# How to use one set of concrete axioms to build another (smaller) set of abstract axioms.

Say you have a system where your only axioms are:

1. All foo are 10.
2. All bar are 20.
3. All baz are 30.

Then from these axioms you can see a pattern, so you say that the underlying axiom is this:

1. All $$x$$ are $$n * 10$$ for $$1 \leq n \leq 3$$.

Something like that. The specifics of the actual axioms shouldn't really be a factor to the question. Just making this example up on the spot. The reality is the axioms could be very complicated and distract from the question.

I'm mainly wondering how you do the following...

Notice how the first set of 3 axioms is a lot simpler to follow than the second single axiom. It's just plain and simple. It starts from concrete stuff, no variables or anything tricky. So you start your argument from this simpler and more concrete "axiom system", and use it as a foundation for "defining" your "main" axiom, "All $$x$$ are $$n * 10$$ for $$1 \leq n \leq 3$$.". You don't want to start your reasoning or education from this "main" axiom because it's a little more abstract and harder to understand (and mentally derive) than if you start from the first, simpler, concrete axioms. At the same time, once you realize that the "main axiom" is what it is, then you can build all the rest of your lemmas and other axioms and such off that, as it just makes things easier and facilitates better comprehension for some future concepts.

So in the end, this mathematical system has 1 axiom, the "main axiom", which is "All $$x$$ are $$n * 10$$ for $$1 \leq n \leq 3$$.". But you don't say that at first. At first you say this mathematical system has 3 axioms, which are "All foo are 10. All bar are 20. All baz are 30.". Then you say "these 3 can be simplified down to a standard "main" axiom which is "All $$x$$ are $$n * 10$$ for $$1 \leq n \leq 3$$."".

My question is, what you do given that these 2 sets of axioms both describe the same system, yet you want to say the final set is the "main" one. Wondering if that's okay to do.

In books you typically see axioms listed out 1-10 sort of thing. But there is only one set. Sometimes they may bundle them into axioms of different "types", but this is just for convenience. What I haven't seen is how to handle this situation, where you are incorporating the "education and simplicity of reasoning" element into the mix.

To summarize, I would like to not do this:

Let's begin. Axiom 1:

All $$x$$ are $$n * 10$$ for $$1 \leq n \leq 3$$.

Lemma/theorem/etc...

I would like to do this, make it smoother:

Let's begin. There are 3 concrete axioms which can be simplified down to 1 axiom. We will therefore state that there is only 1 axiom, but we use these 3 axioms as a guide into understanding the 1 main axiom. So the 3 axioms are:

1. All foo are 10.
2. All bar are 20.
3. All baz are 30.

This simplifies to:

All $$x$$ are $$n * 10$$ for $$1 \leq n \leq 3$$.

We call this the main axiom, so really, there is just one axiom instead of these 3.

The question is like, what you would call this sort of thing, and how you would write it out for print.

The 3 axioms are like "helpers", helper axioms, to guide the reader to the "main" axiom. I don't know if there is a term for that or how to deal with it. Maybe there ends up being two systems:

1. The path to understanding. (3 axioms)
2. The state of understanding. (1 axiom)

In reality, there could be 20 or 100 "simple/concrete" axioms in the path to understanding, and then 5 to 10 or 20 let's say of axioms in the state of understanding. So you want to say the system has only 5, 10, or 20 axioms, not 20 or 100, even though you used them along the way while reasoning. Not sure what really to do here.