Re: What is the 1st order formal system known as PA?


MoeBlee wrote:
> Rupert wrote:
> > Yes, but your hypothesis was just that S and S' are models of your
> > first-order theory. (Or if not, you should have stated it more
> > clearly). His hypothesis is that R1 and R2 are complete ordered fields,
> > which is stronger. His statement is correct; yours is not.
>
> Thank you for your generous replies. I think perhaps you've
> misconstrued me on certain key points, so that perhaps you think I'm
> disagreeing with you, though I am not. But perhaps you are correct that
> I did not state clearly enough. My premise is not that S and S' are
> models. My premise is that they are sets in set theory. My S1 and S2
> are exactly Browder's R1 and R2. There's no conflict among you, me, or
> Browder in this regard.
>
> > > What I mean is that set theory is a first order object level theory,
> > > call it Z0. And the first order formal meta theory is set theory, call
> > > it Z1. Z1 is just like Z0 but a level up. Z0 is first order set theory
> > > at the object level and Z1 is first order set theory at the meta-level.
> > > Why is that not allowed?
>
> > It's allowed, but what do you want to do that for? I would have thought
> > your object theory would be the first-order theory you were talking
> > about and your metatheory would be something like set theory.
>
> Yes, my metatheory for any object thoery is set theory. And if set
> theory itself is the object theory, then my metatheory for it is set
> theory (just one level up).
>
> > It can be, but why do you want a metatheory for set theory? Why do you
> > need that?
>
> Because my object level set theory is in a formal first order language.
> To form that language and to talk about the theory, I need a
> metatheory.
>

Yes, but on this occasion you don't need to talk about that theory.

> > No, your object theory is the theory you are discussing, which is the
> > first-order theory you were talking about. Your metatheory is set
> > theory. You can have a metametatheory if you want, but I don't see the
> > point.
>
> I have a metalanguage and a metatheory to be able to state and prove
> sentences about the object theories.
>
> > > Don't forget, I'm not using any second order language or second order
> > > theory. I have first order object language theories, such as Z0
> > > (Zermelo set theory), PA, axiom sets about fields, etc. Now, I see that
> > > the so-called field axioms or so-called axioms for groups, etc. can
> > > also be not axioms, but rather, definitions, in Z0.
>
> > Yes, but you can't define a complete ordered field to be a structure
> > satisfying some first-order theory. There's no first-order theory whose
> > models are precisely the complete ordered fields.
>
> Since you explained that a few posts ago, I've understood that. I think
> we're in accord here. I'm saying that in set theory, you can define
> 'complete ordered field' and any two complete ordered fields are
> isomorphic with one another. On the other hand, if you "transform"
> those definitions from set theory into axioms for a different theory
> (with a different language) from set theory, call it B, then the class
> of models of B is not the class of complete ordered fields.
>
> > > Then in Z0 we show
> > > that any structure (set theory structure, like an algebraic structure;
> > > not a structure for a language) that satisfies the definition is
> > > isomorphic to any other structure that satisfies the definition.
>
> > What definition are you talking about here?
>
> The definition, in set theory, of 'compelete ordered field'.
>
> > For the tenth time: it is *not* true that any two models of your
> > first-order theory are isomorphic. It *is* true that any two models of
> > the standard, second-order, theory of complete ordered fields are
> > isomorphic, but Loewenheim-Skolem doesn't apply there.
>
> I have not said that they are. We are on exactly the same page. And
> what you said about second order fits too. In set theory, any two
> complete ordered fields are isomorphic. And any two models of second
> order theory of compelte ordered fields are isomorphic. But it is not
> the case that any two models of the first order theory B (axioms for an
> ordered field plus the axiom schema for least upper bound) are
> isomorphic and there are even models of B that are not complete ordered
> fields.
>
> > Browder's proof is *not* a proof about models of your first-order
> > theory.
>
> Exactly. I didn't claim that it is. I was indeed offering it in
> CONTRAST with with a proof about models.
>
> > It is a proof about complete ordered fields, which are the same
> > thing as models of a certain second-order theory. The models of your
> > first-order theory are real-closed fields, not all of them are complete
> > ordered fields. It is not true that any two models of your first-order
> > theory are isomorphic. Browder's proof says nothing about this issue,
> > Browder's proof is talking about complete ordered fields.
>
> Yep. That's what I was getting at.
>
> > > But, reverting back from definitions, I also have a separate first
> > > order object level theory, not set theory, but still a first order
> > > theory, that uses the phi[x] style schema. My set theory definition of
> > > a certain kind of structure is "translatable" back to an object
> > > language theory in a different language (as primitives it has + * <,
> > > rather than set theory which has only e, for 'is an element' while + *
> > > < are defined) in which the definitions are not definitions but rather
> > > axioms. Let's call this B. B is a first order object level theory with
> > > the so-called field axioms and the least upper bound principle as an
> > > axiom schema.
>
> > This is not a translation of the definition of a complete ordered
> > field. The first-order axiom schema fails to capture the full power of
> > the second-order principle. The class of structures satisfying this
> > first-order theory is a larger class of structures than the class of
> > complete ordered fields. In fact it is the class of real-closed fields.
>
> That's why I put "translatable" in quotes. And, as I mentioned, I
> understand the point you are making, and it has given me perspective
> and explanation as I needed it.
>
> > Any two complete ordered fields are isomorphic.
> > But the class of complete ordered fields is not the same as the class
> > of structures which are models for B. The latter class is larger.
>
> Yep, that's what I've learnt.
>
> > > The situation is that the object level set theory structures are
> > > isomorphic but the meta level models are not isomorphic. I'm guessing
> > > that this is a reflection of Lowenheim-Skolem.
>
> > It's nothing to do between the distinction between the object level and
> > the meta level. Forget about having a metatheory for set theory. Just
> > work in set theory. Set theory is your metatheory, the first-order
> > theory you were talking about is your object theory.
>
> Here I disagree. First order theories such as PA, B, the theory of
> groups, etc. have set theory as the metatheory. But first order set
> theory itself also gets a metatheory, which is also first order set
> theory (but a level up).

