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.deBruijn@xxxxxxxxxxxxxx writes:

On 18 okt, 19:52, "Jesse F. Hughes" <je...@xxxxxxxxxxxxx> wrote:

It's not clear to me what the last claim means, but regardless, it
must mean something about "all sets" in this particular model of
ZFC - Infinity and not all sets of ZFC.

No. It's about all sets in (ZFC - Infinity). And the "particular
model" is any implementation, is anything applicable.

I guess you mean ZFC - Infinity + ~Infinity, since there are models
of
ZFC - Infinity that have infinite sets.

It's impossible to have infinity without Infinity. If not, show us such
a model, please.
Any model for ZFC is a model for ZFC - Infinity. Why? Because it
satisfies every axiom for ZFC - Infinity.
As I said, what you must mean is ZFC - Infinity + ~Infinity. That
is,
take ZFC, remove the axiom of infinity and add the negation of the
axiom of infinity.

Not in my article:

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

Look, what you said was simply wrong. You said (piecing things together):

Only the first four axioms are necessary for a constructive build of
all sets of any "implementation" (model?) of ZFC - Infinity.

In the first case, it's not at all clear what this statement means.
But whatever it means, since a model of ZFC is also a model of
ZFC - Infinity, you seem to be claiming that you can build an infinite
set using only the first four axioms.

Obviously, you do not believe that, so I am sure you did not say what
you mean.

--
"The needs of the many outweigh the needs of the few [...] I must
make the same choice as those who came before me without regard to the
impact today, for the sake of the children of humanity, the children
of tomorrow." -- JSH channels Spock and generic politicians everywhere
.

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