[FOM] 293: Concept Calculus 2
Harvey Friedman
friedman at math.ohio-state.edu
Tue Jun 20 18:27:03 EDT 2006
We continue from Concept Calculus 1, June 17, 2006.
Section 7, Multiple Agents, Two States, reworks FMPC postings 0606002 and
0606003, see http://www.cs.uky.edu/fmpc/fmpcMembers.html
which are the same as #291, #292 on FOM, called Independently Free
Minds/Collectively Random Agents, Independently Free Minds/Collectively
Random Agents/more.
***************************************
CONCEPT CALCULUS
by
Harvey M. Friedman
June 20, 2006
Concept Calculus 1 has sections 1,2, and is rather detailed.
We did not make it clear in Concept Calculus 1, from sections 1,2, what the
quickest and simplest way is to arrive at a system in which ZFC can be
interpreted, thus providing a consistency proof for mathematics.
Thus we have added a section 3 (see table of contents below).
We keep sections 1 and 2 in order to indicate just how we can get a variety
of logical strengths. However, we do NOT take such a systematic approach in
the later sections, preferring to go directly after levels >= ZFC. We will
return to more systematic presentations later.
We now start with section 3.
What is emerging is that there are two fundamental principles:
Completeness/Randomness/Creativity (anything that can happen will).
Symmetry/Indiscernibility (various horizon effects).
What is also emerging is RIGOROUS theories of
naive probability.
naive statistics.
naive geometry.
naive physics.
naive theory of agents (minds).
naive differential equations?
naive biology?
naive psychology?
...
naive anything?
We are, of course, avoiding normal mathematical science, with its natural
numbers and real numbers. They are incompatible with our two principles
COMBINED - although they are just fine with these two principles, ALONE.
Yet, surprisingly, mathematics itself (ZFC and even ZFC + large cardinals)
is interpretable, rigorously, in these naïve theories.
Introduction.
1. Varying Quantity, Common Scale.
2. Varying Quantity, Common Scale, Transitions.
3. Varying Quantity - SIMPLIFIED.
4. Two Varying Quantities, Three Separate Scales.
5. Binary Relation, Single Scale.
6. Binary Relation, Two Separate Scales.
7. Multiple Agents, Two States.
8. Structure of Naive Time.
9. Naive Time, Varying Bit.
10. Transitions as Equivalence Relations.
11. Monotonicity, Continuity, Discreteness.
3. VARYING QUANTITY - SIMPLIFIED.
We go directly after a system in which ZFC can be interpreted. Here is the
simplest way we know how to do this.
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.
T(x) is read "x is a transition". They represent time horizons.
Bounded intervals are intervals of the form
(x,y), [x,y], (x,y], [x,y)
where x,y are points. We allow x >= y.
LINEARITY. < is a linear ordering.
TRANSITIONS. Every time is < some transition.
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 3.1. 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. Any true statement involving x,y remains true if
we hypothetically restrict the transitions to those that are <= max(x,y) or
>= any transition w > max(x,y). Here we use L(<,F,T) to present the true
statement.
In the above, we use all formulas in L(<,F,T). See section 2 for a precise
formulation.
THEOREM 3.2. Linearity + Transitions + Arbitrary Bounded Ranges + Tail
Transition Similarity interprets ZFC + "for all x containedin omega, x#
exists" and is interpretable in ZFC + "there exists a measurable cardinal".
This is provable in EFA.
NOTE: In Theorem 2.6, we wrote "This is provable in ZFC". This should be
"This is provable in EFA".
TRANSITION ACCUMULATION. There is a point which is the limit of earlier
transitions.
THEOREM 3.3. Linearity + Transitions + Arbitrary Bounded Ranges + Tail
Transition Similarity + Transition Accumulation interprets ZFC + "there
exists a measurable cardinal kappa with kappa many measurable cardinals
below kappa" and is interpretable in ZFC + "there exists a measurable
cardinal kappa with normal measure 1 measurable cardinals below kappa". This
is provable in EFA.
STRONG TRANSITION ACCUMULATION. There is a transition which is the limit of
earlier transitions.