Yes, but I don't see why we need to refer to a metatheory for set
theory on this occasion.

> But I think my speculation about
> Lowenheim-Skolem is incorrect. I should not have written that after you
> had already explained about real closed fields and complete ordered
> fields. I should have seen from that that it's not just a cardinality
> issue.
>
> > In set theory, we can prove that any two complete ordered fields are
> > isomorphic. But we can also prove that it's not the case that any two
> > models of B are isomorphic. Both these theorems are theorems of your
> > metatheory. There's no contradiction because not every model of B is a
> > complete ordered field. Simple.
>
> I agree except one point, which reflects my feeling that set theory
> itself can be an object theory. We can prove the isomorphism of
> complete ordered fields in both object level set theory and in meta
> level set theory. That redundancy is just a consequence of the fact
> that I view set theory as both an object level theory and as a
> metatheory. This doesn't complicate things as much as you might expect,
> since the object level theory and the metatheory are just like one
> another except one is a level up.
>
> > Your object theory is B. Your metatheory is set theory. In the
> > metatheory, all complete ordered fields are isomorphic but not all
> > models of B are isomorphic.
>
> Right. Additionally, in my way of looking at, in the object level set
> theory, all complete ordered fields are isomorphic.
>
> > > Because one is isomorphism of structures
> > > in a theory, and the other is non-isomorphism among models that are
> > > structures for first order languages.
>
> > Whether or not an isomorphism exists between two structures is a
> > question that gets decided in the metatheory.
>
> I tried to make clear which of the two senses of 'structure' I meant.
> One is a tuple, such as an algebraic structure; the other is a function
> from the parameters of a language into a domain and its relations and
> functions. In object level or meta level set theory (let's just say,
> set theory, since the difference doesn't matter here) we prove
> isomorphisms between structures (in the first sense of the word); in
> model theory (which is part of the metatheory, which is set theory a
> level up) we look at structures (in the second sense of the word) as
> some of these structures are models for a set of axioms or not.
>
> > The phenomenon we're
> > observing is just due to the fact that not every model of B is a
> > complete ordered field. It's nothing to do with the distinction between
> > two levels of theory.
>
> I see that I was probably barking up the wrong tree on that point. I
> see what you're saying.
>
> > > Just like, in first order object
> > > level set theory, the set of reals is uncountable, but in first order
> > > meta level model theory (expressible within first order meta level set
> > > theory) there are countable models if there are infinite models.
>
> > You can have a model of set theory such that a set is uncountable in
> > the model but countable "in the real world". And you can have a model
> > of set theory such that a structure is a complete ordered field in the
> > model but not a complete ordered field "in the real world".
>
> I have a problem with ""real world"".

