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Wikipedia does not elaborate much on the concept of "One" in such branches as graph theory, ring theory, algebra, topology, measure theory, formal logic, etcetera. How can one grasp the concept of "oneness" in diverse branches of mathematics?

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It’s generally taken as a primitive concept not really requiring much discussion. What sort of elaboration do you have in mind? – Brian M. Scott Mar 3 '13 at 21:09
Where does "1" appear in graph theory, topology, measure theory, or formal logic?? – Rahul Mar 3 '13 at 21:10
@Brian M. Scott Isn't $1:=\{\varnothing\}$ in all 'regular' mathematics? – Git Gud Mar 3 '13 at 21:12
@BrianM.Scott In the link it says: 1 is " more generally, in abstract algebra, the multiplicative identity ("unity"), usually of a ring". So my question would be to list similar notions in all branches of mathematics? – Sniper Clown Mar 3 '13 at 21:13
Mahmud, that’s a statement about notation, not about the concept! – Brian M. Scott Mar 3 '13 at 21:14

"One" is typically defined as $S0$ in Peano Arithmetic. $\forall x(Sx=x+S0)$ is a theorem of PA and much weaker theories of arithmetic.

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The number $1$ has several unique properties. First, it is the neutral element with respect to multiplication, meaning that $1\cdot x = x$ for all $x$. This property is used in many algebraic structures to define a $1$.

Another unique property among non-negative integers is $1 \mid x$ for all $x$. This is used in order theory and lattice theory: An element is called the $1$-element if it is less or equal than all other elements.

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