[FOM] 729: Consistency of Mathematics/1

Harvey Friedman hmflogic at gmail.com
Tue Oct 25 13:25:27 EDT 2016

What are the most monumental issues in the foundations of mathematics?
It would be valuable to hear what subscribers think, but the two
obvious classical ones are, in no particularly order:

1. Is mathematics consistent - I.e., free of contradiction? After all,
it deals with objects not accessible to us by observations in any
usual sense. Not only does mathematics use infinitely many objects, it
uses myriads of completed infinite totalities. Can we, and how can we
be assured of the consistency of mathematics? What would constitute
evidence or confirmation of the consistency of mathematics?

2. Is the apparent overwhelming power of the usual foundations for
mathematics through ZFC of any real use in ordinary mathematical
environments? What if anything is there to gain from extending ZFC in
terms of ordinary mathematical environments? Or is there some kind of
completeness of ZFC for typical problems arising from familiar
concrete mathematical investigations?

I have worked and written the most on 2, under Concrete Mathematical
Incompleteness, and this has culminated in the series on Emulation
Theory, titled Large Cardinals and Emulations/1-25 and continuing.
This adventure has taken on the general character of Musical
Composition in the following sense. Works of musical composition of
course do not normally conform to specific previously stated problems
and challenges. But the masterpieces exude a compelling
*inevitability*, a critical aspect of Musical Composition that is
discussed so eloquently in, e.g.,
https://www.youtube.com/watch?v=KI1klmXUER8 See especially 8:20-37,
and 24:00-40. I view myself as searching for an apparently related
kind of *inevitability* in Concrete Mathematical Incompleteness.

For those who take abstract set theory as intrinsically fundamentally
important in its own right, independently of the quality and quantity
of its connections with the rest of mathematics, on this list would be

3. Is the continuum hypothesis true or false? Can we and how can we
obtain evidence or confirmation either way?

Research on 3 could use a refreshing new methodology, which I hope to
be offering in Refuting the Continuum Hypothesis?/1-9 and hopefully

There are some other important issues in f.o.m., including ones that I
would regard as less than monumental; it would be interesting to hear
from subscribers about their views.

There are some monumental, or arguably monumental issues that I would
not classify as foundations of mathematics, bur rather foundations of
computer science. This is way beyond the scope of this series, and so
I will just briefly mention a) the notorious complexity issues
everybody is aware of, and b) Church's Thesis. Actually b) is such a
radically different problem than a), that I prefer to view b) as an
arguably monumental issue in Philosophical Computer Science, which I
like to distinguish from Foundations of Computer Science, and
certainly from Computer Science itself. So perhaps we can add
Philosophical Computer Science to a list of subjects that includes
Philosophical lGeometry - although for Philosophical Geometry I do
have a reasonably coherent and perhaps resonating view as to just what
I mean.

So I have seriously thought about 1,2,3 and b), and written about them
on the FOM. With regard to a), I do not admit to have worked on it.

In this series I want to deal only with 1. I have written about 70
pieces about and surrounding 1, including these:

FROM https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/

#10. The interpretation of set theory in pure predication theory,
August 20, 1997, 64 pages.
#12. The interpretation of set theory in mathematical predication
theory, October 25, 1997, 10 pages.
#13. The axiomatization of set theory by extensionality, separation,
and reducibility, October 28, 1997, 71 pages
#35. A Way Out, August 28, 2002, 49 pages. Also in: A Way Out, in: One
Hundred Years of Russell’s Paradox, ed. Godeharad Link, de Gruyter,
49-86, 2004.
#39. Sentential Reflection, January 5, 2003, 3 pages,
#40. Restrictions and extensions, February 17, 2003, 3 pages.
#41. Elemental sentential reflection, March 3, 2003, 6 pages.
#42. Similar subclasses, March 11, 2003, 3 pages.
#46. Relational system theory, May 26, 2005, 24 pages.
#52. Concept Calculus, October 25, 2006, 42 pages,
#59. 3. Interpreting Set Theory in Ordinary Thinking: Concept
Calculus, 12 pages.
#62. Concept Calculus: Much Better Than, October 31, 2009, 48 pages.
Also in: Concept Calculus: Much Better Than, in: New Frontiers in
Research on Infinity, ed. Michael Heller and W. Hugh Woodin, Cambridge
University Press, 130-164, 2011.
#72. Concept Calculus: universes. October 2, 2012, 33 pages,
#74. A Divine Consistency Proof for Mathematics. December 25, 2012, 70 pages.
#82. Testing the Consistency of Mathematics, 7 pages, July 23, 2014.
#83. Flat Mental Pictures, September 5, 2014, 5 pages.
#85. Conservative Growth: A Unified Approach to Logical Strength,
October 12, 2014, 15 pages.

