lies in the range of points if and only if there is a
point
mass at position x+c. Here we use L(<,P,R,S,T,+) to present
the discrete range of points.
DISCRETE POINT MASS TRANSLATION (future). For any point b
and discrete range of points before b, there exists a
translation distance c such that any point x lies in the
range of points if and only if there is a point mass at
position x+c. Here we use L(<,P,R,S,T,+) to present the
discrete range of points.
TRANSITIONS. Every transition is at the present. Every
present point is earlier than some transition point.
EXTENDED TRANSITION SIMILARITY. Any true statement
involving a range of points before a point before a
transition, and that transition, remains true if we use any
later transition. Here we use L(<,P,R,S,T,+) and any
points, to present the range of points, and L(<,P,R,S,+) to
present the true
statement.
THEOREM 5.5.1. Naive Linearity + Naive Addition + Order
Completeness (present,intermediate,future) + Inner
Expansion + Double Property Preservation + Point Mass
Translation (present,intermediate,future) + Transitions +
Extended Transition Similarity interprets NBG + "there is a
nontrivial
elementary embedding from V into V" interprets ZF + “there
exists a nontrivial elementary embedding from some V(k) into
V(k), where V(k) is an elementary substructure of V
(scheme)”. Hugh Woodin has shown some time ago that ZFC +
“there exists a nontrivial elementary embedding from some
successor rank into itself” is interpretable in this
theory, and hence into Naive Linearity + Naive Addition +
Order Completeness (present,intermediate,future) + Inner
Expansion + Double Property Preservation + Point Mass
Translation (present,intermediate,future) + Transitions +
Extended Transition Similarity.
We have a number of choices of additional axioms, without
changing the results.
i. Discreteness of points.
ii. Density of points.
iii. Discreteness of point masses. 41
iv. Every point is earlier than some present point.
v. There is a point later than all present points.
Of course, if we add ii then we cannot add i. If we add iv
then we cannot add v. 6. TOWARDS THE MEROLOGICAL
In all of the theories thus far, with the exception of the
theories in section 1.2, we have used a unary predicate P
on the time scale, where P(t) means "t is a transition".
The idea is that transition points mark out epochs.
However, we can take the view that changing epochs are not
marked by single points. Thus epochs are regions of time.
We view this as a small step towards a fully mereological
reworking of our results.
Under this approach, instead of the unary predicate T, we
have a binary relation E, where E(x,y) means that x,y are
in the same epoch. Thus we have the axiom
EPOCHS. E forms an equivalence relation, where every
equivalence class under E forms a bounded interval
(however, the endpoints of these equivalence classes may or
may not exist).
Transition Similarity is replaced by
EPOCH SIMILARITY. Let x,y,z,w be three given times, where
z,w reside in an epoch later than x,y. Let a true statement
involving x,y, the relevant notions, and “being in the
epoch in which z resides” be given. Then the statement
remains true about x,y, the relevant notions, and “being in
the epoch in which w resides”.
Tail Transition Similarity is replaced by two different
forms, one weaker than the other:
TAIL EPOCH SIMILARITY (1). Let x,y,z be three given times,
and let a true statement involving x,y, the < relation and
other relevant notions, and “being in an epoch later than
x,y”. The statement remains true involving x,y, the
relevant notions, and “being in an epoch later than x,y,z”. 42
TAIL EPOCH SIMILARITY (2). Let x,y,z be three given times,
and let a true statement involving x,y, the < relation and
other relevant notions, and the relation E restricted to
points in an epoch later than x,y”. The statement remains
true involving x,y, the relevant notions, and E restricted
to points in an epoch later than x,y,z.
Transition Accumulation is replaced by
EPOCH ACCUMULATION. There is an epoch which is a limit of
earlier epochs.
The results conform to the following analogies.
Transitions: Epochs.
Transition Similarity: Epoch Similarity (2).
Tail Transition Similarity: Tail Epoch Similarity.
Strong Transition Accumulation: Epoch Accumulation.
Transition Accumulation does not have an obvious analog in
the epoch formulation.
Fully mereological formulations are expected to be of clear
importance and require the banishment of points in favor of
intervals (where endpoints have no significance). ...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Calculus

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