Is it possible to create my own system of math with a new set of axioms that may or may not be observable in "the real world"?

Question

If the axioms that we know about are true statements that can not be proven and are the foundation of "standard mathematics", would it still be considered mathematics if I create my own set of axioms then derive theorems from those axioms? Despite that the axioms probably wouldn't be observable in the real world just like how 2 apples + 2 apples =/ 3 apples (and thus wouldn't be applicable to anything) but would that still be a valid system you would consider as mathematics?

Answer 1

So, if I understand your question correctly, then the answer is definitely "yes", without doubt.

There are already existing axiomatic systems which are independently studied but mutually inconsistent. Based on your question, I have no information about your current level of familiarity with mathematical structures. So I will use the easiest examples I can think of:

EXAMPLE A: The additive group of integers, denoted by $(\mathbb{Z},+)$, consisting of the numbers $\cdots, -3, -2, -1, 0, 1, 2, 3, \cdots$:

• AXIOM G1: There is an integer $0$ such that $a+0=a$ for any integer $a$.
• AXIOM G2: For any integer $x$ there is an "opposite" integer, denoted by $(-x)$, such that $x+(-x)=0$.
• AXIOM G3: For any integers $a$, $b$, $c$, we have $(a+b)+c=a+(b+c)$.
• AXIOM AB: For any integers $a$ and $b$, we have $a+b=b+a$.
• AXIOM Z1: There is an integer $1\neq0$.
• AXIOM Z2: The integer $1+1+\cdots+1$ is never equal to $0$.
• AXIOM Z3: Every nonzero integer can be written as either $1+1+\cdots+1$ (if it is a positive integer), or $(-1)+(-1)+\cdots+(-1)$ (if it is a negative integer).

EXAMPLE B: The additive group of integers mod 2, denoted by $(\mathbb{Z}_2,+)$, consisting of the numbers $0$ and $1$:

• AXIOM G1: There is an integer $0$ such that $a+0=a$ for any integer $a$.
• AXIOM G2: For any integer $x$ there is an "opposite" integer, denoted by $(-x)$, such that $x+(-x)=0$.
• AXIOM G3: For any integers $a$, $b$, $c$, we have $(a+b)+c=a+(b+c)$.
• AXIOM AB: For any integers $a$ and $b$, we have $a+b=b+a$.
• AXIOM Z1: There is an integer $1\neq0$.
• AXIOM Z4: $1+1=0$
• AXIOM Z5: Every integer is equal to either $0$ or $1$.

These two axiomatic systems are clearly inconsistent with each other, especially axioms Z2 from example A and Z4 from example B. However, both $\mathbb{Z}$ and $\mathbb{Z}_2$ have immeasurable importance in not just math but also computer science and statistics.

Now, it is possible that you meant more by your question. However, I am answering your question as I understand it. I hope this helps!

EDIT: If you are interested in the study of consistency and what can be concluded from sets of axioms (known as "theories"), then I highly recommend researching the continuum hypothesis. Basically, the following are all true in standard ZFC set theory:

• The size of the set of natural numbers is denoted $\aleph_0$.
• The size of the set of real numbers is denoted $2^{\aleph_0}$.
• It is possible that there exists an infinite set $B$ of size $\beta$ such that $\aleph_0 < \beta < 2^{\aleph_0}$. This is denoted $\neg CH$. "There is a size between countable and continuum."
• It is ALSO possible that there does not exist such a set $B$. This is denoted $CH$. "There is no size between countable and continuum."

In other words, even the axioms you select do not completely determine which statements will be true and false.