Re: Implementable Set Theory and Consistency of ZFC

On 17 Okt., 14:13, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:
Jesse F. Hughes wrote:
Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:

MoeBlee wrote:

On Oct 15, 7:18 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> wrote:

Hereditarily finite sets = naturals : implementable set theory
_ Mainstream mathematics : naturals = finite ordinals

So the naturals are a common factor in two theories. And they join the
finite ordinals (that is: axiom of Infinity) with the "set of all sets"
in implementable set theory. The latter does not exist, though.

Doesn't that say something? Isn't there an analogous pattern, somewhere
in common model theory?

There is no principle of model theory or mathemtatical logic that
permits the inference you are trying to make.

Can it be assumed that you are knowledgable enough, so that I can trust
this assertion of yours?

Your argument is ludicrous; it's based on your manifest ignorance and
misunderstanding of the basics of the subject.

The usual mainstream reaction when they are running out of "arguments".

It's also the usual mainstream reaction when faced with someone who
gives a silly argument. You know, like "N is in both theories, but
doesn't exist in one of the theories it's in. Doesn't that say
something?"

Now, how do we tell which situation this is: mainstream in retreat, or
mainstream reacting to a silly person?

This is the new summary of my - updated - article:

With help of a bijection, which was basically discovered by Alexander
Abian, a "simple model", or rather an implementation of the axioms of
ZFC in memory of common digital computers, has been conceived,
in theory as well as in practice.

This is still not correct as the Axiom of Infinity is an axiom of ZFC
and does not hold in your model (which is thus a model of ZFC-
Infinity,
in fact one of ZFC-Infinity+~Infinity).

With the implementation it can be
proved that eight out of the nine axioms of ZFC are consistent,
and that only the first four axioms are necessary for a constructive
build of all sets. A slight modification of the ninth axiom, Infinity,
is needed.

?? "slight modification" as in "prepend it with '~'"?

Which makes that it is not consistent within Implementable
Set Theory, even less when the latter is supposed to allow the Naturals
as a completed infinite set.

The last few sections of the article have been adapted accordingly:

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

Meaning that I'm no longer interested in the question whether ZFC is
consistent within common mathematics. I'm only interested whether it's
consistent within my own mathematics. And I'm trying to be political
correct by not saying such explicitly. To make everybody happy :-)

Han de Bruijn

You're still wasting a lot of paper to state that no infinite set
exists
under the assumption that all sets are finite.

Note that it does not matter whether you take von Neumann or binary
successors. The (or at leas one) essential point of the Axiom of
Infinity
is the existence of a set N and a relational predicate S on N such
that
- N contains a special element 0
- for all a in N there is b in N such that S(a,b)
- S(a,b) and S(a,c) implies b=c
- S(a,b) and S(c,b) implies a=c
- there is no x such that S(x,0)
No finite set can have this property of being (Dedekind) infinite
and you are only dealing with finite set throughout in your model.
Don't you see that people had a *reason* to include Infinity
as otherwise set theory had an useless simple model?

.

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