Re: Implementable Set Theory and Consistency of ZFC

On Wed, 24 Oct 2007 15:57:41 +0200, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

David C. Ullrich wrote:

On Wed, 24 Oct 2007 09:50:42 +0200, Han de Bruijn
<Han.deBruijn@xxxxxxxxxxxxxx> wrote:

MoeBlee wrote:

On Oct 23, 4:23 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

1. Extensionality 5. Specification X. Infinity
2. Empty set 6. Substitution
3. Pairing 7. Power Set
4. Union 8. Foundation
9. Choice

And, as I've said, in this "model", only (1-4) are necessary as axioms,
because (5-9) appear as theorems. And (X) is not part of the "model".

I just saw Virgil's post, which made me realize I overlooked that you
said 5-9 are theorems of 1-4!

You're wrong. None of 5-9 is derivable from all of 1-4.

What you might mean is that in a certain model of 1-4, we have that
5-9 are true in that model. But that does not ENTAIL that any of 5-9
are theorems of 1-4.

You are wrong. Read the article.

Just for fun I downloaded it and looked at it.
The supposed proof of (5) is this:

Think you mean (8) Foundation, throughout the following.

"Theorem. The minimal element in a set, when implemented
as a natural array, is always disjoint from the set.
Proof. This is a direct consequence of the above lemma:
the leftmost bit position of an integer >= 0 has a numeric
value which is always less than the numeric value of the
integer itself.
This completes the proof, of the assertion that the axiom
of Foundation is just a Theorem in our implementable set
theory."

All that stuff about bit positions is specific to your
"model". Sure enough, exactly as Jesse conjectured,
it's _not_ a proof that 5 follows from 1-4 as you
insist, it's just a proof that 5 is true in your
"model".

At most it's a proof that 5 is true of the hereditarily
finite sets. That's not what you've been insisting
it is, in particular it is simply _not_ a proof that
5 follows from 1-4.

Any (implementable) set is a hereditarily finite set, anyway.

Which of course is not surprising, since 5 does not
follow from 1-4.

Once you've entered my universe, you're plain wrong.
But I think much of the confusion rather stems from poor terminology.

http://en.wikipedia.org/wiki/Computable_function

Modified quote. Replacing "function" by "SET":

According to the Church-Turing thesis, computable SETs are exactly the
SETs that can be calculated using a mechanical calculation device given
unlimited amounts of time and storage space. Equivalently, this thesis
states that any SET which has an algorithm is computable. Consequently:
"Implementable Set Theory" has been renamed to "Computable Set Theory".

http://hdebruijn.soo.dto.tudelft.nl/jaar2007/set_theory.pdf

Does that make things better? Is it an improvement?

In fact the idea that you think this helps is sillier than
I realized at first. Because all the sets in your gizmo are
finite - computable sets are not necessarily finite.

Han de Bruijn


************************

David C. Ullrich
.

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