Re: Implementable Set Theory and Consistency of ZFC

Jesse F. Hughes wrote:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

Jesse F. Hughes wrote:

"Jesse F. Hughes" <jesse@xxxxxxxxxxxxx> writes:

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

From my response to Ullrich:

So, even if I don't make use of (5-8), a proof of A from (1-4) is a
proof from (1-8) ? So, even if I say "there exists a Foo", then such
a statement is a valid premise for proving that the integral of 1/t

from 1 to x is ln(x) ? Weird ..

Talking about "a very simple observation" ..

The theorem "There is only one empty set" uses no axiom of ZFC aside

from extensionality. Do you think it's *not* a theorem of ZFC?

Instead, it is only a theorem of the theory (Extensionality)?

I should have said "There is at most one empty set." Proving that an
empty set exists requires more than just extensionality.

Yes. According to Halmos, it requires Specification. But Empty set can
also be introduced as an axiom (and I've learned to prefer the latter).

Whatever. Again, I ask: Do you agree that "There is at most one empty
set" is a theorem of ZFC, *even though* it uses only one of the axioms
of ZFC?

Yes.

Han de Bruijn

.

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