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Euclidian postulates are statements about points, lines, angles, etc. which exist separate from their definitions, created solely based on observation of our universe. Consequently, we have been able to model universes with hyperbolic space, for example, where euclid's postulates don't hold, for the same definitions of points, lines, angles.

However, when I look at the ZFC axioms, or the Peano axioms, they seem to me just definitions of sets, succesors, and equality. Do we have complete definitions of these terms separate from the axioms? If so, can we model algebraic theories where these axioms do not hold, for the same definitions?

dnaik
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    I have observed about as many "2"s in the universe as I've observed "lines". – JonathanZ Dec 16 '22 at 17:12
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    What's $2$ and $3$ and $5$? Axioms are syntactic, so as far as we can tell, these are just symbols. They have no meaning, even if they do have an intended interpretation. – Asaf Karagila Dec 16 '22 at 17:17
  • Axioms will typically be the foundational rules you use to derive results while definitions will relate to some important substructure created by the axioms. I can also use definitions to make queries about existence given some axioms while axioms assume existence. – CyclotomicField Dec 16 '22 at 17:29
  • The point of axiomatic systems like ZFC (and I think Euclidean geometry too) is to avoid relying on "definitions" of the basic nouns (sets/points/lines). Everything we prove about sets is a logical consequence of the properties expressed by the axioms; the theorems of set theory apply to any class of objects that satisfies the axioms, regardless of what those objects "actually are". – Karl Dec 16 '22 at 17:33
  • I think my comment still applies after your edit. Your question makes a distinction between axioms and definitions, but an axiomatic theory begins with no "definitions" in the sense you mean; we just pick a word like "set" to refer to objects governed by the axioms. So the answer to your last question is yes, we can freely vary the axioms and study the logical consequences. We can also freely change the noun if we want. The axioms alone (plus the logical framework) determine the logical content of the theory. – Karl Dec 17 '22 at 03:56
  • When we do make definitions in math, they are "built on top of" and justified by the underlying axioms. E.g. the set theory axioms imply that sets with a certain property exist, so we define terminology to refer to those particular objects (e.g. "infinte sets"). – Karl Dec 17 '22 at 04:01
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    It seems I have misunderstood the order of precedence of definitions and axioms. I believed that definitions were more fundamental than axioms, and definitions were a prerequisite for stating axioms. – dnaik Dec 18 '22 at 04:54
  • $2+3=5$ is a theorem of arithmetic. Its proof needs Arithmetical axioms for $+$ as well as the definitions for $2=s(s(0))$ and similar for $3$ and $5$. In addition, we need rules and axioms for logic with equality. – Mauro ALLEGRANZA Dec 20 '22 at 12:57

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A proper definition should satisfy 2 criteria (Suppes, 'Introduction to Logic', chapter 8):

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criterion of eliminability: if a new symbol $D$ is introduced (the definiendum) by: $D \iff S$ then $S$ ($S$ is called the definiens) should contain only primitives and $D$ should be derivable from axioms and previous theorems of the theory.

Thus one can not define: $x\leq x \iff (x=x) \lor (x<x)$ in the theory of arithmetic where the primitives are $=, >$ since one would not be able to eliminate: $\leq$ for 2 variables.

  1. a definition can not be used as an axiom to prove things that can not be proved by axioms, theorems and primitives of the theory.

this is the criterion of non-creativity.

Thus, definitions are not axioms and usually in mathematics they require a previous theorem.

ryaron
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A definition would be the same as an axiom, apart from the fact that you don't need definitions, you can always rewrite a sentence with defined terms using their definientia (that which was used to define the term).

Yodo
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