I A confusion about axioms and models

[Demystifier]

Suppose that I have a set of axioms in first-order logic. And suppose that I have several inequivalent models for this set of axioms. And suppose that I want to choose one specific model. To choose it, I need to make some additional claims which specify my model uniquely.

My question is the following: What kind of claims these additional claims are? Are they some additional axioms? Or are they claims which are not classified as axioms? If they are not classified as axioms, what property do they have/lack so that they cannot be classified as axioms?

If the question looks too abstract, let me consider an example. Suppose that I start from axioms of group theory. There are many different groups satisfying these axioms. So I choose some specific group, say SO(3), defined by some claims which define that group. Are these claims also axioms? If not, then what property do they have/lack so that they cannot be called axioms?

Could it be that my confusion stems from the fact that the word "axiom" in logic has more narrow meaning than the word "axiom" in the rest of mathematics?

 

[Samy_A]

Interesting question. I wasn't aware until now that I'm confused about this too.

Your last sentence is probably the answer.

There are four axioms of Group theory. But when you claim that SO(3) is a group, you can prove that it satisfies the four axioms. Nothing here is assumed as being true. The axioms of Group theory act more like a definition. Contrast this with the axiom of Choice: you can do Mathematics with it or without it, but you never prove it.

But I also would like to know if there is some formal definition of "axiom" in Mathematics outside of logic and ZF(C).

 

[fresh_42]

But I also would like to know if there is some formal definition of "axiom" in Mathematics outside of logic and ZF(C).

How should this be possible? The Wiki entries on this issue aren't so bad.

"As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A andB) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). "

I'm not happy with term 'non-logical axiom' which you called 'acts like a definition'. To me those are simply a list of assumptions for the following.
Axioms in mathematics belong to logic systems and we usually constrain ourselves to a predicate logic. Euklid's parallel axiom or as you said Zorn's Lemma are axioms. And to me, only those kinds. To distinguish between the 'models' in the OP I would demand to apply distinguishing axioms and call the models theories.

I know that sounds very Hilbertian.

 

[Samy_A]

How should this be possible? The Wiki entries on this issue aren't so bad.

"As used in mathematics, the term axiom is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define (e.g., (A andB) implies A), while non-logical axioms (e.g., a + b = b + a) are actually substantive assertions about the elements of the domain of a specific mathematical theory (such as arithmetic). "

I'm not happy with term 'non-logical axiom' which you called 'acts like a definition'. To me those are simply a list of assumptions for the following.
Axioms in mathematics belong to logic systems and we usually constrain ourselves to a predicate logic. Euklid's parallel axiom or as you said Zorn's Lemma are axioms. And to me, only those kinds. To distinguish between the 'models' in the OP I would demand to apply distinguishing axioms and call the models theories.

I know that sounds very Hilbertian.

Nothing wrong with sounding Hilbertian.

But I think we agree. The use of the term axiom in "usual" (I won't dare write real ) Mathematics seems rather loose and inconsistent. Why are the defining properties of a group "axioms", and the defining properties of a norm "properties"?

I don't like the term "non-logical axiom" either.

 

[Demystifier]

Fresh_42, there seems to exist a much wider notion of "axiom" than you seem to imply. Sometimes, by "axiom" one means any claim which seems reasonable but can be neither proved nor disproved. For instance, the claim
"The Godel sentence (This sentence cannot be proved) is true."
can be regarded as an axiom.

 

[fresh_42]

Why are the defining properties of a group "axioms", and the defining properties of a norm "properties"?

I assume we could formulate everything in logical terms but we couldn't read it anymore.
On another Wiki page is written:

"The example $\forall v_0 \forall v_1 +v_0v_1 ≡ +v_1v_0$ as symbolization of the commutative law in $L_I^{\{0,+,-,≤\}}$ demonstrates that the resulting expressions are often hard to read. Therefore the classical writing $\forall x,y : x+y = y+x$ is used. However, this is not an expression in $L_I^{\{0,+,-,≤\}}$ but merely a message about the original expression in another language, the meta-language in which mathematicians talk about $L_I^{\{0,+,-,≤\}}$"

Interesting here is their distinction between axiom and law! However, they call the laws of their example of Abelian, ordered groups Axioms. Maybe Wiki isn't so good as I first thought. And maybe I'm so happy with my ZFC that I don't like laws outside of it to be called axioms. Nevertheless we started within logic systems and therefore there is no need to talk about different fields. So the remaining question keeps: Differ logical systems alone by different axiomatic systems or can there be additional claims (OP's term) beside to distinguish them?
I find the answer should be 'axioms only' as part of a logical construction. Whether we call additional claims such or law or assumption or definition or whatever is a discussion about the meta-language, i.e. within meta-meta-language which IMO is to blame for confusion.

