A confusion about axioms and models

[microsansfil]

looks like a true syntactic statement to me. Am I missing something?

How do you establish the proposal truth table (in the language of arithmetic) ?

$$\forall x \forall y \forall z \forall n (n \, > \, 2 \,\Rightarrow \, (x^n \, +\, y^n \, \neq \, z^n))$$

(This is the great Fermat theorem, proved by A. Wiles, a few years ago.)

Patrick

 

[WWGD]

That doesn't sound right to me: axioms are the statements we start with in a given theory, while deduction rules are ways to get new statements from old ones.

.

I think you're right, I don't know where I got this name from. My bad.

 

[atyy]

Both semantic and syntax are expressed as a collection of claims. Can a machine distinguish between semantics and syntax?

Would you accept rephrasing this question as: Can a version of the Löwenheim-Skolem Theorem be formally proved within ZFC (in which case a machine should be able to prove the LST which is about the relationship between syntax and semantics)?

Kunen http://logic.wikischolars.columbia.edu/file/view/Kunen,+K.+(1980).+Set+Theory.pdf p143 Appendix 3 talks about formalizing model theory in ZF.

With a little googling, I found https://www.cl.cam.ac.uk/~lp15/papers/Formath/Goedel-ar.pdf which seems to describe a mechanized proof of Goedel's incompleteness theorem, including the semantic version (ie. in which the Goedel statement is true). The underlying programme seems to be something called Isabelle https://isabelle.in.tum.de/overview.html, which seems to be an implementation of higher-order logic.

With even more googling I found http://www.concrete-semantics.org.

 

[Demystifier]

Another non-technical and very illuminating text on related issues
http://lesswrong.com/lw/93q/completeness_incompleteness_and_what_it_all_means/

 

[A. Neumaier]

Thank you all, but my questions are still not answered explicitly and I am still confused. So let me rephrase my questions.
- The additional statements which define a model uniquely, are they axioms or not?

Note that the notion of axioms is relative to a concept. There are axioms for set theory, for projective geometry, for group theory, for natural numbers, etc., and all of them are different and mean different things.

In modern mathematics, the statements one assumes as the basis of a mathematical theory are called the axioms for that theory. Everyone making a new theory or a variation of an old one can choose the axioms. Even for the same disciplines. Thus an axiom is something very relative - except among those who agreed on a particular axiom system for their discipline.

Different books on group theory start with different but equivalent axiom systems. This means that each axiom in one of the books but not the other is derivable as a theorem in the latter, and conversely. If you add to the axiom for groups the additional axiom of commutativity, the resulting axiom system defines a different concept than a group, namely that of an abelian group. This is typical for the Bourbaki approach to mathematics.

Different books on set theory start with axioms that are not necessarily equivalent. This indicates that there are several flavors of such a theory with a slightly different meaning, all covering in their intersection a lot of common ground. For example, you can do set theory in ZF (Zermelo-Frankel), ZFC (ZF with the axiom of choice), or ZFGC (ZF+global choice, Bourbaki's starting point). The three theories are not equivalent, but every axiom in ZF is also an axiom of ZFC, which has as additional axiom the ''axiom of choice'', which is not an axiom of ZF. ZFGC has a stronger form of this axiom, the existence of a choice operator that selects a distinguished choice from each nonempty set. These axiom systems define different (inequivalent) set theories. (For simplicity I only discuss these thee; there are also axioms for set theory that differ essentially from ZF!)

As a consequence, every theorem of ZF is a theorem of ZFC and of ZFGC, and every theorem of ZFC is one of ZFGC - with proper inclusion in each case. Correspondingly, every model of ZFGC is a model of ZFC and of ZF, and every model of ZFC is a model of ZF; again with proper inclusions.

Models are never unique; they can be unique only up to isomorphism (relabeling of the objects in it). Even that is possible only in second-order logic, and then only for special concepts such as the natural number or the real numbers. Or for extremely specific objects such as the cyclic group of order 5 or the monster. But in the latter cases one talks of characterizing properties rather than axioms...

 

[Demystifier]

But this is a string without meaning until you assign a specific one to it. I think you are assuming the standard meaning of "x and y implies x" , but there are many other possible meanings/models to be assigned to the string .

Consider the following string:
"Logic is tricky."
This string has some meaning to me, and probably to you to. But does it have the same meaning to me as it does to you? Probably not exactly the same. And to someone who does not speak english, it dos not have any meaning at all. Does it mean that meaning only exists in the human mind?