Is set theory possible without the concept of zero? How much of mathematics breaks if we don't have zero?

>>16707264 (OP)
>don't even get me started

>>16707270
Here's another question for you to chew on, a far more problematic one. Let's say we have zero, and the concept of zero so we can have the empty set et al. BUT all operations on zero are undefined. Exponentiation, division in any order, multiplication, addition, subtraction, negation, etc. all are UNDEFINED.

>>16707264 (OP)
Point me to an object called “zero” in ZFC axioms. There isn’t even arithmetic on sets. That requires you to talk about groups and rings, which are their own categories.

>>16707294
How can you have the empty set without the notion of zero?

>>16707318
Ehmmm just have it? Zero is an algebraic object; it is the identity element of addition. There is no addition defined on sets a priori.

>>16707320
How many elements does an empty set have?

>>16707323
It doesn’t have any. What does this have to with zero? I reiterate, the notion of a zero comes from an algebraic structure on a set, not the set itself. You can canonically induce a monoid structure of addition on the naturals via the successor function. But you are not required to. And not every set admits such a canonical algebraic structure. For example, given the power set of some set, how do you define addition? You don’t. There’s a different algebraic structure you can define on a power set called a lattice that doesn’t at all act like addition on natural numbers. These are all additional “layers” you put on sets, but set theory itself isn’t concerned with these in the slightest.

>>16707331
>It doesn’t have any
So you're saying it has zero elements

>>16707334
It has cardinality zero, yes. Tautologically so. We just say “a set has cardinality ‘zero’ if there doesn’t exist an injective function to that set”. So you saying “the empty set has zero elements” is you just regurgitating a definition. When people colloquially bring up the concept of zero, they immediately invoke some idea of addition.
>if I have 5 apples, add x apples, and end up with 5 apples, how many apples did I add?
Notice how none of this has you have some pair sets and an injective function between. Everything lives within one set.

And to answer your original question, there are many algebraic structures without an identity you can consider. In picrel those would be magmas, semigroups, and quasigroups.

>>16707320
By zero, I'm not just referring to the algebraic object, but any mathematical object with a property that can be equated to nil and nothing.

>>16707264 (OP)
there is no concept of zero in set theory.
sets have set membership. a set with no members is the empty set. that's it.

>>16707377
So an initial object in a category. Yes, plenty of categories without an initial object exist. For example, any unbounded poset is such a category. There is no requirement for a category to have an initial object. But a lot of "nice" properties like cocompleteness hinge on its existence.

>>16707393
>a set with no members is the empty set
express this without implying the existence of zero

>>16707417
[math]\exists\varnothing:\forall x(x\not\in\varnothing)[/math]

>>16707318
axiom 1. there exists a set
axiom 2. for every set A and every formula P there is a set {x in A : P(x) is true}

let A be any set, and let P be some retarded shit

>>16707264 (OP)
sure but the bigger problem is treating every zero the same.