[FOM] 295: Concept Calculus 4

[Harvey Friedman]

We continue from Concept Calculus 3, June 25, 2006,
http://www.cs.nyu.edu/pipermail/fom/2006-June/010630.html

CONCEPT CALCULUS
by
Harvey M. Friedman
July 3, 2006

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. Varying Bit.
9. Persistently Varying Bit.
10. Naive Time. 
11. Epochs Replacing Transitions.
12. Discrete Point Masses in One Dimensional Space.
13. Discrete Point Masses with End Expansion.
14. Discrete Point Masses with Inner Expansion.
15. Point Masses with Inner Expansion.
16. Discrete Point Masses with Inner Expansion Revisited.
17. Observers, Objects, Observations.

Concept Calculus 1 http://www.cs.nyu.edu/pipermail/fom/2006-June/010616.html
has sections 1,2. Concept Calculus 2
http://www.cs.nyu.edu/pipermail/fom/2006-June/010622.html has sections 3-7.
Concept Calculus 3 http://www.cs.nyu.edu/pipermail/fom/2006-June/010630.html
has sections 8-11. 

We begin with section 12.

NOTE: We indicated previously that we understated the logical strength of
the systems with Strong Transition Accumulation. I am in some correspondence
verifying that "a measurable cardinal above infinitely many Woodin
cardinals" is more than enough for an upper bound, and "a measurable
cardinal kappa of order >= kappa" is a lower bound.

12. 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.

Consider the case of point masses whose distribution is unrestricted. This
is exactly the case of a Varying Bit treated in section 8. See the NOTE here
before section 12. 

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 9, but is a little bit
different. So we restate the results.

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).

Formally, let y,z not appear in phi. The formula "phi is discrete on the
variable x" is 

(forall x)(phi implies (therexists y,z)(y < x < z and (forall w)((y < w < z
and phi[x/w]) implies w = x))).

DISCRETE POINT MASS TRANSLATION. For any point b and discrete range of
points, there exists a translation distance c such that any point before b
lies in the range of points if and only if there is a point mass at position
b+c. 

DISCRETE POINT MASS TRANSLATION. (phi is discrete on the variable x) implies
(therexists c)(forall y < b)(phi iff P(y+c))), where phi is a formula of
L(<,T,P,+) in which c,y,z do not appear.

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. 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(<,T,P,+) 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 12.1. Linearity + Addition + Order Completeness + Transitions +
Discrete Point Mass 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 12.2. Linearity + Addition + Order Completeness + Transitions +
Discrete Point Mass Translation + 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.

THEOREM 12.3. Linearity + Addition + Order Completeness + Transitions +
Discrete Point Mass Translation + 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 12.4. Linearity + Addition + Order Completeness + Transitions +
Discrete Point Mass Translation + Tail Transition Similarity + Strong
Transition Accumulation interprets ZFC + "there exists a measurable cardinal
kappa of order >= kappa" and is interpretable in ZFC + "there exists a
measurable cardinal above infinitely many Woodin cardinals". This is
provable in EFA.

With regard to Theorem 12.4, see the NOTE before section 12.

As in section 9, we can add the naïve time principles reformulated as naïve
one dimensional space principles, in the sense of section 10, 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.

13. 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.

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.

DISCRETE POINT MASS TRANSLATION (present). For any point b in the present,
and discrete range of points in the present, there exists a translation
distance c in the present such that any present point before b lies in the
range of points if and only if there is a point mass at position b+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, there exists a translation distance c such that any point before
b lies in the range of points if and only if there is a point mass at
position b+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 13.1. 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 naïve time principles reformulated as naive one dimensional
space principles, in the sense of section 10, 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.

14. DISCRETE POINT MASSES WITH INNER EXPANSION.

We carry out the development in section 13, 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 13.

NAIVE LINEARITY. < is a linear ordering with left endpoint and no right
endpoint.

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,+) 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 in the present, there exists a translation distance c in the
present such that any present point before b lies in the range of points if
and only if there is a point mass at position b+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, there exists a translation distance c such that any point before b
lies in the range of points if and only if there is a point mass at position
b+c. Here we use L(<,P,R,+) to present the range of points.

THEOREM 14.1. 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.

Of course, if we add ii then we cannot add i. If we add iv then we cannot
add v.

15. POINT MASSES WITH INNER EXPANSION.

Here we carry out the development of section 14 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, 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,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.

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, there exists a translation distance c in the
present such that any present point before b lies in the range of points if
and only if there is a point mass at position b+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, there exists a translation distance c such that any point before b
lies in the range of points if and only if there is a point mass at position
b+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 15.1. Naive Linearity + Naïve Addition + Order Completeness
(present,future) + Inner Expansion + Property Preservation + Point Mass
Translation (present,future)  + Transitions + Extended Transition Similarity
interprets NBG + "there is a nontrivial elementary embedding from V into V"
and is interpretable in NBG + "there is a nontrivial elementary embedding
from V into V" + "V is almost ineffable". Here V is almost ineffable means
that "for all class functions f:On into V, where each f(alpha) containedin
V(alpha), there exists a proper class X containedin On such that for all
alpha < beta, f(alpha) containedin f(beta). Actually, we only need that X
have order type omega. Hugh Woodin has shown some time ago that ZFC + "there
exists a nontrivial elementary embedding from a rank into itself" is
interpretable in NBG + "there is a nontrivial elementary embedding from V
into V".

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.

16. DISCRETE POINT MASSES WITH INNER EXPANSION REVISITED.

We carry out the development in section 15, 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.

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, there exists a translation
distance c in the present such that any present point before b lies in the
range of points if and only if there is a point mass at position b+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, there exists
a translation distance c in the intermediate such that any intermediate
point before b lies in the range of points if and only if there is a point
mass at position b+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, there exists a translation distance c such that any point before
b lies in the range of points if and only if there is a point mass at
position b+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 16.1. Naive Linearity + Naive Addition + Order Completeness
(present,intermediate,future) + Explosion + 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" and is interpretable in NBG + "there is
a nontrivial elementary embedding from V into V" + "V is almost ineffable
cardinal". Here V is almost ineffable means that "for all class functions
f:On into V, where each f(alpha) containedin V(alpha), there exists a proper
class X containedin On such that for all alpha < beta, f(alpha) containedin
f(beta). Actually, we only need that X have order type omega. Hugh Woodin
has shown some time ago that ZFC + "there exists a nontrivial elementary
embedding from a rank into itself" is interpretable in NBG + "there is a
nontrivial elementary embedding from V into V".

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.

Of course, if we add ii then we cannot add i. If we add iv then we cannot
add v.

17. OBSERVERS, OBJECTS, OBSERVATIONS.

To be continued.

**********************************

I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 295th 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
293: Concept Calculus 2  6/20/06  6:27PM
294: Concept Calculus 3  6/25/06  5:15PM

Harvey Friedman