THEOREM 3.4. Linearity + Transitions + Arbitrary Bounded Ranges + Tail
Transition Similarity + Strong Transition Accumulation interprets ZFC +
"there exists a measurable cardinal kappa with normal measure 1 measurable
cardinals below kappa (order >= 2)" and is interpretable in ZFC + "there
exists a measurable cardinal kappa of order >= 3". This is provable in EFA.
4. TWO VARYING QUANTITIES, THREE SEPARATE SCALES, TRANSITIONS.
In most contexts from naïve thinking, a quantity varies over time, where the
scale for that quantity is not appropriately identified with the time scale.
In mathematical science, it usually is, because real numbers are normally
used for all scales.
We can carry out a development that is analogous to that of section 3,
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 3).
We refer to this language as L[3,T]. If we do not use T, then we write L[3].
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
section 1,
*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."
A more explicit form 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.
Thus we can say this:
#the comprehension axiom scheme or separation axiom scheme can be viewed as
a kind of informal, intuitive, naïve probability theory - a naive form of
(one of the commonly stated forms of) the law of large numbers.#
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 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 n-ary 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 n-1 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 several points in the
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 4.1. 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.
TAIL TRANSITION SIMILARITY. Any true statement involving several points in
the three scales remains true if we hypothetically restrict the transitions
to those that are not later than all of these points together with those
that are at least any given transition later than all of these points. Here
we use L[3,T] to present the statement.
THEOREM 4.2. Linearity + Boundedness + Transitions + Arbitrary Bounded
Ranges + Arbitrary Bounded Cross Sections + Tail Transition Similarity
interprets ZFC + "for all x containedin omega, 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 4.3. Linearity + Boundedness + Transitions + Arbitrary Bounded
Ranges + Arbitrary Bounded Cross Sections + Tail Transition Similarity +
Transition Accumulation interprets ZFC + "there exists a measurable cardinal
kappa with kappa many measurable cardinals below kappa" and is interpretable
in ZFC + "there exists a measurable cardinal kappa with normal measure 1
measurable cardinals below kappa". This is provable in EFA.
NOTE: In Theorem 2.6, I wrote "provable in ZFC". This should be "provable in
EFA".
STRONG TRANSITION ACCUMULATION. There is a transition which is the limit of
earlier transitions.
THEOREM 4.4. Linearity + Boundedness + Transitions + Arbitrary Bounded
Ranges + Arbitrary Bounded Cross Sections + Tail Transition Similarity +
Strong Transition Accumulation interprets interpret ZFC + "there exists a
measurable cardinal kappa with normal measure 1 measurable cardinals below
kappa (order >= 2)" and is interpretable in ZFC + "there exists a measurable
cardinal kappa of order >= 3". This is provable in EFA.
5. 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 naïve Cartesian plane, with a naively random
pointset. Or we can think of it as naïve 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 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 naïve plane, or naïve two dimensional space, then
transitions are like the demarcations of physical or spatial horizons. For
example, consider
a. Looking out over the ocean.
b. Looking out over the solar system.
c. Looking out over the galaxy.
d. Looking out over the local group.
e. Looking out over the visible universe.
f. "Looking" out over the universe.
g. "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.
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 formulas
from the language. 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. Any true statement involving x,y remains true if
we hypothetically restrict the transitions to those that are <= max(x,y)
together with those that are >= any given transition > x. Here we use
L(<,R,T) to present the true statement.
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. 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.
THEOREM 5.2. Linearity + Transitions + Arbitrary Bounded Ranges (symmetric)
+ Tail Transition Similarity interprets ZFC + "for all x containedin omega,
x# exists" and is interpretable in ZFC + "there exists a measurable
cardinal". This is provable in ZFC.
THEOREM 5.3. Linearity + Transitions + Arbitrary Bounded Ranges (symmetric)
+ Tail Transition Similarity + Transition Accumulation interprets ZFC +
"there exists a measurable cardinal kappa with kappa many measurable
cardinals below kappa" and is interpretable in ZFC + "there exists a
measurable cardinal kappa with normal measure 1 measurable cardinals below
kappa". This is provable in EFA.
THEOREM 5.4. Linearity + Transitions + Arbitrary Bounded Ranges (symmetric)
+ Tail Transition Similarity + Strong Transition Accumulation interprets ZFC
+ "there exists a measurable cardinal kappa with normal measure 1 measurable
cardinals below kappa (order >= 2)" and is interpretable in ZFC + "there
exists a measurable cardinal kappa of order >= 3". This is provable in EFA.
