Can two indifference curves cross?

An indifference curve maps out all the different combinations of consumption which yield the same utility.  Consider the following example with oranges and apples.

The line curve tells us that consuming 2 oranges and 3 apples gives the consumer the same utility as consuming 4 oranges and 2 apples.

two-indifference-curves

 

Now let's answer the question of whether it is possible for two indifference curves to cross. One of the assumptions we make in economics is non satiation which means that "more is better than less". This means that we can conclude that "3 apples and 3 oranges" must be preferred to "3 apples and 2 oranges", which means that the consumption bundle of "3 apples and 3 oranges" must lay on an indifference curve further from the origin (the further the indifference curve from the origin, the better). This means that we can conclude that the entire area shaded in light blue gives us more utility than what we would receive on our current blue indifference curve.

Now if two indifference curves cross that must mean that they both give us the same amount of utility, as an indifference curve is defined as all the different combinations of a good which give us the same amount of utility. However, as we can see with the red indifference curve, if the two curves cross, than the red curve must intersect that light blue zone where consumption is higher, which violates our assumption that "more is better". If we look at the red curve, we see that the consumption bundle "3 apples and 3 oranges" is possible on that curve. If the red and blue curve both gave us the same amount of utility, that would mean that consumers would be indifferent between "3 apples and 2 oranges" and "3 apples and 3 oranges" but since they cannot be more is better, this must mean that it is not possible for two indifference curves to cross. 

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