[FOM] Is ZFC consistent?

[Harvey Friedman]

There has been a number of thought provoking postings recently 
touching on the question of why most people believe in the 
consistency of ZFC, and related matters.

In particular, there has been a discussion of the role of the actual 
experience with working in ZFC and studying ZFC, and there being no 
hint or trace of a difficulty with ZFC.

I would like to state my own view of this situation. I.e., the 
situation that arises out of the apparent attractiveness of ZFC, no 
hint or trace of any difficulty with ZFC, and that one is not going 
to prove that ZFC is consistent within ZFC. If one did prove that ZFC 
is consistent within ZFC, one would immediately also have an 
inconsistency proof of ZFC, and one would discard the former in favor 
of the latter.

As usual, I don't have anything novel to say directly about this situation.

What I think about, instead, is: what can be done about this 
situation? I.e., what f.o.m. theorems can be proved that bear on this 
situation?

Here are a few ideas that come to mind. This is not my complete list.

1. Find general philosophical principles of a fundamental character 
that transcend set theory and have a deeper philosophical meaning, 
that are perhaps compelling as well, and that prove the consistency 
of ZFC. One also looks for them to be equivalent to the consistency 
of ZFC, or at least provably consistent within ZFC + large cardinals. 
Also, one wants to do this for ZFC + large cardinals as well.

This project is off the ground in a serious way. See

a. From Frege to Friedman: A Dream Come True?, revised version, John 
Burgess,  http://www.princeton.edu/~jburgess/anecdota.htm

b. E Pluribus Unum, revised version, John Burgess, 
http://www.princeton.edu/~jburgess/anecdota.htm

c. A Way Out, Harvey Friedman, http://www.mathpreprints.com/math/Preprint/show/

d. Sentential Reflection, Harvey Friedman, 
http://www.mathpreprints.com/math/Preprint/show/

e. Elemental Sentential Reflection, Harvey Friedman, 
http://www.mathpreprints.com/math/Preprint/show/

f. Similar Subclasses, Harvey Friedman, 
http://www.mathpreprints.com/math/Preprint/show/

g. Restrictions and Extensions, Harvey Friedman, 
http://www.mathpreprints.com/math/Preprint/show/

2. Prove that ZFC and/or related systems are in an appropriate sense 
"complete". I.e., there are no fundamental "axioms" that are missing. 
To make a long story short, I have a general simplicity project, 
which has been discussed in bits and pieces from time to time on the 
FOM. I will post shortly on this in greater depth.

The simplicity project with regard to set theory should be relevant 
to this project. E.g., see

h. Three Quantifier Sentences, Harvey Friedman, 
http://www.mathpreprints.com/math/Preprint/show/

i. Primitive Independence Results, Harvey Friedman, 
http://www.mathpreprints.com/math/Preprint/show/

In this approach, it should be a theorem that the CH is not simple.

3. Show that ZFC and/or related systems are in an appropriate sense 
the slavish importation of known and well accepted facts about the 
finite universe, into the infinite universe. I.e., just add the axiom 
of infinity. Again, I am expecting incomparably better results than I 
have now. See

j. Transfer Principles in Set Theory, Harvey Friedman, 
http://www.math.ohio-state.edu/~friedman/

4. The contemplation of fundamental mathematical concepts results in 
an altered state of consciousness. When contemplating ZFC, most 
experienced logicians, particularly set theorists, experience a 
distinctly comfortable state of consciousness that is not too 
different in many qualitative senses than what they experience by 
contemplating, say, hereditarily finite set theory, or Peano 
Arithmetic. Analyze such altered states of consciousness from several 
points of view, including brain wave data, and other physiological 
responses. Make sense of the data, by contrasting it sharply with the 
corresponding data obtained when contemplating other formal systems 
and various mathematical structures - e.g., ZFC + CH and ZFC + notCH. 
Investigate the role of simplicity, particularly its effect on the 
brain wave and other physiological data.