[FOM] 730: Consistency of Mathematics/2
Harvey Friedman
hmflogic at gmail.com
Thu Nov 17 21:50:26 EST 2016
THIS POSTING IS ENTIRELY SELF CONTAINED
In http://www.cs.nyu.edu/pipermail/fom/2016-October/020140.html I promised
"In the next posting, we will start with revisiting the Communicating
Minds approach. I now think of it as Expanding Minds. Also I will
consolidate it with the Real/Transcendental approach connected with
theology."
I now have made good on the first sentence above. (Sorry for being
gone so long). I have put this on my website:
91. Expanding Mind Theory, November 17, 2016, 17 pages.
https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
In keeping with my general new policy, I will now put complete proofs
of major claims on my website, except for those truly enormous proofs
like for Emulation Theory which will be done next year.
Here is the Informal initial segment of the above manuscript.
EXPANDING MIND THEORY
by
Harvey M. Friedman
Distinguished University Professor of Mathematics, Philosophy, and
Computer Science Emeritus
Ohio State University
Columbus, Ohio 43235
November 17, 2016
DRAFT
Abstract. We present a theory EM (expanding minds) of a Younger and
Older Mind that contemplates objects and unary/binary/ternary
relations on objects. This inserts a subjective element into a small
fragment of the usual theory of types pioneered by Bertrand Russell.
The resulting formal systems interpret the usual ZFC axioms for
mathematics, and are interpretable in ZFC augmented with a certain
large cardinal hypothesis (1-extendibility). The results can be viewed
as a consistency proof for mathematics relative to this Expanding Mind
Theory EM. The thesis is that EM embodies principles that have a
plausibility that is independent of that of ZFC. EM can be naturally
strengthened in various ways so that it interprets certain large
cardinal extensions of ZFC. This development suggests a new kind of
Philosophy of Mind with deep interactions with Philosophy and
Foundations of Mathematics.
1. The Expanding Mind - Informal.
2. EM in ZF + 1-extendible.
3. ZFC in EM.
3.1. EMX in EM.
3.2. ZFC in EMX.
4. Extensions.
1. THE EXPANDING MIND - INFORMAL
The plan is to interpret the usual ZFC axioms for mathematics - and
more - in formalizations that represent some arguably compelling extra
mathematical intuition. It may not be more compelling than ZFC. But
the thesis is that it represents an alternative kind of intuition with
a prima facie legitimacy. We will be interested in minimalism - i.e.,
we strive for limiting the level of commitment needed to interpret the
formalizations.
A particular aspect of what our minds grasp are the objects and the
unary/binary/ternary relations on objects. We will simply refer to
these as unary/binary/ternary relations, where it is understood that
they apply onto to objects and not to relations. There is a strict
separation between objects and relations as in a simple theory of
types going back to Bertrand Russell.
Of course, there are many other aspects of what our minds grasp, and
maybe these other aspects will also be subject to related
developments. Here we are referring to the objects and
unary/binary/ternary relations on objects grasped by our minds.
Now hopefully, as we grow older, our minds become more powerful, and
we grasp more. We now take a snapshot of our minds at two points of
time, call them the Younger Mind and the Older Mind.
The Older Mind will grasp more objects than the Younger Mind. The
Older Mind will also grasp more unary/binary/ternary relations than
the Younger Mind. Any unary/binary/ternary relation grasped by the
Younger Mind will remain grasped by the Older Mind, but objects
grasped by the Older Mind and not grasped by the Younger Mind might
fall under the relation. In fact this happens for any unary relation R
which holds of all objects, as R is grasped by the Younger Mind and R
holds of all objects grasped by the Older Mind.
IMAGINATION asserts that ny unary/binary/ternary relation that can be
defined by the Older Mind, with references allowed to specific objects
and unary/binary/ternary relations grasped by the Older Mind, can
actually be grasped by the Older mind. Definitions are allowed to use
all of the primitives being discussed here. Having such a definition
is a particularly clear form of imagination. We need formulate
IMAGINATION only for the Older Mind, in light of the STUBBORNENESS
discussed below.
We are STUBBORN as we age. We don't change our opinions (we don't
change our minds). STUBBORNNESS asserts the following. Any statement
that refers to any particular objects and unary/binary/ternary
relations grasped by the Younger Mind (parameters), and quantifying
over all objects and unary/binary/ternary relations grasped by the
Younger Mind, remains true for the Older Mind, referring to the same
particular objects and unary/binary/ternary relations grasped by the
Younger Mind (parameters), and quantifying over all objects and
unary/binary/ternary relations grasped by the Older Mind.
