On 02 Jun 2015, at 20:10, Brian Tenneson wrote:
Grammatical systems just might be the type of thing Tegmark is
looking for that is a framework for all mathematical structures...
or at least a large class of them.
I am still exploring the idea of grammatical system induction.
I am not sure what you mean by grammatical induction. Is that not
equivalent with the omega-induction principles, like with PA?
I believe it can be used to provide an induction principle that
allows one to prove something about all sets in ZFC (or any set
theory).
You will need transfinite induction.
But with the usual (omega) induction, you get already Löbianity, and
the self-reference logics will not been changed with addition.
Applying the general grammatical system induction to formal systems,
I believe there is a way to prove something about all theorems
within a formal system,
Yes, PA can prove that ZF proves things. For proving that a formal
system does not prove something, you will need strionger systems, and
by incompleteness such negative statements cannot be axiomatized in
once system.
perhaps providing a little insight into truth in general.
Also, an induction principle applies to all proofs if one wants to
prove something about all proofs in a formal system.
The document in the first post has been updated to include all of
this. There are some words I need to change so just notice the
essence...
Any feedback is appreciated!
If you assume computationalism, simple (omega) induction is enough to
get the machine psychology and theology, and to justify why machines
will build more and more induction rules, but none will get the
"whole" truth, which is beyond axiomatization and formalization (even
assuming computationalism).
You seem to try to do what the logicians have already done. You might
study the little book by Torkel Franzen on the "Inexhaustibility",
which makes rather clear the elusive character of truth.
Bruno
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