[FOM] 729: Consistency of Mathematics/1
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
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