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What is the reasoning for labeling the axiom of extensionality as an implies operator rather than an if... then statement? For example, I have seen the axiom of extensionality written as an if...then statement and not an implies operator as described in the Axiom of extensionality Wikipedia page.

Also, I understand that an if... then statement that is a tautology is the same thing as an implies operator. I think I am missing something simple, and any help would be greatly appreciated.

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  • $\begingroup$ As you say yourself, if ... then and implies is the same. So what is your question? $\endgroup$ – Christian Matt Mar 25 '17 at 15:07
  • $\begingroup$ Not clear... if it is about the symbol, it is only the choice to use $\Rightarrow$ instead of $\to$ or $\supset$. Unfortuantely, there is no stable international "standard". $\endgroup$ – Mauro ALLEGRANZA Mar 25 '17 at 15:08
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    $\begingroup$ "if... then statement that is a tautology" ? Of course the Axiom of Extensionality is not a tautology $\endgroup$ – Mauro ALLEGRANZA Mar 25 '17 at 15:10
  • $\begingroup$ The axiom states a condition on two sets that licenses us to conclude with their equality: if the sets $A$ and $B$ have the same elements, then they are equal. $\endgroup$ – Mauro ALLEGRANZA Mar 25 '17 at 17:15
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"If $A$ then $B$" means the same thing as "$A$ implies $B$". The use of one rather than the other is just a choice of language - they both mean the same thing, "$A\implies B$".

Meanwhile, I don't understand what tautologies have to do with this, but as Mauro commented the axiom of extensionality is not a tautology. For instance, consider the $\{\in\}$-structure whose domain is $\{a, b\}$, and whose $\in$-relation is empty. This fails the axiom of extensionality - $a$ and $b$ have the same elements (namely, none) but $a\not=b$. So extensionality is not a tautology, since it is not true in every structure.

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  • $\begingroup$ Implies is slightly different then if A then B regarding the two definitions, but I see what you mean by it not being a tautology. $\endgroup$ – Will Mar 25 '17 at 23:37
  • $\begingroup$ @Will "Implies is slightly different then if A then B regarding the two definitions" No, it's really not - "$A$ implies $B$" and "if $A$, then $B$" mean exactly the same thing. What difference do you think there is? $\endgroup$ – Noah Schweber Mar 26 '17 at 0:19
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    $\begingroup$ @NoahSchweber - maybe the reason for the misunderstanding is the "eternal querrel" between "implication" as conditional (the connective) and "implication" as logical consequence (i.e. entailment) :-) $\endgroup$ – Mauro ALLEGRANZA Mar 26 '17 at 8:18
  • $\begingroup$ Exactly see page 8 on this link people.math.gatech.edu/~ecroot/2406_2012/basic_logic.pdf. How do you tell the difference between these two? $\endgroup$ – Will Mar 26 '17 at 20:53
  • $\begingroup$ Rather, in this context it is used as a conditional correct? $\endgroup$ – Will Mar 26 '17 at 21:14

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