We now consider
SYMMETRY. R is symmetric. I.e., R(x,y) iff R(y,x).
This makes good sense if we are thinking of two communicating agents - i.e.,
two agents communicating over time.
THEOREM 5.5. Theorems 5.1-5.4 hold if we add Symmetry to all of the
theories. We can also use Arbitrary Bounded Ranges (asymmetric).
6. 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.
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.
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.
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. Any true statement involving x1,x2 in the first
scale remains true if we hypothetically restrict the transitions to those
that are <= max(x1,x2) or >= any transition x > max(x1,x2). Any true
statement involving y1,y2 in the second scale remains true if we
hypothetically restrict the transitions to shoe that are <= max(y1,y2) or >=
any transition y > max(y1,y2). Here we use L(<1,<2,R,T1) to present the true
statements in the first assertion, and L(<1,<2,,T2) to present the true
statements in the second assertion.
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. There is a transition in the
second scale which is the limit of earlier transitions.
THEOREM 6.1. 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 6.2. Linearity + Transitions + Arbitrary Bounded Ranges + Tail
Transition Similarity interprets ZFC + "for all x containedin omega, x#
exists" and is interpretable in ZFC + "there exists a measurable cardinal".
This is provable in ZFC.
THEOREM 6.3. Linearity + Transitions + Arbitrary Bounded Ranges + Tail
Transition Similarity + Transition Accumulation interprets ZFC + "there
exists a measurable cardinal kappa with kappa many measurable cardinals
below kappa" and is interpretable in ZFC + "there exists a measurable
cardinal kappa with normal measure 1 measurable cardinals below kappa". This
is provable in EFA.
THEOREM 6.4. Linearity + Transitions + Arbitrary Bounded Ranges + Tail
Transition Similarity + Strong Transition Accumulation interprets ZFC +
"there exists a measurable cardinal kappa with normal measure 1 measurable
cardinals below kappa (order >= 2)" and is interpretable in ZFC + "there
exists a measurable cardinal kappa of order >= 3". This is provable in EFA.
7. MULTIPLE AGENTS, TWO STATES.
This is a reworking of material from the following, in order to put it in
line with the other sections in Concept Calculus:
FMPC postings 0606002 and 0606003, see
http://www.cs.uky.edu/fmpc/fmpcMembers.html
which are the same as #291, #292 on FOM, called Independently Free
Minds/Collectively Random Agents, Independently Free Minds/Collectively
Random Agents/more.
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".
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
creation. Although it is natural to do so, we will not assume that every
agent has a birthdate. However, the axiom of Continual Creation asserts that
every time is a 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. Any true statement involving times t1,t2 remains
true if we hypothetically restrict the transitions to those that are <=
max(t1,t2) together with those that are >= any transition t > max(t1,t2).
Here we use L(<,A,T) to present the true statement.
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 7.1. 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 7.2. Linearity + Transitions + Continual Creation + Unrestricted
Activity + Tail Transition Similarity interprets ZFC + "for all x
containedin omega, x# exists" and is interpretable in ZFC + "there exists a
measurable cardinal". This is provable in ZFC.
THEOREM 7.3. Linearity + Transitions + Continual Creation + Unrestricted
Activity + Tail Transition Similarity + Transition Accumulation interprets
ZFC + "there exists a measurable cardinal kappa with kappa many measurable
cardinals below kappa" and is interpretable in ZFC + "there exists a
measurable cardinal kappa with normal measure 1 measurable cardinals below
kappa". This is provable in EFA.
THEOREM 7.4. Linearity + Transitions + Continual Creation + Unrestricted
Activity + Tail Transition Similarity + Strong Transition Accumulation
interprets ZFC + "there exists a measurable cardinal kappa with normal
measure 1 measurable cardinals below kappa (order >= 2)" and is
interpretable in ZFC + "there exists a measurable cardinal kappa of order >=
3". This is provable in EFA.
8. STRUCTURE OF NAIVE TIME.
To be continued in Concept Calculus 3.
**********************************
I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 293rd 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 6/12/06 11:01AM
291: Independently Free Minds/Collectively Random Agents (more) 6/13/06
5:01PM
292: Concept Calculus 1 6/17/06 5:26PM
Harvey Friedman
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