 

[fresh_42]

Fresh_42, there seems to exist a much wider notion of "axiom" than you seem to imply. Sometimes, by "axiom" one means any claim which seems reasonable but can be neither proved nor disproved. For instance, the claim
"The Godel sentence (This sentence cannot be proved) is true."
can be regarded as an axiom.

But wasn't our way out of the dilemma to establish meta-systems?

Edit: "Sometimes, by "axiom" one means any claim which can be neither proved nor disproved." is what I think is an axiom. Only that I would say "... cannot be derived from previous axioms."

 

[fresh_42]

Nothing wrong with sounding Hilbertian.

It's pre-Goedel. It just sounds better.

 

[Samy_A]

I find the answer should be 'axioms only' as part of a logical construction. Whether we call additional claims such or law or assumption or definition or whatever is a discussion about the meta-language, i.e. within meta-meta-language which IMO is to blame for confusion.

Agreed.

After taking a non-scientific sample of the literature about Groups all I can see is that some confusion indeed reigns.

My good old algebra book (Algebra from Serge Lang) doesn't talk about axioms. Monoids are defined as sets with an associative law of composition and with a unit element. A group is defined as a monoid in which each element has an inverse.

http://mathworld.wolfram.com/Group.html defines groups using the term property.

Taking an online resource at random (first hit when searching for Group theory pdf), I found these lecture notes from J.S. Milne. The term axiom first appears three times in a nice quote at the start of chapter 1:

The axioms for a group are short and natural... Yet somehow hidden behind these axioms is the monster simple group, a huge and extraordinary mathematical object, which appears to rely on numerous bizarre coincidences to exist. The axioms for groups give no obvious hint that anything like this exists.
Richard Borcherds, in Mathematicians 2009..

The text defines a group as a set with a binary operation satisfying three conditions (page 7).
Then, in discussing finite groups, these conditions are called axioms (page 12).

Wikipedia talks about group axioms, and gives Herstein as reference.
In Topics in Algebra, Herstein defines a group in the same way as Lang and Milne, listing the properties of the binary operation (page 28). In an example on page 32 these properties are then called "the requisite four axioms which define a group".

I probably could do the same exercise with other definitions.

I never gave this question any thought until today, and clearly neither did the two excellent authors cited above. The term axiom is used informally and inconsistently in the meta-language.

 

[fresh_42]

As you've mentioned Serge Lang. I guess Bourbaki is the best we can get in a discussion about formalism!

EDIT: Your "My good old algebra book ..." drove me to my bookshelf. I have a German translation of A.G.Kurosh's Group Theory which is quite the opposite of a Bourbaki-style written book. You can hardly find symbols and formulas. I've also looked up the definition of a group by B.L.van der Waerden's algebra textbook. Both are speaking of conditions or requirements or laws at best. Kurosh not without mentioning counterexamples to each condition. They both avoid the term 'axiom'.

 

[Samy_A]

As you've mentioned Serge Lang. I guess Bourbaki is the best we can get in a discussion about formalism!

I didn't know he was part of Bourbaki.
But looking at the book, I could have guessed.

EDIT: Your "My good old algebra book ..." drove me to my bookshelf. I have a German translation of A.G.Kurosh's Group Theory which is quite the opposite of a Bourbaki-style written book. You can hardly find symbols and formulas. I've also looked up the definition of a group by B.L.van der Waerden's algebra textbook. Both are speaking of conditions or requirements or laws at best. Kurosh not without mentioning counterexamples to each condition. They both avoid the term 'axiom'.

I prefer that approach to the Wikipedia one.

 

[Demystifier]

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?
- If they are not, can it be seen from the syntax of the statements?
- If they are, how can it be compatible with the Lowenheim-Skolem theorem, which says that no system of first-order axioms can uniquely determine an infinite model?

 

[fresh_42]

I'm not sure whether this helps nor if I understood it correctly. I find it interesting anyway.
(Source: https://de.wikipedia.org/wiki/Satz_von_Löwenheim-Skolem#Das_Skolem-Paradoxon; Translation by me.)

