What is the purpose of axiomatic systems?

Question

I am a beginner in logic and I am a bit confused on what the purpose of axiomatic systems is.

• Are the axiomatic systems developed to prove all theorems of a given theory. If yes, then does this mean that set of axioms for a given theory are (can be) amended once an statement cannot be proved or disproved using current set of axioms. This seems to be the case with the development of set of (second-order) axioms for real numbers, where to prove theorems like intermediate value theorm (IVT), axiom of completeness has been added. This also seems to be the case of development of axioms for the Euclidean geometry

OR

• Are the axiomatic systems developed to capture our intuitive understanding of a collection of objects and how they work? For example, this seems to be the case with the (second-order) Peano axioms for the Natural numbers.

If the former is correct, then does this mean Peano axioms were developed over a period of time as people were trying to prove theorems about the Natural numbers?

If latter is correct, does this mean that the axioms are chosen in a way that their model fits our intuitive understanding of the structure? In other words, do mathematicians have the model in mind when they are proposing the axioms? If this is the case then all the axiomatic system must be consistent because, they were initially proposed having a model in mind?

For example, the axioms of Natural numbers are chosen in a way that the numbers that we intuitively call Natural numbers are the model of the axioms, or the axioms of real numbers are chosen such that what we intuitively call real numbers ( I guess our intuitive understanding of real numbers comes from its association with straight line. ) becomes the model of the axioms?

However, for the above (refering to the latter) to be true , it seems that the set and the objects of the set should be well defined. For example, the Peano axioms define the objects using the successor function $\mathbf{S}$ and a first element $\mathbf{0}$, which are both undefined, and the induction axiom ensures that the sets that do not correspond to our intuition about the Natural numbers are excluded, e.g, $\mathbb{N}~\cup~\{a,b,c\}$. Similarly, we can define the objects of real numbers as cuts of rational numbers (Dedekind cuts), and then all the sets that are not complete are excluded. The undefined terms for this axiomatic system are the rational numbers.

So the questions are (i) is this (the sets and object need to be defined) a correct observation?, (ii) why is this needed and how is this more formally stated( what are the right terminologies from mathematical logic)?

In summary:

1. What is the purpose of axiomatic system? ( Is it the former or the latter?)
2. If the latter, are axiomatic system proposed having a model in mind?
3. If the latter, why is there a need to define the objects and sets accurately?

Answer 1

My take: in the progression of mathematics, it starts with the latter and eventually needs to address the former for the sake of consistency. That's the key word that answers your questions.

People understood for a long time what the natural numbers are, and how we create them, how we manipulate them, and the limits of them (i.e. negative numbers are not natural numbers).

Axiomatic methods came around for a couple of reasons:

1. When doing proofs, you have to start somewhere and assume some things are true. What's the minimum set of things you need to assume as true before you can prove things?
2. Once you have #1 around, how do you know that it's consistent - i.e. how do you know that you can't prove the statement 0=1 from the axioms you just enumerated?

Model Theory is the study of these things, and Gödel's Completeness Theorem nicely brings it all together by saying

If there's a model for a set of axioms, then the set of axioms is consistent.

So Peano's axioms are shown to be consistent through the natural numbers. There are plenty of other axiomizations for the natural numbers, by the way.

Finally, axiomizations help us understand what can't be proven. For example, check out all of the things that can't be proven in ZFC, which is the standard model for set theory.

Answer 2

Certainly not all mathematicians always have models in mind when proposing an axiomatic system: Quine's "New Foundations", proposed in a 1937 article, have still not been modeled (meaning relative consistency is shown with ZFC).

The answer to your question depends on the axiomatic system. For set theories, the answer is the former. They are used to interpret "all of mathematics" into so as to ensure accountability among all mathematicians and their proofs.

For axiomatic systems defining mathematical structures e.g. "the theory of a group", "the theory of a category", etc. of which particular groups, categories, etc. are models the answer is typically the latter. The set of axioms defining a Quillen model category is written down for the reason of formalizing our intuition about what should be "a homotopy theory."