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Let's assume on the contrary that the indifference curve is "thick" or crosses. We can only assume the four axioms: completeness, transitivity, reflexivity and continuity. We do not assume strict monotonicity of preferences.

Define the IC as $\{y \in X: y \sim x\}$ where $X$ is a (possibly non-finite) set of consumption bundle and $x$ is arbitrarily chosen from $X$.

I do not see a way to prove this without assuming monotonicity at all.

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A counterexample

Suppose the preference relation $\preceq$ is such that for all bundles $x,y$ we have $x \sim y$. This relation satisfies transitivity and continuity (it satisfies all four "axioms") but does not satisfy non-satiation.

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  • $\begingroup$ Could you please re-explain this? I don't understand. Are we considering a set of indifferent bundles? $\endgroup$
    – not tdm's twin
    Commented Jun 23, 2021 at 9:14
  • $\begingroup$ We are considering the preference relation which is indifferent between all bundles. $\endgroup$
    – Giskard
    Commented Jun 23, 2021 at 11:01
  • $\begingroup$ Yes, I meant that- the consumer is indifferent between the bundles. Thanks for clarifying! $\endgroup$
    – not tdm's twin
    Commented Jun 23, 2021 at 11:33

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