[FOM] 298: Concept Calculus 5

Harvey Friedman friedman at math.ohio-state.edu
Fri Jul 14 05:40:16 EDT 2006 

We continue from Concept Calculus 4, July 3, 2006,
http://www.cs.nyu.edu/pipermail/fom/2006-July/010631.html

CONCEPT CALCULUS
by
Harvey M. Friedman
July 14, 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. Better Than, Real.
18. Better Than, Real, Conceivable.
19. Better Than, Real, Intermixed.

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. Concept Calculus 4
http://www.cs.nyu.edu/pipermail/fom/2006-July/010631.html has sections
12-16.

In Theorems 15.1 and 16.1, we can obtain an interpretation of ZF + "there
exists a nontrivial elementary embedding from some V(kappa) into V(kappa),
where V(kappa) 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 in the Concept Calculus theories of Theorems 15.1 and
16.1.

We begin with section 17.

17. BETTER THAN, REAL.

We use a one sorted predicate calculus with equality, with a binary relation
symbol > for "better than", and a unary predicate P for "being real".

The idea we are dividing the objects up into those that are real and those
that are imaginary. Here we mean real vs. imaginary in terms of the level of
goodness. Specifically, see the last item under Basic below.

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. The only things that a real
thing can be better than, are 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
true statement. 

THEOREM 17.1. 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 (< x), but with arbitrary 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 true statement.

If we use this restricted form of Real Existence, then Theorem 17.1 remains
unchanged.

18. BETTER THAN, REAL, CONCEIVABLE.

>From some points of view, one can criticize Real Existence of section 17 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
17.1, 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
Ordinary 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 P1, P2. Here P1(x)
means "x is real". P2(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.

"Even the lowest level real things, collectively, have something to offer."
This has economic, political, and social ramifications.

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 corresponding statement is true
involving the two real things, "better than", and "being conceivable". Here
we use L(>,P1) to present the true statement.

In Conceivable Existence, the range is given by a formula phi all of those
parameters are real things, and whose quantifiers all range over real
things.

THEOREM 18.1. 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. 

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

I use http://www.math.ohio-state.edu/%7Efriedman/ for downloadable
manuscripts. This is the 297th 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
295: Concept Calculus 4  7/3/06  2:34AM
296: Order Calculus  7/7/06  12:13PM
297: Order Calculus/restatement  7/11/06  12:16PM

Harvey Friedman

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