Re: Torkel Franzen on truth

herbzet wrote:

MoeBlee wrote:
herbzet wrote:
MoeBlee wrote:
herbzet wrote:
MoeBlee's assertion that it is a theorem that every structure is a
model of some theory is a new thought to me.
What is difficult about it?
(1) By definition, M is model of a set of sentences G iff M is a
structure for the language of G and every member of G is true in M. So
every model is a structure, by definition.
Right.

(By the way, in another post I think I might have allowed G to be a
set of formulas. But I think G should be a set of sentences.)
OK.

(2) Every structure M is a model of the set of valid sentences in the
language that M is a structure for.
Yup.

So every structure is a model.
Of any validity, yes. Can we say further that every structure is a
model of a theory (other than the null theory containing only validities)?
I'm not familiar with the expression 'null theory' to refer to the set
of validities, but I'll go along with it.

I probably just made it up, although it seems right. It's the theory
with no non-logical axioms (ie only validities as theorems). It seems
to me that this is a degenerate case of the word "theory": usually
I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic
and to have some non-logical axioms/theorems.

As to the question, I'd have to think about it.

Bingo. That's what's "a new thought to me": that an arbitrary structure
is a model of some non-null theory. I don't happen to know whether
that's true.
Shall we agree as to what sets of sentences constitute "a theory" first?
For simplicity, let's confine to classical first order.

Authors such as Enderton take a theory to be any set of sentences
closed under entailment (which, thanks to the completeness theorem, is
a set of sentences closed under provability). Authors such as Chang &
Keisler take a theory to be any set of sentences. And some authors
take a theory to be a pair <S E> where S is a set of sentences and E
is the set of sentences entailed by S (which, thanks to the
completeness theorem, is the set of theorems of S).

I adopt Enderton's definition.

OK.
I usually think of "a theory" as having a recursive (or at least r.e.)
set of axioms.
That is a recursively axiomatized theory. There are theories that are
not recursively axiomatized.

Yes, they seem a little imaginary to me, but I'm willing to go along
with it.

And as being consistent (i.e. having a model)!
Those are consistent theories. There are theories that are not
consistent.

It would seem that there is but one such theory over any signature:
the theory that consists of all sentences in the language. This
seems like another degenerate case of "theory". I'm willing to
accept the informal locution "inconsistent theory"; I'm willing
to accept an "inconsistent theory" as existing under the Enderton
definition of "theory".

But the extremal cases of "theories" that contain only validities,
or that contain every sentence, do seem a little silly to me. I
guess it's the price you pay for definitional elegance.

Even if it is so,
it does seem to me to be ill-advised to use "structure" and "model"
interchangably (although I'm as slack as anyone else on using
"structure" and "interpretation" more or less interchangeably).
In such contexts as I mentioned, what is the harm of using 'model' and
'structure' interchangably, especially the context I mentioned?:
If I recall correctly, this started when you said to Nam:

Okay, in a technical sense, '2=1+1' is true relative to models because
it's true in some models but not in others.
To which George objected:

No, true in some interpretations but not in others.
which might seem like a rather pedantic distinction (or, as you seem
to think, a false distinction), but I think that in this area with
Nam you have to be very precise, and the distinction is merited.
Anyone may state definitions and explicate a discussion on the basis
of those definitions. Meanwhile, what I said is precisely correct
given ordinary defintions:

'2=1+1' is true in some models and not true in other models.

That is precisely correct.

What is cloudy is the terminology 'true relative to models', which is
Nam's terminology. I don't use that terminology. All I said (or meant
to convey) in the passage you mentioned is that I could see a sense in
which that terminology could be understood.

So, again, to be clear:

My terminology is 'true in a model' and it is precisely correct that
'2=1+1' is true in some models and not true in other models. Nam's
terminology is 'true relative to a model', and though I do not in any
way claim to arbitrate what HE means by that, my point was just to say
that IF he means 'true in a model', then yes, of course, '2=1+1' is
true in some models and not true in other models. Then, there were the
rest of my remarks.

Yuh, I guess the argument turns on whether you want to affirm

a) 2 = 1 + 1 is false in no model (of PA)

or

b) 2 = 1 + 1 is false in some models (of the language of PA).

It seems to be a simple ambiguity to be resolved. I personally would
prefer (a) and would assume that you are using "model" in the sense
of (a) in the absence of an explicit definition otherwise. I concede
that it is possible that Nam uses the word "model" in some sense
other than (a).

For what it's worth, I actually don't use a different definition of the
word "model" here, in FOL context. The phrase that seems difficult to some
to understand is:

(1) 'true relative to a model'

Now suppose you believe PA is consistent and let M be a model of PA,
then the following meta level statement would be true:

(2) '2=1+1' is true in M.

