[FOM] 807: Beyond Perfectly Natural/11

Harvey Friedman hmflogic at gmail.com
Sun Apr 29 21:19:06 EDT 2018

There is a new link at


105. Concrete Mathematical Incompleteness Highlights 4/29/18, 7 pages,
April 29, 2018. State of the art incorporating 104. This is intended
to close Phase 1 CMI (at least temporarily). Some discussion of the
Principle of Invariant Maximality. CMIstatus042918-1qbaf1x

Here is the cover page.

Harvey M. Friedman
Distinguished University Professor of Mathematics, Philosophy,
Computer Science Emeritus
Ohio State University
Columbus, Ohio
April 29, 2018

Abstract. The Concrete Mathematical Incompleteness (CMI)
project was founded in 1967, with the ambitious intention
of touching mathematical thinkers at all levels in terms of
their fundamental attitudes toward mathematical objectivity
and reality. The initial impetus for CMI is of course the
famous Gödel Incompleteness Theorems from the early 1930s
which, at least initially, did touch mathematical thinkers
at all levels in this way. Yet much more is needed to keep
Incompleteness at the forefront of mathematical thought. We
are at a seemingly advanced stage of Phase 1 CMI, where 1
refers to the use of essentially no mathematical machinery
in the formulation of the Incompleteness. In fact, we use
only sets of rational tuples and sometimes only finite sets
of integer tuples, with only the usual ordering of the
rationals and sometimes only the usual ordering of finite
initial segments of the nonnegative integers. The plan has
been to first go as deeply as we can (or know how to) in
this pure context to uncover the necessarily new purely
combinatorial structure. In latter Phases, the plan is to
start involving very elemental structure such as addition
on the nonnegative integers. We present four different
Incompleteness results and variants in sections 2-5. All
are implicitly Π01, whereas some are explicitly finite and
even explicitly Π01. All correspond to the SRP hierarchy of
large cardinals, except for one that corresponds to the
much higher HUGE hierarchy of large cardinals. Each will
resonate more or less according to the experience and
orientation of various mathematical thinkers.


1. Introduction.
2. Maximal Emulation.
3. Inductive Upper Shift.
4. Finite Upper Image.
5. Internal Inductive Upper Shift.
6. Formal Systems Used.
7. Principle of Invariant Maximality

My website is at https://u.osu.edu/friedman.8/ and my youtube site is at
This is the 807th 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 #1-799 can be found at

800: Beyond Perfectly Natural/6  4/3/18  8:37PM
801: Big Foundational Issues/1  4/4/18  12:15AM
802: Systematic f.o.m./1  4/4/18  1:06AM
803: Perfectly Natural/7  4/11/18  1:02AM
804: Beyond Perfectly Natural/8  4/12/18  11:23PM
805: Beyond Perfectly Natural/9  4/20/18  10:47PM
806: Beyond Perfectly Natural/10  4/22/18  9:06PM

Harvey Friedman

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