A confusion about axioms and models

In summary, there is a distinction between logical axioms and non-logical axioms in mathematics. Logical axioms are statements that are taken to be true within a system of logic, while non-logical axioms are substantive assertions about the elements of a specific mathematical theory. The use of the term "axiom" in mathematics can be loose and inconsistent, and can refer to any claim that seems reasonable but cannot be proved or disproved. Different logical systems can be distinguished by the axioms they use, and additional claims may be called laws, assumptions, definitions, or other terms depending on the context.
  • #1
Demystifier
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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?
 
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  • #2
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).
 
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  • #3
Samy_A said:
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.
 
  • #4
fresh_42 said:
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 :angel:) 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
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
Samy_A said:
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.
 
  • #7
Demystifier said:
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."
 
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  • #8
Samy_A said:
Nothing wrong with sounding Hilbertian.
It's pre-Goedel. It just sounds better. :sorry:
 
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  • #9
fresh_42 said:
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.
 
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  • #10
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'.
 
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  • #11
fresh_42 said:
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. :oldsmile:
fresh_42 said:
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.
 
  • #12
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
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.
 
  • #14
Demystifier said:
- 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.
 
  • #15
@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
Demystifier said:
@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.
 
  • #17
@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
Demystifier said:
@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.
 
  • #19
micromass said:
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"?
 
  • #20
Demystifier said:
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.
 
  • #21
micromass said:
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?
 
  • #22
Demystifier said:
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).
 
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  • #23
micromass said:
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?
 
  • #24
Demystifier said:
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.
 
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  • #25
Thanks @micromass ! I think I finally get it. :woot:

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?
 
  • #26
micromass said:
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:
Demystifier said:
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?
 
  • #27
Demystifier said:
Thanks @micromass ! I think I finally get it. :woot:

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!
 
  • #28
Demystifier said:
Thanks @micromass ! I think I finally get it. :woot:
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?
 
  • #29
Samy_A said:
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.
 
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  • #30
Demystifier said:
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.
 
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  • #31
micromass said:
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?
 
  • #32
Demystifier said:
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.
 
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  • #34
fresh_42 said:
@micromass Thank you for the enlightening discussion.
From me too, though the conclusion is rather underwhelming (not that micromass is to blame for that).
 
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  • #35
micromass said:
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?
 
<h2>1. What are axioms and models in scientific research?</h2><p>Axioms are statements or principles that are accepted as true without needing to be proven. Models are simplified representations of complex systems or phenomena used to make predictions and understand relationships.</p><h2>2. How do axioms and models relate to each other?</h2><p>Axioms serve as the foundation for creating models, as they provide the starting assumptions or principles for understanding a particular system or phenomenon. Models then use these axioms to make predictions and test their validity.</p><h2>3. Can axioms and models change over time?</h2><p>Yes, as scientific knowledge and understanding evolves, axioms and models may also change. New evidence or discoveries may require adjustments to previously accepted axioms or models in order to better explain and predict phenomena.</p><h2>4. How do scientists determine which axioms and models to use?</h2><p>Scientists use a combination of evidence, logic, and peer review to determine which axioms and models are most accurate and useful for explaining and predicting phenomena. They may also consider the simplicity and consistency of a particular model.</p><h2>5. Are axioms and models used in all areas of science?</h2><p>Yes, axioms and models are used in all areas of science, from physics and biology to social sciences and economics. They are essential tools for understanding and making sense of the natural world and human behavior.</p>

1. What are axioms and models in scientific research?

Axioms are statements or principles that are accepted as true without needing to be proven. Models are simplified representations of complex systems or phenomena used to make predictions and understand relationships.

2. How do axioms and models relate to each other?

Axioms serve as the foundation for creating models, as they provide the starting assumptions or principles for understanding a particular system or phenomenon. Models then use these axioms to make predictions and test their validity.

3. Can axioms and models change over time?

Yes, as scientific knowledge and understanding evolves, axioms and models may also change. New evidence or discoveries may require adjustments to previously accepted axioms or models in order to better explain and predict phenomena.

4. How do scientists determine which axioms and models to use?

Scientists use a combination of evidence, logic, and peer review to determine which axioms and models are most accurate and useful for explaining and predicting phenomena. They may also consider the simplicity and consistency of a particular model.

5. Are axioms and models used in all areas of science?

Yes, axioms and models are used in all areas of science, from physics and biology to social sciences and economics. They are essential tools for understanding and making sense of the natural world and human behavior.

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