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Topic: 10^7-ZFC
Replies: 30   Last Post: May 2, 2012 2:53 PM

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 ross.finlayson@gmail.com Posts: 642 Registered: 2/15/09
Re: 10^7-ZFC
Posted: May 1, 2012 2:08 PM

On Apr 30, 11:47 am, MoeBlee <modem...@gmail.com> wrote:
> On Apr 30, 1:06 pm, Zuhair <zaljo...@gmail.com> wrote:
>

> > What I mean by syntatically consistent here is: non existence of
> > contradictory theorems, which is what you refer to as limited
> > consistency.

>
> I DON'T refer to any terminology "limited consistency".
>
> And we don't need the word 'syntactically' to qualify.
>
> Here are the definitions and some theorems I'm using (all for ordinary
> first order logic):
>
> (1) Def. A sequence S of formulas is a derivation from G iff S is
> finite and every entry of S is either a member of G, a logical axiom,
> or follows by a rule of inference from previous entries.
>
> (2) Def: A set G of formulas is consistent iff there is not both a
> derivation from G of a formula P and a derivation from G of a formula
> ~P.
>
> (3) Def: A set of formulas G entails a formula P iff every structure
> and assignment for the variables that satisfies G also satisfies P.
>
> (4) Def: A sentence is a formula with no free variables.
>
> (5) Def: A theory is a set of sentences closed under entailment.
>
> (6) Thm: A theory is set of sentences closed under derivability.
>
> (7) Thm: A theory T is consistent iff there is no sentence P such that
> both P and ~P are members of T.
>
> (8) Def: A model M is a model of a set of sentences G iff every member
> of G is true in M. (And we say a set of sentences G has a model iff
> there is an M such that M is a model of G.)
>
> (9) Thm: A set of sentences G is consistent iff there is an M that is
> a model of G.
>
> Now you also have these notions, which I'll give names:
>
> (8) Def: A set of formulas G is self-compatible iff there is no
> formula P such that both P and ~P are members of T.
>
> (9) Def: A set of sentences G is an n-limited closure of a set of
> formulas H iff G is the set of sentences derivable from H in less than
> n characters of proof.
>
> (10) Thm: A theory G is self compatible iff G is consistent.
>
> (11) Thm: A theory G is self compatible iff G has a model.
>
> However:
>
> (12) It is NOT the case that for all G we have that if G is a self-
> compatible set of sentences then G is consistent.
>
> (13) It is NOT the case that for all G we have that if G is self-
> compatible then G is consistent.
>
> (14) It is NOT the case that for all G we have that if a set of
> sentences G is self-compatible then G has a model.
>
> (15) It is NOT the case that for all G, H and all n we have that if G
> is a self-compatible n-limited closure of a set of formulas H then G
> is consistent.
>
> (16) It is NOT the case that for all G, H and all n we have that if G
> is a self-compatible n-limited closure of a set of formulas H then G
> has a model.
>
> MoeBlee

You have "A set G of formulas" then "a set of formulas G".

Where you have "self-compatible" I have "non-contradictory". A
consistent theory is not contradictory. So I would say 13 is false -
not that it's complete, but that it's consistent (with itself) as any
other theory you define above is.

Here I don't think you'd admit non-deterministic formulas and agree
that they were true. You might agree to evaluate them together (what,
that) as expecting they may or may not be true - these formulas that
for however deterministic they are, are non-deterministic. But,
that's not the same condition where you are also providing a model.
And here both times you say n-limited closure it is a model and the
same, 14-16.

Also 12 there is false -

"However:

(12) It is NOT the case that for all G we have that if G is a self-
compatible set of sentences then G is consistent.

(13) It is NOT the case that for all G we have that if G is self-
compatible then G is consistent.

(14) It is NOT the case that for all G we have that if a set of
sentences G is self-compatible then G has a model.

(15) It is NOT the case that for all G, H and all n we have that if G
is a self-compatible n-limited closure of a set of formulas H then G
is consistent.

(16) It is NOT the case that for all G, H and all n we have that if G
is a self-compatible n-limited closure of a set of formulas H then G
has a model." -- MoeBlee

If g is a self-compatible, i.e., non-self-contradictory set of
sentences or formula, that is the definition of consistency for it.

So, no?

What's the difference between an n-limited closure and a model?
What's the difference between self-compatible and non-contradictory?