Well, you shouldn't. Saying something holds "in the real world" is just
saying that it holds without any restriction on the quantifiers.

> The way I think of it is that in
> set theory, as a formal theory, we prove a sentence that we READ as
> saying there are uncountable sets; literally we have a proven formula,
> which in our intuitive, informal interepretation, says there are
> uncountable sets. In our metatheory, we see that Lowenheim-Skolem tells
> us that there is a countable and an uncountable model. So there are
> FORMAL interpretations (by means of structures for the language) some
> of which have countable domains and some of which have uncountable
> domains. (One wrinkle is that to prove the existence of these models,
> our meta level set theory has to be a stronger version of our object
> level set theory, right?). But at the formal meta level, we still read
> the formulas with our intuitive, informal interpretaion, which is
> occuring in the informal meta-meta level. So we formalize the meta-meta
> theory, ad infinitum. So instead of relying on a notion of a real world
> (an abstract mathematical real world), I rely on an infinite escalation
> of meta theory. The advantage of positing a real world (an abstract
> mathematical real world) is simplicity; the disadvantage is that it is
> not formal, or for, non-platonists, simply unacceptable.

You can be a formalist instead of a platonist if you want to be. I
don't mind. But you still have to pick which theory you're working in.
When I say "there are models in which there are uncountable sets which
are countable 'in the real world'", I am saying something which can be
given as a perfectly well-defined theorem of set theory. If you're
working in that theory, you shouldn't have any problem with what I say.

> The advantage
> of an infinite escalation of meta theory is that it is always formal
> and mathematical, or can be, each time we reach up to formalize. The
> disadvantage of an infinite escalation of meta theory is that it keeps
> deferring - there may not be an ultimate settlement of certain
> questions that would be settled by "looking directly" at the "real
> mathematical world" just to see what is or is not the case in that
> world. As I understand, Godel talks about the possiblity that these
> questions may be settled by finding axioms (perhaps large cardinal
> axioms) that settle them. But I don't see why he has any hope that the
> axioms would be evidently true in the real mathematical world when the
> sentences that we want to settle for truth or falsehood are themselves
> not evidently true or false. What hope do we have that a large cardinal
> axiom would be evidently true when the continuum hypothesis, which is
> to be settled by a large cardinal axiom, is neither evidently true or
> false?
>
> > But these considerations are irrelevant to what we are discussing.
> > There are models of B which are not complete ordered fields in any
> > (transitive) model of set theory. What we are talking about is nothing
> > to do with exotic models of set theory which look different from the
> > inside than from the outside. We are just talking about the simple fact
> > that not every model of B is a complete ordered field.
>
> Yes, I think I was on the wrong track there.
>
> Thanks again for your posts. I am trying my best to be clear, within
> reasonable time for composing and editing.
>
> MoeBlee

.

Relevant Pages

  • Re: What is the 1st order formal system known as PA?
    ... His hypothesis is that R1 and R2 are complete ordered fields, ... My premise is that they are sets in set theory. ... And the first order formal meta theory is set theory, ... >> at the object level and Z1 is first order set theory at the meta-level. ...
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  • Re: What is the 1st order formal system known as PA?
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  • Re: Interesting (IMO) question about the reals...
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  • Re: Interesting (IMO) question about the reals...
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  • Re: What is the 1st order formal system known as PA?
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