Re: Implementable Set Theory and Consistency of ZFC

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

Jesse F. Hughes wrote:

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

Jesse F. Hughes wrote:

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

Jesse F. Hughes wrote:

I did not ask about your needs. I asked whether the formula ~Infinity
is also a theorem of (1)-(4), where ~Infinity is the negation of the
axiom of Infinity.

(~Infinity) is _not_ a theorem of (1)-(4), in this article:

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

Why not?
You show that foundation is true in your model. You conclude (1)-(4)
entail foundation.
~Infinity is also true in your model. Why do you not conclude that
(1)-(4) entail ~Infinity?

It is remarkably difficult to get an answer from you sometimes.

Really?

Really.

I simply cannot "prove" anything (constructively) about something I can
not understand (constructively).
As I just indicated, if your argument for (5) is a proof of (5) from
(1)-(4), then my argument for ~Infinity is a proof of ~Infinity from
(1)-(4). They are essentially the same argument.
Your response seems utterly beside the point. You *do* agree that
every set in your model is finite, yes? Then why isn't this a proof
of ~Infinity?

I have given you the answer. Why isn't this a proof of ~(exists a
Foo)?

I don't follow your so-called answer at all.

Here's my statement of ~Infinity:

Every set has a finite number of members. In other words, every set
can be put in one-to-one correspondence with some set {0,1,...,n}.

Proof:

Each set n has at most log_2(n)+1 members (except 0, which has 0 members
of course).

Seems to me that this proof is very similar to the proof of (5) found
in your paper. So what's wrong with concluding that (1)-(4) also
entail ~Infinity?

--
"[T]he Cantorian pseudomathematicians are defending a religion, and
they really can't see what monsters they have become. What the
Cantorians are doing is nothing less than a crime against
humanity. What they are doing is evil." -- David Petry, victim.
.

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