# I A confusion about axioms and models

1. Jan 27, 2016

### 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?

Last edited: Jan 27, 2016
2. Jan 27, 2016

### Samy_A

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).

Last edited: Jan 27, 2016
3. Jan 27, 2016

### Staff: Mentor

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.

4. Jan 27, 2016

### Samy_A

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.

5. Jan 27, 2016

### 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.

6. Jan 27, 2016

### Staff: Mentor

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.

7. Jan 27, 2016

### Staff: Mentor

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."

Last edited: Jan 27, 2016
8. Jan 27, 2016

### Staff: Mentor

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

9. Jan 27, 2016

### Samy_A

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 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.

10. Jan 27, 2016

### Staff: Mentor

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'.

Last edited: Jan 27, 2016
11. Jan 28, 2016

### Samy_A

I didn't know he was part of Bourbaki.
But looking at the book, I could have guessed.
I prefer that approach to the Wikipedia one.

12. Jan 28, 2016

### 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?

13. Jan 28, 2016

### Staff: Mentor

I'm not sure whether this helps nor if I understood it correctly. I find it interesting anyway.

"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.

14. Jan 29, 2016

### micromass

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 wont 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.

15. Jan 29, 2016

### 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?

16. Jan 29, 2016

### micromass

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. Its much more fun to describe a lot of different objects with axioms (like groups, or set theories). So Id 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.

17. Jan 29, 2016

### 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.

18. Jan 29, 2016

### micromass

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.

19. Jan 29, 2016

### Demystifier

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"?

20. Jan 29, 2016

### micromass

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.

21. Jan 29, 2016

### Demystifier

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?

22. Jan 29, 2016

### micromass

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).

23. Jan 29, 2016

### Demystifier

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?

24. Jan 29, 2016

### micromass

I dont see why you would say that the real numbers form a model of ZFC?? If thats what youre asking, then yes this was definitely wrong.

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.

25. Jan 29, 2016

### 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?