From Newsgroup: sci.logic
On 03/31/2019 05:26 PM, Ross A. Finlayson wrote:
On Sunday, March 31, 2019 at 5:07:32 PM UTC-7, Peter Percival wrote:
Ross A. Finlayson wrote:
On Sunday, March 31, 2019 at 3:27:14 PM UTC-7, Peter Percival wrote:
Ross A. Finlayson wrote:
On Sunday, March 31, 2019 at 1:06:14 PM UTC-7, Peter Percival wrote: >>>>>> Ross A. Finlayson wrote:
No, you've confused Tarski's definition of truth as
"provable, in the theory" with that distinction, i.e.,
as in "the" theory, the working theory, usually.
After just three lines I haven't a clue what you're talking about. Do >>>>>> you write your posts on slips of and then pick them out of a hat at
on slips of paper and...
random and type whatever turns up? It's not just this post, it's every >>>>>> post of yours I've ever read.
Also, it's Pete who thinks that true means provable ('in the theory' >>>>>> seems to be an optional extra for him), not Tarski and not me.
If I've got confused, I look forward to the readers who know what
they're talking about correcting me.
Ah, then it seems as simply
"don't confuse Tarski's truth
with incompatible definitions
as (all of their) derivations,
inductive and deductive".
"Provable, inside the theory, is
no different than true, inside
the theory."
I don't know what true inside a theory means. Pete does. Well, maybe >>>> not, he seems to have as much trouble understanding his own posts as he >>>> does understanding other people's.
Tarski-esque: "Fine by me,
familiar with Goedel's results
I let truth be outside, too".
See also https://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem ,
"Tarski's undefinability theorem (general form): Let (L,N) be
any interpreted formal language which includes negation and
has a G||del numbering g(x) such that for every L-formula A(x)
there is a formula B such that B rao A(g(B)) holds in N. Let T*
be the set of G||del numbers of L-sentences true in N. Then
there is no L-formula True(n) which defines T*. That is,
there is no L-formula True(n) such that for every L-formula A,
True(g(A)) rao A is itself true in N. "
It seems that if there is to be that there is only Goedel
number of L-sentences true in N, it seems a way to see
that there aren't Goedel numbers of sentences not true in
T, i.e., here that there is a "standard" model, of the
objects of the language of arithmetic.
It's otherwise usually the idea I think that
"if there exists a model, there exists a standard
model". Where the system is strong enough to
support (all of) arithmetic, there are ideas
that's not always so, and here that it's not.
"No standard models of N, or T"?
What is the standard model of T?
Most would agree numbers and theory have models.
For example data or information, these are often formalized.
But, if Goedel and Tarski say "no standard models
of N, or T", and that's: "nor" T, has then that
models of T for example ZF are not standard.
These doesn't mean they aren't uniform: the theories,
just not standard, to "ordinary arithmetic", here as
that that's not algebra.
There's still that all the standard theories, are the
standard, it's that all together, "that" standard theory,
is to them not regular in ordinary well-foundedness, the
regularity of ZF set theory.
That's logical, usually, the composition of sets in set
theory, then for example constructing their products:
the sets', these are the usual Cartesian functions.
That's, "the sets'", it's the possessive of "the sets".
--
"He who will not reason is a bigot;
he who cannot is a fool;
he who dares not is a slave."
- Sir William Drummond
Then, some "standard model of T", here "the theory of
all theories or ''meta-'' theory", is also no standard models,
in T.
This has that to be a model of the meta-theory, that is
the theory or model of T, the theory, whatever other
theory it is that T models, the content mostly or structurally,
"there's isn't a standard model of T" for the same reason
that besides there _is_ a model of T, that there isn't,
a _standard_ model. (Of course here T is strong enough
for arithmetic.)
(Of T, or "in T, of T".)
Here the usual example of a theory strong enough to
support arithmetic is ZF set theory (and its language,
whatever that is and here only what it is).
Because, otherwise it would be.
Of course there's nothing but a standard model available
in theories like ZF set theory, of descriptive set theories
of models of theory, here with ZF sets in ZF set theory.
Here that there's not, from outside or Tarski's logic where
any theory he has besides that other one is still classically
true, Tarski may or may not have the resources to structurally
recognize another theory as so, he acknowledges, while still
yet what knowledge Tarski has in his theory he cares to maintain.
That
"there exists a standard model"
and
"there exists a model"
aren't the same thing is relevant
in theories for example ZF set theory
where there is for example Goedel's
argument following Cantor, and that
the space is large the infinite space,
of words, that it's large enough to be
the constructible universe, besides that
for no instance of "not all" the words
together, does it suffice the language
to "structurally encode" the universe.
This is where ZF set theory and other
theories strong enough for arithmetic
and for example finite combinatorics
and many various bounded type theories,
clearly there are models of systems
in finite combinatorics, if essentially
memoryless, yet closed in counting
arguments.
(Those are standard models.)
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