Math Myth: The rationals are numbers

[trees and plants]

How can you prove that $\frac{12}{12}\neq 1$? Show me a proof if you know.What is your definition of equality?

 

[jbriggs444]

The sets are not equal

In mathematics, a set is a collection of distinct elements.[1][2][3] The elements that make up a set can be any kind of things: people, letters of the alphabet, numbers, points in space, lines, other geometrical shapes, variables, or even other sets.[4] Two sets are equal if and only if they have precisely the same elements.[5]

12/12 has 12 elements 1 has 1, could you use that?

I thought that 12/12 had infinitely many elements, each element being an ordered pair. Each ordered pair would naturally be equivalent to every other ordered pair in the set under our standard equivalence relation for the rationals. An ordered pair, following the Kuratowski definition would be a two-element set {a,{a,b}}. In the case of this particular equivalence class, each ordered pair would collapse to {a,{a}}.

So the infinite set would be something like { {1,{1}}, {2,{2}}, {3,{3}}, ... {12,{12}}, ... }

We would likely adopt the notation where 0 = {}, 1 = {0}, 2 = {0,1}, etc. So, for instance, 12 would be a twelve element set.

Of course, this is assuming that "12/12" is to be interpreted as a ratio of naturals. If it is to be a ratio of signed integers, there are additional layers to the construction.

Meanwhile, we all understand that none of this folderol is relevant to what someone means when they write 12/12=1.

 

[Mark44]

The thread seems to have run its course, and is now closed.