The question though, if we change the inference rules and come up with
a different reasoning framework, say (FOL)', then would (2) still be
true? The answer is "Not necessarily", depending on what changes of
rules of inference that have taken place of course. Now if we consider
the formal system PA as just a *collection of axioms* then syntactically
PA is the same. But what we consider as M might or might not be an absolute
model of PA, right? In that context then, '2=1+1' is being true in M is
relative to the fact M might or might not be a model, depending on who's
doing the reasoning. In that context, then the following statement
would make sense:

(3) '2=1+1' is true, relative to a model.

which is not different in nature from the statement:

(3') the speed of the train is 100 km/hour, relative to the framework M.

At any rate, '2=1+1' is no more absolute than 'the speed of the train is
100 km/hour'.

Those who believes otherwise would not realize the relativity nature of
human mathematical reasoning (through FOL at least).

Now there is another "more technical" way to demonstrate the model-relativity
connoted in (3), and I did allude to this way a couple of times in the past.
Basically, if we consider a FOL system sPA ("super" PA) whose languages
contains *infinite* symbols:

L(0, S,+,*,<, S',+',*',<', S'',+'',*'',<'', S''',+''',*''',<''', ...)

In other words, besides symbol 0, the rest would be grouped together and
each group, when coupled with 0, would form a language we could use to
formalize a theory we'd name "PA". [This alone would signify the relativity
of "PA" and associated models - and (3). Wouldn't it?]

The (infinite number of) axioms of sPA would be the union of those individual
axioms per each ("PA") group mentioned above.

Now let's examine the anatomy of a model M of a general FOL formal system T.
In a nutshell, the major components of M are:

c1: a set S of individuals of which certain n-ary relations would exist.
c2: a collection of n-ary relations, each of which would correspond to an
n-ary symbol of the language.
c3: collection of (subjective) interpretations, each of which would predicate
a theorem-formula as true.

Now MoeBlee suggested above:

"Anyone may state definitions and explicate a discussion on the basis
of those definitions."

So in this context here and now, let's call S a "structure" (and temporarily
forget if some text books reserve this word for something else). Then it's
not hard to see that the relativity of a model M would come from component c3!
For example, if the formula is "a < b", S = {a',b'}, and the n-ary relation
is {(a',b')}, I still could at my subjective willingness interpret (or predicate)
"a < b" as false while you or any other as true. And so relative to whose
predicating or interpreting, M would be or *not* be a model of say T = {a < b},
or for that matter of T = {~(a < b)}.

Now back to sPA, let S be the structure (i.e. the _set_) of individuals of a
model of the integers (i.e. not the natural numbers). The long and the short
of it is out of S, we know there exist _uncountably many_ "successor" functions
S()'s, hence uncountably many n-aries "addition", "multiplication", and "less-than".
Put if differently, not only S is a structure for sPA, it would be the very same
structure for _uncountably many_ models of each "PA" theory (written in L(sPA)).
But it's not hard to demonstrate that due to the subjective interpretation in c3,
if one interpret S as a model of a "PA", others might disagree and (re)interpret S
as *not* a model this "PA".

Of course when we talk about "PA" we typically talk about it outside the context
of this orangutan sPA system. But it should not matter! Given *any* L(PA), one could
consider it as part of L(sPA). And given a model - over a structure S - of *any*
"PA" theory one could interpret this structure S as a non-model of this "PA".

In summary, a model is always (at least) *relative* to:

- which exact theory in what exact language that's under consideration
- which exact *subjective interpretation* (which would make a formula true).

I know it's a bit long way to explain all this, and I don't think "typical" text
books would care to give a discussion. But unless one could come up with some
credible counter arguments, I'd think we'd have no choice but accept the
relativity nature of reasoning in general.

.

Relevant Pages

  • Re: My talk about Godel to the post-grads.
    ... I didn't change any context. ... not have communicated well the idea of relativity to you. ... since the very word is absolute. ... This is not what I would call subjectivity. ...
    (sci.logic)
  • Re: Id like to understand this better.
    ... evident lack of any genuine content in his response to me, ... :::Relativity is related to the gravitational potential within the context ... without any need for him to logically argue towards such a categorization. ...
    (sci.physics.relativity)
  • Re: Torkel Franzen on truth
    ... context that we choose. ... consistent" have a determinate truth value; ... not an absolute one. ... I might add though there are 2 kinds of contexts where relativity of mathematical ...
    (sci.logic)
  • Re: relativistic thermodynamics (temperature in inertial frames)
    ... the context of Special Relativity in a much more elementary analysis. ... monograph on the relativistic classical dynamics. ...
    (sci.physics.relativity)
  • Re: what follows from denying an axiom
    ... Chris Menzel is the first person I have ever heard say "second-order ... language purely syntactically. ... Formal system has no semantics. ... to the resultant perception-phenomenon known as relativity, ...
    (sci.logic)