Re: Implementable Set Theory and Consistency of ZFC

On Oct 10, 7:00 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
With the implementation, it can be proved that eight out of the nine
axioms of ZFC are consistent, that only four axioms are needed for a
constructive build of all sets, and that common ZFC, with the axiom of
Infinity included, is not consistent.

Although HdB is wrong about Foundation here, he does raise some
interesting questions about ZF-Infinity.

1) HdB points out that the Axiom of Choice is not needed for finite
sets at all. In other words:

ZF-Infinity+~Infinity |- AC (where |- denotes turnstile)

(i.e., if every set is finite, then AC is automatically provable).

Using the Deduction Theorem, the above implies:

ZF-Infinity |- (~Infinity -> AC)

Replacing the right side by its logical contrapositive:

ZF-Infinity |- (~AC -> Infinity)

Finally, using the converse of the Deduction Theorem:

ZF-Infinity+~AC |- Infinity

But I can hardly imagine what a proof of the Axiom of
Infinity would look like in ZF-Infinity+~AC. (But I do
agree with HdB on this point.)

2) HdB tells us that Powerset is not needed either.

ZF-Infinity+~Infinity-Powerset |- Powerset

This time, I can sort of imagine the proof. To prove that
any set x must have a powerset, we induct on the
cardinality (finite, of course, by ~Infinity) of x.

Clearly P(empty) exists by the Axiom of Pairing, and
pair the empty set with itself, of course.

Suppose P(x) exists, and we wish to construct
P(x union {y}), with y not in x. Now we use the Axiom
Schema of Replacement, and replace each element
of P(x) (i.e., each subset of x) with the same subset
union {y}. We then use the Axiom of Union by taking
the union of the above result and the original P(x). The
final result is now P(x union {y}).

I don't know how rigorous the above proof is, and
besides, one would think that Powerset would be
provable without the Axiom Schema of Replacement.

And speaking of Replacement:

3) HdB avoids mentioning how Replacement is
derivable from other axioms. Of this, I am highly
skeptical, for how can one derive a full _schema_ from
Extensionality, Empty, Pairing, and Union (HdB's
four axioms) anyway?

Notice that if HdB is right, he'll have shown that
ZF-Infinity is finitely axiomatizable, which doesn't
seem right at all (since after all, not even PA is
finitely axiomatizable).

.

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