"Taken ZF is free of contradictions, every finite axiomatic system in ZF has a countable model (Löwenheim-Skolem).
On the other hand you can give a finite axiomatic system $ψ$, such that the existence of an uncountable set can be derived.

This contradiction can be resolved by clarifying what countable with respect to a model means:
Let $M$ be a system derived from $ψ, A ∈ M$ an uncountable set within the model of $M$.
Then there is no surjection $f: N → A$ within this model.

The set $N$ denotes the set of 'natural numbers' constructed in the model of $M$.
This does not mean the set $M$ itself is uncountable from the meta-language's point of view." (*)

*) I've tried to stay close to the original. So it might be the case that some expressions are a bit unlucky.

Remark: They mention that Skolem's generalization of Löwenthal's theorem needs the Axiom of Choice. I don't know why they've written ZF in the remark on the paradox instead of ZFC.

 

[micromass]

- If they are, how can it be compatible with the Lowenheim-Skolem theorem, which says that no system of first-order axioms can uniquely determine an infinite model?

In first order theory! For example, I can very easily make some axioms that defines $\mathbb{R}$ uniquely (this is called categorical). Any analysis book will do this. But this won`t establish a first order theory.

What is an axiom? Exactly any collection of statements that you wish to take as axioms. If those statements are first-order statements, then there won't be a unique model, otherwise there might be.

 

[Demystifier]

@micromass, if higher-order theory is necessary for unique definition of mathematical models, then why most textbooks on logic say almost nothing about higher-order logic and claim that first-order logic is sufficient for most of mathematics?

 

[micromass]

@micromass, if higher-order theory is necessary for unique definition of mathematical models, then why most textbooks on logic say almost nothing about higher-order logic and claim that first-order logic is sufficient for most of mathematics?

My guess would be that is because set theory can be defined as a first-order model. This implies there is no unique model of set theory of course, but still most of mathematics can be done in a model of set theory.
A lot of other structures like groups can be described with first-order logic too. But for example, a definition of the real numbers cannot be given. In some sense, a categorical property is boring, since we would then describe only one object with axioms. It`s much more fun to describe a lot of different objects with axioms (like groups, or set theories). So I`d say that is the point of logic.

So if you say that most of mathematics can be described as first order logic, then that really needs an explanation.

 

[Demystifier]

@micromass, are you saying that textbooks on logic which say that "first-order logic is sufficient for most of mathematics" are, strictly speaking, wrong or at least imprecise?

If this is what you are saying, that would make sense to me. But it would also be very disappointing, because logicians are the last ones from whom I would expect such wrong or imprecise statements.

Anyway, if you ask logicians why do they prefer first-order logic, they will not tell you that this is because there is more fun without categoricity. They will rather tell you that it is because of the compactness theorem.

 

[micromass]

@micromass, are you saying that textbooks on logic which say that "first-order logic is sufficient for most of mathematics" are, strictly speaking, wrong?

The statement is meaningless to me without any further explanation. If the statement is that I can encode over 90% of all mathematics in first-order statements, then that is correct since ZFC is a first-order theory. See http://us.metamath.org/ for a very huge list of theorems encoded in first-order statements.

 

[Demystifier]

If the statement is that I can encode over 90% of all mathematics in first-order statements, then that is correct since ZFC is a first-order theory.

That sounds strange to me. If you cannot define real numbers, then I would think that you cannot encode any statement about a whole branch of mathematics such as analysis. How can you encode 90% of all mathematics without analysis? Or am I taking a wrong measure on the set of "all mathematics"?

 

[micromass]

That sounds strange to me. If you cannot define real numbers, then I would think that you cannot encode any statement about a whole branch of mathematics such as analysis. How can you encode 90% of all mathematics without analysis? Or am I taking a wrong measure on the set of "all mathematics"?

I can perfectly define the real numbers in ZFC set theory by first order statements. The first order statements are set theoretic in nature.
What I cannot do is give axioms of real numbers outside a set theory such as ZFC. I can do this with groups, but not with a categorical structure like $\mathbb{R}$.

The issue is that to encode something in mathematics, one usually needs to take an entire detour along set theory and ZFC (which can not even be shown to be consistent!). But if we do that, then we can encode it in first-order theory. We cannot avoid the use of ZFC most of the time however.

