Re: 1-1/2+1/3-1/4+1/5-1/6+1/7

On 2008-02-19, in sci.math, Virgil wrote:
Isn't it worthwhile investigating the consequences of assuming ZFC as
long as it is not known to be internally inconsistent (and possibly even
after)?

Investigating what follows from the usual principles of set theory is
indeed worthwhile.

What I was commenting on was the not uncommon idea that since no-one
has yet found a contradiction in ZFC we should assume ZFC is not
inconsistent. On that reasoning, since much more effort has gone into
proving Goldbach's conjecture than to finding a contradiction in ZFC,
and yet Goldbach's conjecture remains unproven, we should similarly
assume that Goldbach's conjecture is unprovable in ZFC. But of course
we do no such thing.

So if one wishes to maintain that our confidence in the consistency of
ZFC is a result of reflecting on the failure of mathematicians to find
a contradiction it must be explained what the relevant difference is
between the two mathematical assertions, "ZFC is consistent" and
"Goldbach's conjecture is unprovable in ZFC".

Predictably, I must now also remind you that consistency is a piddling
condition. In particular, the consistency of a theory T does not
guarantee that if it proves "the algorithm A terminates on every
input" the algorithm A terminates on every input, or that if it proves
the twin conjecture, the twin prime conjecture is true, and so on. So
if we accept set theoretic proofs of such statements as establishing
their truth, we are implicitly committed to more than just the
consistency of ZFC.

These reflections of course do not imply that we should not be
confident in the consistency, arithmetical soundness and so on, of
ZFC, only that the "inductive" argument and obsessing over mere
consistency are silly. Consistency, arithmetical soundness etc. are
simply consequences of the soundness of ZFC. If one accepts that there
exists an inductive set, that every set has a powerset, and so on, as
most working in mathematics do, one should just as willingly accept
that the axioms of ZFC are true, since this amounts to nothing more
than accepting that there is an inductive set, every set has a poweset
and so on. If one does not accept these principles, one shouldn't
accept proofs in ZFC either, and studying ZFC might or might not be
somewhat pointless.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
.

Relevant Pages

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