[FOM] Pi01 Incompleteness/search for truth

[Harvey Friedman]

On Tue, Sep 2, 2014 at 8:10 AM, Colin McLarty <colin.mclarty at case.edu> wrote:
>
>> Writing about his view that V=L might be more attractive in most
>> mathematics than its negation, On Fri, Aug 29, 2014 at 6:04 PM, Harvey
>> Friedman <hmflogic at gmail.com> concluded:
>>
>> > Of course, there is a monkey wrench in all this, particularly if the
>> > following Thesis is verified - and we are not quite there yet.
>> >
>> > THESIS. Corresponding to every interesting level in the interpretation
>> > hierarchy referred to above, there is a Pi01 sentence of clear mathematical
>> > interest and simplicity. I.e., which is demonstrably equivalent to the
>> > consistency of formal systems corresponding to that level, with the
>> > equivalence proved in EFA (or even less). There are corresponding
>> > formulations in terms of interpretations and conservative extensions.
>> >
>> > Then what?
>>
>>
>> Why is this a "monkey wrench"?  Why is it not just a reason to continue
>> pursuing low complexity sentences of clear mathematical interest equivalent
>> to consistency of various formal systems?

Because at least a substantial number of the best mathematicians in
the world, and many other mathematicians, view themselves as pursuing
objective truth. So the question for them is: are these statements
true or not? Thus they will inexorably be drawn into the philosophical
and the foundational.

I am in reasonably steady contact with good to great mathematicians
way outside logic, and will be raising this issue with them on a
regular basis. The examples are just about good enough for me to
effectively raise the issue - just barely. Of course, I will get
clearer ideas about how they think as the examples progress in variety
and quality.

Harvey Friedman