 

[Demystifier]

I can perfectly define the real numbers in ZFC set theory by first order statements.

This turns me back to my initial confusion. ZFC contains models different from real numbers. So I must make some additional statements which tell me that the set I am talking about is the set or real numbers. But these statements cannot be first-order axioms, due to the Lowenheim-Skolem theorem. Yet, you say that these statements are first-order statements. What am I missing?

 

[micromass]

This turns me back to my initial confusion. ZFC contains models different from real numbers.

No, I can define a unique model of $\mathbb{R}$ within ZFC (for example, I can define $\mathbb{R}$ as the collection of Dedekind cuts). I can state and prove categoricalness of $\mathbb{R}$ within ZFC. The issue is that there are many inequivalent models of ZFC. But once I choose a particular model of it (which cannot be done with first-order logic), then I have a unique model of the reals.

So I must make some additional statements which tell me that the set I am talking about is the set or real numbers.

True. And this can be formalized as first-order statements within ZFC, but not outside of ZFC.

So are the reals unique? Once we choose a model of ZFC, then yes, we have a unique model of the reals. But if we only work with the first-order ZFC axioms, then I cannot describe a unique model of ZFC. Rather, I have many models of ZFC, and thus also many models of the reals (one for each set theory).

 

[Demystifier]

No, I can define a unique model of $\mathbb{R}$ within ZFC (for example, I can define $\mathbb{R}$ as the collection of Dedekind cuts). I can state and prove categoricalness of $\mathbb{R}$ within ZFC. The issue is that there are many inequivalent models of ZFC. But once I choose a particular model of it (which cannot be done with first-order logic), then I have a unique model of the reals.

True. And this can be formalized as first-order statements within ZFC, but not outside of ZFC.

So are the reals unique? Once we choose a model of ZFC, then yes, we have a unique model of the reals. But if we only work with the first-order ZFC axioms, then I cannot describe a unique model of ZFC. Rather, I have many models of ZFC, and thus also many models of the reals (one for each set theory).

Hm, it seems to me that I am starting to see the source of my confusion. So far I was thinking like that:
ZFC has many models; one model is the set of natural numbers, another model is the set of real numbers, yet another model is the set of complex numbers, etc ...

Now you are telling me that it was all wrong and that I have to modify my picture of all that. So let me try to explain my new picture of all that, after which you can tell me if that is correct.

ZFC defines the class of all sets. (Not the set of all sets, to avoid the Russell antinomy.) In this class we have the set of natural numbers, the set of real numbers, etc. For example, real numbers can be defined as collection of Dedekind cuts, and the statement "set X is collection of Dedekind cuts" is a first order statement.

Is that correct?

 

[micromass]

ZFC has many models; one model is the set of natural numbers, another model is the set of real numbers, yet another model is the set of complex numbers, etc ...

I don`t see why you would say that the real numbers form a model of ZFC?? If that`s what you`re asking, then yes this was definitely wrong.

Now you are telling me that it was all wrong and that I have to modify my picture of all that. So let me try to explain my new picture of all that, after which you can tell me if that is correct.

ZFC defines the class of all sets. (Not the set of all sets, to avoid the Russell antinomy.) In this class we have the set of natural numbers, the set of real numbers, etc. For example, real numbers can be defined as collection of Dedekind cuts, and the statement "set X is collection of Dedekind cuts" is a first order statement.

That is right. And to iterate what I said; you cannot find a unique model of the reals outside some other structure where-in it is embedded. So you might be able to find a model of the reals in set theory, or even in geometry. But not outside and independent of these structures. Not while using first-order logic that is.

 

[Demystifier]

Thanks @micromass ! I think I finally get it.

And concerning different models of reals within ZFC, in one model we have $2^{\aleph_0}=\aleph_1$, in another model we have $2^{\aleph_0}=\aleph_2$, etc. Right?

 

[Samy_A]

What is an axiom? Exactly any collection of statements that you wish to take as axioms. If those statements are first-order statements, then there won't be a unique model, otherwise there might be.

Does this mean that the correct answer to the question in the first post:

If the question looks too abstract, let me consider an example. Suppose that I start from axioms of group theory. There are many different groups satisfying these axioms. So I choose some specific group, say SO(3), defined by some claims which define that group. Are these claims also axioms? If not, then what property do they have/lack so that they cannot be called axioms?

is yes, if Demystifier likes to name these claims axioms?

