Hierarchy of Arithmetics

[Newberry]

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.]

[Antonio Speltzu]

And what about TC = T U T' U T'' U T''' ..... ?
Obviously TC is consistent if T is.
Could it be TC = TC + Con(TC)?
Could TC be complete?

[Julio Di Egidio]

On Saturday, 1 October 2022 at 04:43:33 UTC+2, 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.]

IIANM, that sequence (standardly!) has no limit, i.e.
as long as the (standard!) set of ordinals has no limit.

As an aside:

And what would the "limit of Goedelization" be
anyway? That I propose is the statement "you fail":
<< "You fail" might express more than the abstract
sentence ~Bf ("consis"; can read "not-believe-that"),
where belief is [standardly!] the foundational meaning
behind logical negation and falsehood. >> (*)

(*) The unified theory of all we can do with it (seriously)
<https://groups.google.com/g/sci.math/c/rWSbPUGsDFU/m/M0ItgdR4mVEJ>
(Of course the original post is not to be found anywhere.)

> 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.]

There are essentially two ways to state consistency:
one is model theoretic, as in "there exists a model for
*this theory*", the other is via a formula stating "it is
not possible in *this theory* to derive both P and ~P".
In either case, we have a statement about the
present theory (say T), not a statement about the
theory plus the consistency statement (say T'),
so I would think what you ask is *not* possible.
But I might be wrong/missing something here...

Julio

[Ross A. Finlayson]

Goedel omega?

(Goedel's omega'th function?)

[Ross A. Finlayson]

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.

[Ross A. Finlayson]

There's no point intuitionism except to constructivism.

Here the "arithmetization" provided by Goedel functions,
is about as simple as induction and all under it in type theory.

Then at the end these hier and hier towers, are also what
need to exist to support the elements of all the replete sets
under types, then still the _incompleteness_. is about fixed
point or compact, in that certainly the fixed point or compact,
most represents the limit point that is still the universe of
things, for example everything in its language.

(The theory's, "its", ....)

Yes, so, indeed, that's the definition of fixed point here.

[Ross A. Finlayson]

The limit point that to all points of perspective is infinity? Fixed-point?

[Ross Finlayson]

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

[Ross Finlayson]

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.

[Ross Finlayson]

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.