1.2: What is a model?

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The Euclidean plane can be defined rigorously the following way:

Define a point in the Euclidean plane as a pair of real numbers \((x, y)\) and define the distance between the two points \((x_1, y_1)\) and \((x_2, y_2)\) by the following formula:

\[\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2.}\]

That is it! We gave a numerical model of the Euclidean plane; it builds the Euclidean plane from the real numbers while the latter is assumed to be known.

Shortness is the main advantage of the model approach, but it is not intuitively clear why we define points and the distances this way.

On the other hand, the observations made in the previous section are intuitively obvious — this is the main advantage of the axiomatic approach.

Another advantage lies in the fact that the axiomatic approach is easily adjustable. For example, we may remove one axiom from the list, or exchange it to another axiom. We will do such modifications in Chapter 11 and further.