 

[micromass]

Thanks @micromass ! I think I finally get it.

And concerning different models of reals within ZFC, in one model we have $2^{\aleph_0}=\aleph_1$, in another model we have $2^{\aleph_0}=\aleph_2$, etc. Right?

Yes. Exactly. And both of those can be taken as a further axiom!

 

[Demystifier]

Thanks @micromass ! I think I finally get it.

Or maybe not yet. Now I have a question which can be thought of as a version of Skolem paradox. If ZFC has a countable model, does it mean that real numbers have a countable model? If so, can one say explicitly what that model is?

 

[micromass]

Does this mean that the correct answer to the question in the first post:
is yes, if Demystifier likes to name these claims axioms?

Yes, I think so. Any set of `defining properties` are axioms to me.

 

[micromass]

Or maybe not yet. Now I have a question which can be thought of as a version of Skolem paradox. If ZFC has a countable model, does it mean that real numbers have a countable model? If so, can one say explicitly what that model is?

Yes, we can say (reasonably) explicit what that model is, but it`s complicated.

But Skolem`s paradox is the problem that countability is inherent to the set theory you`re looking at. Inside a specific set theory, the reals are always uncountable. The problem starts when you start comparing two models of set theories. This is quite technical.

Compare it with the following situation. We have the axioms of group theory. these are a first order theory, and there are many models of this. Fine. Now we choose one model at random to work in. Now consider the statement `This group is commutative`. This is an unprovable statement. We choose to work in a specific model which might or might not be commutative. We don`t know. We can however find a submodel of our specific model which is commutative (the trivial group). This shows that commutativity cannot be disproven for groups. This also shows that commutativity is relative, submodels won`t respect commutativity or lack of it.
This is an awkward way to look at group theory, since it is so easy to describe models of it. But it is a rewarding look since it makes set theory much clearer. Indeed, in set theory we can never describe a specific model. We have only the axioms to work with. We imagine as having chosen one specific model. But there are others out there. Whether your specific model satisfies $2^{\aleph_0} = \aleph_1$ is something you don't know. But you might be able to describe a submodel that satisfies it (and you can). The issue with Skolem`s paradox is that countability is not necessarily preserved under submodels.

 

[Demystifier]

I don`t see why you would say that the real numbers form a model of ZFC?? If that`s what you`re asking, then yes this was definitely wrong.

Additional question: Since real numbers are not a model of ZFC, this means that real numbers do not satisfy some of the axioms of ZFC. Can you say what these un-satisfied axioms are? Obviously, one of them is the existence of the empty set (because empty set is not a real number), but what are the others?

 

[micromass]

Additional question: Since real numbers are not a model of ZFC, this means that real numbers do not satisfy some of the axioms of ZFC. Can you say what these un-satisfied axioms are? Obviously, one of them is the existence of the empty set (because empty set is not a real number), but what are the others?

The entire language doesn`t match. ZFC is a model of first-order logic with two relations: $in$ and $=$.
The real numbers can be modeled as a logic with as relations $=$, $<$ and with as operations $+$ and $\cdot$.
So the two are incomparable.

 

[fresh_42]

@micromass Thank you for the enlightening discussion.

 

[Samy_A]

@micromass Thank you for the enlightening discussion.

From me too, though the conclusion is rather underwhelming (not that micromass is to blame for that).

 

[Demystifier]

We have the axioms of group theory. these are a first order theory, and there are many models of this. Fine. Now we choose one model at random to work in. Now consider the statement `This group is commutative`. This is an unprovable statement. We choose to work in a specific model which might or might not be commutative. We don`t know. We can however find a submodel of our specific model which is commutative (the trivial group). This shows that commutativity cannot be disproven for groups. This also shows that commutativity is relative, submodels won`t respect commutativity or lack of it. This is an awkward way to look at group theory, since it is so easy to describe models of it.

But from a standard point of view, it seems rather absolute that group SO(2) is commutative and that SO(3) is not. So what exactly is the difference between the standard and this awkward point of view?

 

[fresh_42]

But from a standard point of view, it seems rather absolute that group SO(2) is commutative and that SO(3) is not. So what exactly is the difference between the standard and this awkward point of view?

The part of the comparison to group theory was the one I had difficulties with, too.
Then rereading it, I interpreted it this way: the starting point was group theory alone, not group theory within ZFC.
Otherwise I had the same question.

