[FOM] Arithmetic Incompleteness/Baez

[Harvey Friedman]

John Baez on his blog asked whether there are, or why there are not,
debates about the truth value of independent arithmetic sentences,
like there is about CH.
https://golem.ph.utexas.edu/category/2016/03/foundations_of_mathematics.html

I wrote the following reply there:

Let me try to clarify the situation with regard to this crucially
important matter for foundations of mathematics.

1. From the standard set theoretic point of view, we have the so
called preferred model of set theory normally written as (V,epsilon),
where epsilon is membership.

2. Set theorists generally believe, or like to wear the hat, that
(V,epsilon) has an objective reality, and it is NOT to be thought of
as some special construction within any more general framework. We can
debate if this is the case, and if it is thought to be not the case,
then is this "more general framework" really philosophically coherent,
or is it a mirage, etcetera...... Not the topic of this post.

3. Cantor raised his famous continuum hypothesis, usually written CH.
Set theorists and others, notably Hilbert, tried to resolve this
question within ZFC, and also with ZFC extended by strongly
inaccessible cardinals, and measurable cardinals, and large cardinal
hypotheses generally, and not only failed, but actually provably
failed. In what sense?

4. In the important sense that people found natural constructions of
models of ZFC and more, other than (V,epsilon), in which they could
prove that CH holds, and other they could prove that CH fails.

5. There is a little bit of a fib in 4. Goedel constructed a very
natural model of ZFC in which he showed that CH holds. Cohen
constructed a very natural collection of continuumly many models of
ZFC in which CH fails. Some refinements pretty naturally construct a
countable family of countable models of ZFC in which CH fails. If
algebraic models (Boolean valued models) are allowed, then Cohen did
construct a single individual natural model of ZFC in which CH fails.

6. It is apparently impossible to construct a single natural model of
ZFC in which CH fails, like Goedel did for CH holding (in the usual
sense, not allowing algebraic model). HOWEVER, the family of models of
ZFC in which CH fails are ESSENTIALLY ALL EQUIVALENT TO EACH OTHER,
AND SO THERE IS A GOOD SENSE IN WHICH COHEN ACTUALLY GIVE A "ONE"
NATURAL MODEL OF ZFC IN WHICH CH FAILS.

7. This pattern repeats itself for ALL so called intensely set
theoretic statements that we have studied. Namely, there is a natural
single model of ZFC in which they hold, and a natural (essentially)
single model of ZFC in which they fail. Sometimes we can eliminate
"essentially", but not in the case of CH.

8. And people have argued in many situations in 7, which model or
models do they accept as "correct" or "like the real one (V,epsilon)",
etcetera.

9. I am painting in 1-8 a philosophically coherent picture of what
generally is going on, but there are some subtle points known to
experts that I have glossed over, and I would like them not to think I
am cheating or that I am  stupid.

10, Now let's come to statements that are not set theoretic, and in
fact, reasonably concrete. This notion is subject to analysis in
standard f.o.m. normally in terms of the standard complexity measures
for mathematical statements. I don't want to get into this crucially
important part of standard f.o.m. in this posting.

11. Suffice it to say this. First of all, so called arithmetic
sentences are way way down lower than the intensely set theoretic
statements referred to above, especially CH.

12. The crucial point is this. IN ALL OF THE NATURAL CONSTRUCTIONS WE
KNOW OF MODELS OF SET THEORY, AND ESPECIALLY THOSE THAT WE VIEW AS
CANDIDATES FOR THE REAL (V,epsilon), WE KNOW A PRIORI THAT THEY ALL
SATISFY THE EXACT SAME NON SET THEORETIC STATEMENTS. IN PARTICULAR, WE
KNOW THAT THEY ALL SATISFY THE SAME ARITHMETIC STATEMENTS.

13. So unlike the situation with CH, in dealing with arithmetic
statements that might not be decided in ZFC, we are going to have to
work with provability with extensions of ZFC, and not, like in the
case of CH, go grab and lobby for some model of ZFC being preferred in
some way.

14. OK, so now we are looking at an arithmetic sentence, and its
provability in various set theories (or, speaking of the devil, maybe
in various alternative foundational frameworks, see below).

15. So, like CH, maybe we are going to have the following situation.
Alice's set theory proves arithemtic sentence A is true, Bob's set
theory proves A is false. And then people "decide" which, if any, has
a correct set theory.

16. Well, here is what is actually going on. FOR ANY TWO SET THEORIES
ANYBODY THINKS ARE NATURAL, WE (GENERALLY) KNOW THAT EITHER EVERY
ARITHMETIC SENTENCE PROVABLE IN THE FIRST IS PROVABLE IN THE SECOND,
OR EVERY ARITHMETIC SENTENCE PROVABLE IN THE SECOND IS PROVABLE IN THE
FIRST.

17. So the only actual criteria for choosing set theories with regard
to arithmetic sentences (and more) that anybody can do anything with,
is simply: which proves more of them - mine or yours?

18. Therefore, given the path that we are on, WE ARE NEVER GOING TO
DIFFER IN OUR OPINION OF THE TRUTH VALUE OF ANY ARITHMETICAL
SENTENCES. WE WILL ONLY DIFFER AS TO WHETHER WE SHOULD ACCEPT THEM AS
HAVING BEEN SETTLED.

19. This phenom is intimately connected with the matter of
INTERPRETATIONS BETWEEN THEORIES. In general, any interpretation of
one natural set theory in another also immediately proves that every
arithmetic sentences provable in the first is provable in the second.

20. We have found out that for any two natural set theories, either
the first is interpretable in the second, or the second is
interpretable in the first.

21. In fact, there is a stronger observed fundamental phenom. For any
two natural set theories, either the first is interpretable in the
second, for the first proves the consistency of the second.

2. FURTHERMORE, seriously alternative foundational frameworks are not
going to change this situation at all, as people are and want to
continue establishing interpretations between them and roughly ZFC. So
whatever good alternative foundational frameworks are, they are not
coming to the rescue here.

22. Another difference between set theoretic statements like CH and
arithmetic sentences.

23. Although there are tons of very natural set theoretic statements
known to be independent of ZFC, there isn't a single natural
*mathematical* arithmetic statement known to be independnt of ZFC.
(For finite set theory, there are, Goodstein, Paris/Harrington, and
pretty much superseded by the Adjacent Ramsey Theorem, and some
others). None for ZFC except for one researcher, with the very latest
being written up by him/her because the naturalness looks perfect.

24. This is a very good place for me to stop.

Harvey Friedman