FROM https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/

#35. Concept Calculus, Mathematical Methods in Philosophy, Banff,
Canada, February 21, 2007, 9 pages.
#37. Concept Calculus’ APA Panel on Logic in Philosophy, APA Eastern
Division Annual Meeting, Baltimore Maryland, January 2, 2008, 17
#39. Concept Calculus, Carnegie Mellon University, March 26, 2009,
Pure and Applied Logic Colloquium
#43. Concept Calculus, Department of Philosophy, MIT, November 4,
2009, 22 pages.
#61. Concept Calculus, Institutional Honorary Doctorate, Ghent
University, Ghent, Belgium, September 5, 2013,  26 pages.

FOM POSTINGS. See FOM Archives, http://www.cs.nyu.edu/pipermail/fom/

#41: Strong Philosophical Indiscernibles
#90: Two Universes 6/23/00 1:34PM
#110: Communicating Minds I 12/19/01 1:27PM
#111: Communicating Minds II 12/22/01 8:28AM
#112: Communicating MInds III 12/23/01 8:11PM
#116: Communicating Minds IV 1/4/02 2:02AM
#122: Communicating Minds IV-revised 1/31/02 2:48PM
#155: A way out  8/13/02  6:56PM Also in: A Way Out, in: One Hundred
Years of Russell’s Paradox, ed. Godeharad Link, de Gruyter, 49-86,
#156: Societies  8/13/02  6:56PM
#157: Finite Societies  8/13/02  6:56PM
#158: Sentential Reflection  3/31/03  12:17AM
#159. Elemental Sentential Reflection  3/31/03  12:17AM
#160. Similar Subclasses  3/31/03  12:17AM
#161: Restrictions and Extensions  3/31/03  12:18AM
#200: Advances in Sentential Reflection 12/22/03 11:17PM
#245: Relational System Theory 1  5/16/05  12:24PM
#246: Relational System Theory 2  5/15/05  9:57PM
#248: Relational System Theory 2/restated  5/26/05  1:46AM
#256. NAME:finite inclusion theory  11/21/05  2:34AM
#257. FIT/more  11/22/05  5:34AM
#283: A theory of indiscernibles  5/7/06  6:42PM
#290: Independently Free Minds/Collectively Random Agents 6/12/06
#291: Independently Free Minds/Collectively Random Agents (more) 6/13/06
#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
#298: Concept Calculus 5  7/14/06  5:40AM
#444: The Exploding Universe 1  11/1/10  1:46AM
#509: A Divine Consistency Proof for Mathematics  12/26/12  2:15AM
#511: A Supernatural Consistency Proof for Mathematics   1/10/13  9:19PM
#513: Five Supernatural Consistency Proofs for Mathematics  1/14/13  1:13AM
#515: Eight Supernatural Consistency Proofs For Mathematics  1/19/13  2:40PM
#524: Testing Consistency of Math  7/23/14  6:49AM
#527: Some Mental Pictures i  8/11/14  11:13AM
#547: Conservative Growth – Triples  9/29/14  11:34PM
#549: Conservative Growth – beyond triples  10/6/14  1:31AM
#556: Flat Foundations 1  10/29/14  4:07PM


Given the number of these items, it is important for me to give a
guide as to what is here.

