On 10/02/2022 08:02 PM, Ross A. Finlayson wrote:
> On Sunday, October 2, 2022 at 4:36:15 PM UTC-7, Ross A. Finlayson wrote:
>> On Friday, September 30, 2022 at 7:43:33 PM UTC-7, Newberry wrote:
>>> Let T be a theory of Peano Arithmetic. We can construct the following
>>> hierarchy
>>>
>>> T
>>> T' = T + Con(T)
>>> T" = T' + Con(T')
>>> T"' = T" + Con(T")
>>> . . .
>>> Does this hierarchy have a fixed point in some sense? ['Con(T)' means T
>>> is consistent, it is equivalent to Gödel's forrmula.]
>>>
>>> RELATED QUESTION:
>>> Is it possible to formulate this
>>>
>>> T' = T + Con(T') ?
>>>
>>> I.e. Con() applies to T' itself. [I know the resulting theory is
>>> inconsistent, and does not have a model.]
>> Goedel omega?
>>
>> (Goedel's omega'th function?)
>
> I just define induction as Goedel functions in the beginning
> and use the completeness part, of Goedel's completeness,
> then build higher order "under" that.
>
>

It's to be thanked that Goedel left logic open
with "incompleteness".

The cumulative hierarchy is the familiar terms of "orders of higher
orders",
here about the consistency strengths of arithmetics,
there's much to be said for both the inner and outer,
the deconstructive and combinatoric, then that what
really is for the bridge results of the transfer principle,
that "what to add to ZF" should be framed as for
"what is a give and take with ZF and a theory with the illative",
because it's really not one without the other,
that "large cardinals" aren't sets nor cardinals,
they're just "model witnesses of higher-order constructs
what must bring foundations to a full consistency".

Adding arbitrary stipulations obviously is just chasing-the-tail.
The goal is that the dog catches its tail.

For the dog chasing its tail,
a usual symbol is that
the snake already caught its tail,
for the Ouroboros as a usual
notion of the cyclical,
as of otherwise the schematic.

Here it's that a theory for incompleteness
of arithmetic sort of must have started
with an incomplete theory of arithmetic,
that either arithmetic arises from numbers existing,
or that it all entirely rulial, here for all
what remains in the middle according to
usually finitely axiomatized arithmetizations.