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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?

MaroonOwl
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    Yes, it would be considered mathematics. But good luck finding a new set of consistent axioms! – David G. Stork Nov 22 '21 at 02:09
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    Your example of $2+2=3$ is unique because it is not only false in the standard integers $\mathbb{Z}$ but also in $\mathbb{Z}n$ for any $n\geq 2$. As a more interesting example, consider the statement $2+2=1$. This statement is still false in $\mathbb{Z}$, but it is true in $\mathbb{Z}{3}$, that is, the integers with the axiom that $3=0$. – olkomat Nov 22 '21 at 02:25
  • If you’re willing to ditch the axiom of choice and classical logic, you can add any one of the following axioms to normal set theory: (1) all functions $\mathbb{R} \to \mathbb{R}$ are continuous, (2) all functions $\mathbb{N} \to \mathbb{N}$ are computable, (3) there is an infinite sequence of sets $S_1, S_2, \ldots$ of real numbers, each of which has at least one element, but there is no sequence of real numbers $x_1, x_2, \ldots$ such that $x_1 \in S_1, x_2 \in S_2, \ldots$, (4) $\mathbb{R}$ is the countable union of countable sets. There are many other “bizarre” possibilities. – Mark Saving Nov 22 '21 at 03:14
  • From the point of view of formalism, mathematics is just a boring game of printing new string of symbols from old ones starting from so-called axioms, following certain rules such as syllogism. You can just program your own printer in whatever way you'd like to print whatever meaningless sentences you can think of. – Just a user Nov 22 '21 at 03:19

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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.

olkomat
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