#82 https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
and FOM posting #524 are in a category by themselves, as they do not
provide a consistency proof of ZFC and large cardinal extensions, but
rather discusses computer based confirmation of such consistency. In
this series, we will be updating these to relate more closely to
Emulation Theory.

#74 https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
and arguably FOM postings #509, #511, #513, #515, are also in a
category by themselves, as they specifically use axioms directly
motivated by classic writings in Theology. The first of these is at
roughly at the level of a measurable cardinal (a little below), and
the four FOM postings are at various levels, below and above a
measurable cardinal. More specifically, the first of these is based on
the concept of positive/negative property (good to have the positive
side, bad to have the negative side), used explicitly by Goedel, and
at least implicitly by Leibniz and St. Anselm. The other four use a
distinction between real and transcendental objects, and also a
mapping from real objects to generally transcendental objects as in
Doppelgängers - https://en.wikipedia.org/wiki/Doppelgänger

A principal theme in the previous paragraph and all of the other
approximately 70 pieces cited is "flatness". The opposite of flatness
is of course the cumulative hierarchy of sets. A typical flat setup is
to have a sort for objects (none of which are sets) and sets of
objects - but if we just use objects and sets of objects we generally
want a pairing function on objects. Or at least a sort for objects and
a sort of binary (or higher arity) relations on objects. There can of
course be more primitives, and maybe a sort for one higher type.

There is a group of pieces titled Concept Calculus where the
primitives are motivated by naive physics of space and time. The
driving intuitive principle is that "anything that can happen will
happen" or "any configuration that is possible is realized". This idea
goes back to Aristotle with Plenitude. (There are some other
directions titled Concept Calculus that are not based on the naive
physics of space and time - see below.) For those that are based on
the naive physics of space and time, see
#52, #72, https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
everything under
#292 6/17/06 #293 6/20/06 #294: 6/25/06 #295: 7/3/06 #298: 7/14/06
#444: 11/1/10, http://www.cs.nyu.edu/pipermail/fom/

There are some pieces titled Concept Calculus that are not motivated
by naive physics of space and time. They use the primitives better
than and much better than, an example of concept amplification.
#52, #62, https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
everything under
#292 6/17/06 #293 6/20/06 #294: 6/25/06 #295: 7/3/06 #298: 7/14/06,
I think of "much better than" as an example of concept amplification.

There is the A Way Out approach to Russell's Paradox, 335,
and #155, http://www.cs.nyu.edu/pipermail/fom/ . This is a direct
natural weakening of the full comprehension scheme that takes the
form: Every virtual set forms a set, or, outside any given set, has
two inequivalent elements, where all elements of the virtual set
belonging to the first belong
to the second.The resulting formal system is mutually interpretable
with certain large cardinals.

There are some pieces that focus on two minds. The basic idea is that
the first mind is dominated by the second mind, which has considerably
more objects than the first mind. However, they have a fundamental
compatibility that allows them to communicate effectively: they agree
on (certain) statements that involve only the objects of the first
mind. This leads to associated formal systems of various logical
strengths some going to extremely large cardinals. The ones about
societies are different, and are focused on the "like" relation among
people, which leads to some strong theories that are natural under the
"like" interpretation, motivated by the A Way Out approach.
#110: 12/19/01 #111: 12/22/01 #112: 12/23/01 #116: 1/4/02 #122:
1/31/02 #156: 8/13/02 #157: 8/13/02 #290: 6/12/06 #291: 6/13/06,

There are some pieces that provide a kind of unified thematic approach
that climbs from the logically weak to the logically very strong.
These include
#83, #85, https://u.osu.edu/friedman.8/foundational-adventures/downloadable-manuscripts/
#527: 8/11/14  #547: 9/29/14 #549: 10/6/14 #556: 10/29/14

There are some other pieces not included in this summary, but I want
to stop here.

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

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 729th 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

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

Harvey Friedman

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