Re: The meaning of set?


Aatu Koskensilta wrote:
Rupert wrote:
Aatu Koskensilta wrote:
Rupert wrote:
The good thing about ZFC is that we have an algorithm for generating
all the sentences we accept as theorems.
Really? What is that algorithm?

What I have in mind is an algorithm for listing every theorem of ZFC.
Such an algorithm exists because this set is recursively enumerable.

Sure.

I'm not sure what your concern is. Perhaps you find my use of the
phrase "the sentences we accept as theorems" to be imprecise.

That was my concern. The recursive enumerability of the set of theorems
of ZFC gives us no clue as to what algorithm it is, if any, that
generates all the sentences we accept as theorems (in general, or of
ZFC), unless "we accept as theorems" is intended to be read as "for
which formal derivations exist in ZFC". So I wonder what the relevance
of your observation was, if it was just that the set of theorems of ZFC
is recursively enumerable?


Perhaps recursive enumerability is not the main point. But I think it's
important to have some precise characterization of what you accept as a
sound argument and what you don't. Arguments in set theory are in
principle formalizable, and then it is machine-checkable whether an
argument is sound or not. So we have a precise characterization of what
counts as correct reasoning and what doesn't. I was encouraging Zuhair
to try to give a precise characterization of what he would be prepared
to accept as a sound argument.

Zuhair's "different approach to sets and analysis based on them" might
well be pure waffle, as the case appears to be, but not because he
doesn't present an algorithm for generating everything that he would
accept as a theorem - whatever sense of accepting is meant here. We
might just as well object that set theory is unacceptable because we
can't given an explicit algorithm listing all the theorems acceptable on
basis of the informal principles. What Zuhair needs to do is to present
a coherent and interesting account of sets - he can worry about precise
axiomatization, let alone formalization, later. I'll be holding my breath.

--
Aatu Koskensilta (aatu.koskensilta@xxxxxxxxx)

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

.

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