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Question: What is the difference between an Axiom and a Definition?

Example Terence Tao's Analysis I, which is intentionally and carefully written to develop theory in small incremental steps from basic axioms has

  • Set Union as an Axiom (3.5 in the 4th ed book)
  • Set Intersection as a Definition (3.1.22 in the 4th ed book)

I can't understand why one if an axiom and one is a definition. Why are they both not axioms?

Discussion

I (think) I understand there is a hierarchy of definitions, with axioms being the most fundamental and which have no supporting definitions. And as such they can't be proved, they are the basic for other proofs.

In my mind a definition is for convenience. That is, a definition uses notions established in axioms to define a shortcut for a new concept.

And then propositions, lemmas and theorems are "taller" constructions based on axioms and definitions, with the only difference between these three being their "height" above the basic axioms.

Penelope
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    The axiom of union guarantees that unions exist. You don't need another axiom to guarantee intersections exist, because that can be proved from the other axioms. Both the union and the intersection must be defined in a definition, not an axiom. – Joshua Tilley Oct 19 '23 at 00:43
  • "Set Union as an Axiom"? What does that even mean. One thing is the definition of the union set, that is a description of what a union set is. But once you gave a definition of the the union set you would like to guarantee the existence of such thing, for that you need the axiom of union. – jjagmath Oct 19 '23 at 01:43

2 Answers2

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Long comment

Compare with Axiom 3.5: "(Axiom of specification, page 40). Let $A$ be a set, and for each $x \in A$, let $P(x)$ be a property pertaining to $x$x (i.e., $P(x)$ is either a true statement or a false statement). Then there exists a set, called ..."

I've highlighted the key-point: the axioms is a statement asserting the existence - under specific conditions - of an object: a set.

With Definition 3.1.23 of Intersection (page 41), the author is not enough rigorous: he has to prove that for each pair of sets the set "described" by the formula exists and it is unique.

This is the gist of the observation: "Note that this definition is well-defined (i.e., it obeys the axiom of substitution, see Section A.7)".

Due to this fact, we may introduce a "name" and the corresponding symbol for that set.

Thus, the intersection of two sets is not postulated by a dedicated axioms because its existence is a theorem.

Note: Axiom 3.4 (Pairwise union, page 39) is needed because we cannot prove, using only the Specification axiom, that the union of two sets exists.

Mauro ALLEGRANZA
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Do you have a quote of what Tao is saying? I'd put union and intersection both at a definition level.

You'll always have undefined terms because otherwise you'd get an infinite regression of terms. Then there are defined terms. Axioms typically establish a relationships between terms, both defined and undefined. So I'd put a rough hierarchy as from more fundamental to less, 1) undefined terms, 2)defined terms, 3) axioms. Though I don't think this is rock a rock solid hierarchy.

Consider Euclid's 5 Axioms. He made attempts to define point, line, and plane. A point is "that which has no part". But what does that mean? A line is "that which has no breadth".

Axiom I: A line can be drawn between any two points.

So point and line are not defined there.

Axiom III: Given two points A and B, a circle can be drawn centered at point A through point B.

Again the axiom only has meaning with terms defined before hand.

Axiom V: When two lines m and n cross a third line, l, lines m and n intersect on that side of l where the alternate interior angles add up to less than two right angles.

What's intersection? What are right angles?

But consider the axioms that define the real number system.

The reals are an ordered field having The Least Upper Bound Property having The Rationals as a property subset.

The definition of field relies on notions of sets, elements, and functions by way of binary operations. Ultimately those rely on undefined terms. Without some meaning of sets and elements, none of the axioms make sense. On the other hand, you are not dealing with the real numbers unless you stipulate the Least Upper Bound Property or some equivalent. It's an axiom that's needed as part of the definition of the Reals. So there's basic definitions, then axioms, from which you compose a definition.

Perhaps another difference between a definition and an axiom is definitions are always bidirectional logical statements. Axioms are typically unidirectional logical statements.

Definition: A natural number is even if and only if it is divisible by 2.

Compare this to Euclid's first axiom. While given two points, you can draw a line between them it doesn't necessarily follow all lines have two points. Consequently Hilbert stated this as the third of his postulates of geometry.

Another difference, the negation of a definition doesn't make much sense."A number is not even if and only if it is divisible by two". If we take this to be true, we are radically changing the meaning of "even number".

If any part of a definition fails, you are talking about some other entity.

If any part of a postulate fails, you are typically talking about some other relationship while preserving the meaning of the terms involved. Given a line and a point not on that line with multiple parallel liens going through that point, we negate Euclid's fifth postulate while maintaining the meaning of all terms involved.

TurlocTheRed
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  • I think Tao's treatment involves developing set theory itself somewhat axiomatically, and there it's standard (eg that's what ZF does) to have an axiom asserting the existence of unions, but there is no need to have such an axiom for intersections, because this follows easily from the axiom of separation. Having an axiom for unions is necessary if you're working in ZF, for example. – Izaak van Dongen Oct 20 '23 at 16:40