COMPLETENESS asserts that for every unary/binary/ternary relation
grasped by the Older Mind, there is a unary/binary/ternary relation
grasped by the Younger Mind, which is equivalent as far as objects
grasped by the Younger Mind are concerned. In this sense, the Younger
Mind and the Older Mind do not differ with respect to the objects that
the Younger Mind grasps.
We can view COMPLETENESS as another form of STUBBORNNESS where we
don't change our minds about the unary/binary/ternary relations seen
by the Younger Mind, as far as objects seen by the Younger Mind are
concerned.
There is an alternative view of COMPLETENESS based on an analogy
between COMPLETENESS and the usual completeness of the real number
system in mathematics. We formed the real number system through
Dedekind or Cauchy Completeness relatively early in the modern history
of mathematics, and relatively early in mathematics education (at the
University level), and we later go on to build more and more
sophisticated mathematical systems. This doesn't affect the
completeness of the original real number system, which survives.
TRANSCENDENCE asserts that the Older Mind is transcendentally more
powerful than the Younger Mind. This is formulated in the following
way. There is a binary/ternary relation R seen by the Older Mind, such
that every unary/binary relation seen by the Younger Mind is
(extensionally equal to) a section of R (obtained by fixing the first
argument to be an object grasped by the Older Mind). We shall see that
TRANSCENDENCE implies that there is an object grasped by the Older
Mind but not grasped by the Younger Mind. We can also think of the R
in TRANSCENDANCE as a kind of naming relation, where each object x is
viewed as "naming" the cross section of R at x (multiple names
allowed). Thus there is a language theoretic interpretation of
TRANSCENDENCE.
The above describes the informal basis for the system EM.
Obviously it is premature at this stage to go more deeply into various
philosophical issues that are raised by these informal presentations.
E.g., the challenge of further developing a theory of grasping and
defining.
***********************************************
My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
https://www.youtube.com/channel/UCdRdeExwKiWndBl4YOxBTEQ
This is the 730th 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 #1-699 can be found at
http://u.osu.edu/friedman.8/foundational-adventures/fom-email-list/
700: Large Cardinals and Continuations/14 8/1/16 11:01AM
701: Extending Functions/1 8/10/16 10:02AM
702: Large Cardinals and Continuations/15 8/22/16 9:22PM
703: Large Cardinals and Continuations/16 8/26/16 12:03AM
704: Large Cardinals and Continuations/17 8/31/16 12:55AM
705: Large Cardinals and Continuations/18 8/31/16 11:47PM
706: Second Incompleteness/1 7/5/16 2:03AM
707: Second Incompleteness/2 9/8/16 3:37PM
708: Second Incompleteness/3 9/11/16 10:33PM
709: Large Cardinals and Continuations/19 9/13/16 4:17AM
710: Large Cardinals and Continuations/20 9/14/16 1:27AM
711: Large Cardinals and Continuations/21 9/18/16 10:42AM
712: PA Incompleteness/1 9/2316 1:20AM
713: Foundations of Geometry/1 9/24/16 2:09PM
714: Foundations of Geometry/2 9/25/16 10:26PM
715: Foundations of Geometry/3 9/27/16 1:08AM
716: Foundations of Geometry/4 9/27/16 10:25PM
717: Foundations of Geometry/5 9/30/16 12:16AM
718: Foundations of Geometry/6 101/16 12:19PM
719: Large Cardinals and Emulations/22
720: Foundations of Geometry/7 10/2/16 1:59PM
721: Large Cardinals and Emulations//23 10/4/16 2:35AM
722: Large Cardinals and Emulations/24 10/616 1:59AM
723: Philosophical Geometry/8 10/816 1:47AM
724: Philosophical Geometry/9 10/10/16 9:36AM
725: Philosophical Geometry/10 10/14/16 10:16PM
726: Philosophical Geometry/11 10/17/16 4:04PM
727: Large Cardinals and Emulations/25 10/20/16 1:37PM
728: Philosophical Geometry/12 10/24/16 3:35PM
729: Consistency of Mathematics/1 Tue Oct 25 13:25:27 EDT 2016
Harvey Friedman
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