 

[micromass]

But from a standard point of view, it seems rather absolute that group SO(2) is commutative and that SO(3) is not. So what exactly is the difference between the standard and this awkward point of view?

That`s the difference between set theory and group theory. In group theory, you can exhibit a very specific model. Then you can prove everything you want about that model. In set theory however, we can never describe a model at all. We can only go by the axioms.
Try to do group theory like this: you know you`re working in some group, meaning that the usual axioms are satisfied. But outside of this, you have no clue about the specifics of the group. You cannot construct groups other than the one you have now. That is the point of view with which we do set theory.

 

[micromass]

The part of the comparison to group theory was the one I had difficulties with, too.
Then rereading it, I interpreted it this way: the starting point was group theory alone, not group theory within ZFC.
Otherwise I had the same question.

I understand it`s difficult. But the analogy between set theory and group theory is a really really good one. It helped me significantly.
Your interpretation is correct by the way. You start only from one single group, for which you know the axioms are valid, and nothing else.

 

[Demystifier]

In set theory however, we can never describe a model at all.

Then why don't they tell us that in the textbooks?

But here is a more interesting question. What if I remove the axiom which claims existence of an infinite set? Would it be possible to explicitly describe a model for such a reduced ZFC?
Or even more interestingly, is it possible to remove some other axiom (while retaining infinite-set axiom), such that again a model can be described explicitly?

 

[micromass]

Then why don't they tell us that in the textbooks?

But here is a more interesting question. What if I remove the axiom which claims existence of an infinite set? Would it be possible to explicitly describe a model for such a reduced ZFC?

Depends on what you mean with describe, but I`d say yes. You can explicitely describe all the sets in such a universe.

Or even more interestingly, is it possible to remove some other axiom (while retaining infinite-set axiom), such that again a model can be described explicitly?

That seems much less likely.

 

[fresh_42]

Wiki states that Goedel's completeness theorem says: each syntactically consistent theory (set of closed formulas without contradiction) has a model. Shouldn't this apply to any reduced ZFC system as long as consistency can be proved?

 

[micromass]

Wiki states that Goedel's completeness theorem says: each syntactically consistent theory (set of closed formulas without contradiction) has a model. Shouldn't this apply to any reduced ZFC system as long as consistency can be proved?

But consistency can`t be proved. Otherwise, yes.

 

[Demystifier]

But consistency can`t be proved.

I know that one can't prove consistency of ZFC or consistency of Peano axioms. But consistency of a weaker system, such as Presburger arithmetic, can be proved
https://en.wikipedia.org/wiki/Presburger_arithmetic#Properties
I think this is what fresh_42 had in mind.

 

[fresh_42]

I thought it answers your question, at least to some extend as it supplies a sufficient condition.

Or even more interestingly, is it possible to remove some other axiom (while retaining infinite-set axiom), such that again a model can be described explicitly?

(To be honest: I haven't thought about whether every reduction of ZFC cannot be proven consistent.)

 

[atyy]

http://www.columbia.edu/~hg17/nonstandard-02-16-04-cls.pdf
[PLAIN]http://lesswrong.com/lw/g0i/standard_and_nonstandard_numbers/[/PLAIN]
http://lesswrong.com/lw/g0i/standard_and_nonstandard_numbers/

 

[atyy]

@micromass, if higher-order theory is necessary for unique definition of mathematical models, then why most textbooks on logic say almost nothing about higher-order logic and claim that first-order logic is sufficient for most of mathematics?

First order logic has the Goedel "Completeness" Theorem, which second order logic does not. The notion of "Completeness" in the "Completeness Theorem" is not the same as that in the "Incompleteness Theorem". Mathematicians have much worse terminology than physicists :P

https://en.wikipedia.org/wiki/Gödel's_completeness_theorem

 

[Samy_A]

Mathematicians have much worse terminology than physicists :P

Now that I have seen how mathematicians use the term "axiom", I have (with great sadness) to somewhat agree.

 

[atyy]

Now that I have seen how mathematicians use the term "axiom", I have (with great sadness) to somewhat agree.

They justify it with computer science https://en.wikipedia.org/wiki/Operator_overloading

 

[stevendaryl]

Suppose that I have a set of axioms in first-order logic. And suppose that I have several inequivalent models for this set of axioms. And suppose that I want to choose one specific model. To choose it, I need to make some additional claims which specify my model uniquely.

