Am I correct in understanding Axiom of Extension?

Question

Halmos mentions the following:

Axiom of extension: Two sets are equal if and only [emphasized] if they have the same elements.

My understanding of Axiom of Extension as presented above is as following:
Axiom of Extension is independent of the Axioms of Equality in first-order logic with equality. A set might (or must always?) also have an intension which determines its extension. So two equal sets (by Equality Axioms) logically implies that they must have the same intension as well as extension, doesn’t it? Now, Axiom of Extension makes a logically unprovable remark that a set’s extension is all that matters.

Please fix my understanding.

Also, is “if and only if” required in Halmos’ statement? Cuz according to Jech,

If $X$ and $Y$ have the same elements, then $X=Y$ …
The converse … is an axiom of predicate calculus.

I’ve never come across this axiom. So help!

"I’ve never come across this axiom." Yes, you have. You just cited it. It's an instance of the substitution for formulas axiom. — Derek Elkins left SE, Jun 11 '19 at 18:40
Does first-order logic then include the notion of belonging? — Atom, Jun 11 '19 at 18:43
Oh! I just realized that we *can* include $\in$ as a binary relation. — Atom, Jun 11 '19 at 18:46
But then, $\in$ is the “principal primitive notion” in axiomatic set theory (see https://math.stackexchange.com/questions/3257748/is-belonging-not-defined-in-axiomatic-set-theory). — Atom, Jun 11 '19 at 18:49
See also the post What is the standard first-order language to formalize ZFC? — Mauro ALLEGRANZA, Jun 11 '19 at 18:52
"Axiom of Extension makes a logically unprovable remark that a set’s extension is all that matters." What does it mean "logically unprovable" ? The axiom is called Extensionality Ax exactly because it formalize the idea that what matters in "identifying" a set are exactly its elements (i.e. its extension). — Mauro ALLEGRANZA, Jun 11 '19 at 18:55
@MauroALLEGRANZA By “logically unprovable”, I mean that it can’t be a logical consequence. — Atom, Jun 11 '19 at 18:57
It is a mathematical axiom, i.e. it is not a "logical truth" i.e. a valid formula (true in every interpretation). — Mauro ALLEGRANZA, Jun 11 '19 at 18:59
No; in axiomatic set theory $\in$ is a primitive symbol (the only one except $=$, in the case of predicate logic with equality). Its meaning is "descrbed" the theory through the axioms. — Mauro ALLEGRANZA, Jun 11 '19 at 19:07

Mauro ALLEGRANZA · Accepted Answer · 2019-06-11T19:00:26.317

4

According to Jech, the converse of the Extensionalty Axiom "is an axiom of predicate calculus."

If the underlying logic is predicate calculus with equality, we have the substitution axiom for formulas :

$x = y → (\varphi → \varphi')$,

where $\varphi'$ is obtained by replacing any number of free occurrences of $x$ in $\varphi$ with $y$.

Thus, considering the formula $(z \in x)$ as $\varphi$, we have :

$x=y \to (z \in x \to z \in y)$.

edited Jun 11 '19 at 19:00

answered Jun 11 '19 at 18:49

Mauro ALLEGRANZA

94,169

Please see the comments. – Atom Jun 11 '19 at 18:51

Am I correct in understanding Axiom of Extension?

1 Answers1