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.

Erm, that's silly. As people keep saying over and over,
if you're using words to mean something other than
what they usually mean (for example if words have
different meanings in "your universe") you need to
explain what those meanings are. Or better, use
different words.

In fact 5-8 do not follow from 1-4. The explanation
for the problems with your "proof" that they do is
exactly as people have been saying - showing that
5-8 are true in _a_ model that happens to satisfy
1-4 does not prove that they follow, for that you
need to show that 5-8 hold in _every_ model of 1-4.
Many people explain this over and over - instead
of trying to understand their perfectly correct
corrections you complain about the fact that they
haven't read your "proof" because they're lazy.
This is just silly.



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?

Huh? It has no bearing on the question of whether 5-8
follow from 1-4. They don't.

Han de Bruijn


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

David C. Ullrich
.

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