[FOM] 729: Consistency of Mathematics/1

Dmytro Taranovsky dmytro at mit.edu
Tue Oct 25 22:55:25 EDT 2016

Harvey Friedman asked:
 > What are the most monumental issues in the foundations of mathematics?

I think the most important problem in foundations of mathematics is 
finding a true reasonably complete axiomatization of set theory.  It may 
be like projective determinacy for second order arithmetic but covering 
first order truth in V and beyond.  Set theory provides the universal 
language for mathematics, but basic questions about V remain independent 
of the current axioms.
Note: Formalists (if they remain formalists) will recognize the solution 
not as metaphysically true but as making set theory (and mathematics) 
much better and more beautiful as a theory.  Currently, most formalists 
understandably have doubts about whether the solution exists.

However, it might be that there is something fundamental that we are 
missing, and once we discover it, we realize it as the most important 
advance in foundations of mathematics.
For example, if effective P=NP holds, then P vs NP (or a similar 
question) is the most important problem in foundations of mathematics. 
And even a theorem that P is not NP might be comparable to Godel's 
incompleteness theorem in importance to foundations of mathematics:  It 
will be a fundamental advance in the theory of computation, and hence 

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

I think that GCH is true, and that once we sufficiently understand set 
theory, it will become obvious. However, it is possible that no 
consensus on CH will be reached until we have a true reasonably complete 
axiomatization of third order arithmetic.  I wrote about CH in previous 
FOM postings ("On the Continuum Hypothesis, part 1" and "On the 
Continuum Hypothesis, part 2"):
In simple terms, GCH makes for a more natural and coherent theory, and 
arguments against CH usually depend on applying intuitions beyond their 
domain of applicability.

 > 1. Is mathematics consistent - I.e., free of contradiction?

One way to address consistency is to investigate theories that under 
current knowledge are close to the edge of inconsistency.  An example is 
ZFC + for every ordinal kappa there is a transitive kappa-closed model 
of ZF + "there is a set S such that every binary relation whose domain 
extends S is nontrivially self-embeddable". (Use of ZF might seem 
arbitrary, but I expect that we get an equivalent statement if we 
replace "ZF" with anything reasonable (without choice) such as ZF + 
proper class of inaccessibles. Also, one (possibly equivalent) extension 
is to make the embedding nontrivial on S.)

If there is inconsistency, it might be easiest to find in such a theory, 
and conversely, failure to find inconsistency supports the view that the 
notion of the cumulative hierarchy with elementary embeddings is consistent.

Another way to help rule out inconsistency is to identify and study 
well-understood models of set theory.  Fine structure theory gives 
strong evidence of consistency of Woodin cardinals, but not yet 
consistency of supercompact cardinals.  Ordinal representation systems 
go qualitatively beyond fine structure in the level of detail they 
provide.  I think that pursuing ordinal notation systems is promising, 
but currently there is no simple/reasonable ordinal notation system that 
has been proved to go beyond a fragment of second order arithmetic (the 
strength of the notation systems I proposed remains conjectural).

 > [is the strength of] ZFC of any real use in ordinary mathematical 

The phrase 'ordinary mathematics' is relative.  If there is sufficient 
interest in uncountable sets, their study becomes ordinary mathematics.  
That said, I do not know how important large cardinal axioms will be to 
number theory.

Dmytro Taranovsky

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