My question is the following: What kind of claims these additional claims are? Are they some additional axioms? Or are they claims which are not classified as axioms? If they are not classified as axioms, what property do they have/lack so that they cannot be classified as axioms?

If the question looks too abstract, let me consider an example. Suppose that I start from axioms of group theory. There are many different groups satisfying these axioms. So I choose some specific group, say SO(3), defined by some claims which define that group. Are these claims also axioms? If not, then what property do they have/lack so that they cannot be called axioms?

Could it be that my confusion stems from the fact that the word "axiom" in logic has more narrow meaning than the word "axiom" in the rest of mathematics?

I spent a fair amount of time pretending to do mathematical logic, and I think I can answer this question.

The difference between a theory and a model is that a theory is a collection of mathematical statements, while a model is a mathematical object. Of course, to reason about a mathematical object, you need a characterization of that object, and that requires statements, too. But usually, what in mathematical logic is considered a "model" is some specification of a mathematical structure in a way that is believed to uniquely characterize that structure (or sometimes, up to isomorphism).

Coming up with a model is often along the lines of a construction--a concrete recipe for building that object. Of course, the recipe often involves steps that cannot be carried out by a human being, but would require some kind of godlike being that can do infinitely many things in a finite amount of time.

Maybe it's best to see an example:

To axiomatize the natural numbers, you might give the following (Peano axioms): (I'll use x' to mean the next number after x)

1. x+0 = x
2. x+y' = (x+y)'
3. x*0 = 0
4. x*y' = (x*y) + x
5. 0 \neq x'
6. x' = y' \Rightarrow x=y

There's actually an induction schema, too, but I'll skip that for now.

I think it's obvious that the interpretation of the domain as the natural numbers 0, 1, 2, ..., and the interpretation of x' as x+1, and the interpretation of "+" as ordinary addition, and "*" as ordinary multiplication works. But the question is, do those axioms uniquely specify a mathematical structure? The answer is no. To see this, you can see that the axioms don't rule out "hyperfinite" objects that are infinitely far removed from 0. More concretely, let's add a new constant symbol, \infty, and add the (infinitely many) axioms:

\infty \neq 0
\infty \neq 0&#039;
\infty \neq 0&#039;&#039;
\infty \neq 0&#039;&#039;&#039;
etc.

(Having an infinite number of axioms might seem like cheating, but when they follow a simple pattern, as they usually do with axiom schema's, an infinite number of axioms is no harder to use in theorem-proving from than a finite number)

It's clear that this new theory is not about the (finite) natural numbers, since it has an object, \infty, that is not anyone of the natural numbers. But the original set of axioms didn't rule out the possibility of such an object, so the original axioms didn't uniquely characterize the natural numbers, either.

It turns out that you can't possibly uniquely characterize the natural numbers using only first-order language. That is, using only statements involving +, *, 0, x', etc., you can't axiomatize the truths about the natural numbers.

But we can (allowing godlike operations) construct a model of the natural numbers. Roughly speaking:

Let A be any set, and let f: A \rightarrow A be any unary function on A that is one-to-one, but not onto. Let Z be any element of A that is not in the image of f. If A&#039; is a subset of A, call it "closed" if Z is in A&#039; and whenever x is in A&#039;, so is f(x). Finally, we define N = the set of all x that are in every closed subset of A.

Then N is a model of the natural numbers. It was definitely not obtained by adding more axioms to the Peano axioms. The language used to "construct" N doesn't even mention the original axioms.

A model is definitely not a collection of axioms. It's a mathematical object.

 

[microsansfil]

The difference between a theory and a model is that a theory is a collection of mathematical statements, while a model is a mathematical object.

IMHO, we can also claim : A theory is a set of mathematical formula provable from a set of chosen axioms and the logic used (here first-order logic). Thus It is also a mathematical object. A mathematical syntactic object. A model is use to give a semantic (e.g. an interpretation ) to the theory. The model theory is the semantic of the formal discourse.

For example from the group axioms (syntax) https://proofwiki.org/wiki/Definition:Group_Axioms we can built models (semantic) that we call group which is a "concrete realization" of the axioms. There are many different groups satisfying the group axioms, and they come in all shapes and sizes.

The Gödel's[/PLAIN] completeness theorem give a relationship between theory and model.

Patrick
https://en.wikipedia.org/wiki/First-order_logic