Hey /sci/, so I've been wondering, since axioms by definition

>>8280248

You can always define some addition operator, say +', so that 2 +' 2 = 5.

However, to make this operator interesting and analyzable, you would need to define it for all real or complex numbers and see what properties it has. Depending on how you do this, you can figure out whether or not you can define some form of Euclidean geometry with these numbers, etc.

>>8280254
So essentially, you could do this but it wouldn't be very interesting because it probably wouldn't be consistent?

>>8280263

Maybe it would be consistent; I invite you to make the attempt. The requirement that 2 +' 2 = 5 is not a stringent one, it just depends on how you extend that feature to everything else. A positive indicator is if your new definition of addition and multiplication, acting on the real numbers, forms a vector space.

>>8280273

hell the result of using the +' operator could just always be 5

>>8280248

yes, and we do.

However not all axioms work. For example if we say that sqrt(-1) is i, we get useful results but if we say that 1/0 = j we get no useful results out of it.

2+2=4 isn't an axiom though. You can derive/prove it from a set of axioms (such as the Peano axioms). It would be really problematic if every addition of every pair of natural numbers had to be defined axiomatically, we'd have to have an infinite list of axioms just to add two numbers.

>>8280248
>couldn't we potentially base a mathematics around axioms like 2 + 2 = 5?
of course. you can also go with 2 + 2 = 1 (in a modulo 3 class)

>>8280299
So then how do we get the primitive concept of "2" if not by defining it as "1+1=2"

>>8280775
Skipping all the set theory bullshit: natural numbers are defined by the successor function and zero. That is, S(S(0)) = S(1) = 2, where S(x) takes a natural x and returns x+1.

>>8282282
this, though some people prefer to start with 1