# Representing a Lexicographic Preference in a Natural X Natural Choice Space With Utility Function

my current thinking is i have to dis/prove two things

1. cardinality
2. continuity but im not sure about how it would apply since the above is a natural X natural choice space

I know cardinality of natural choice space = cardinality of rational numbers, but im not sure how I can relate that to representation in a utility function

Im not sure about continuity because epsilon balls around numbers in a natural space doesnt include anything else

• Have you tried writing down a few bundles and seeing if there is a way to compare them? Try out out some examples and it should be clear if such a representation is possible. Jan 25, 2022 at 18:58
• No it doesn't. The link you provide shows an example where they are comparing real to rational as a proof, whereas my question is set in the natural choice space @Giskard Jan 25, 2022 at 19:03
• @WalrasianAuctioneer yeah, the lexicographic preference can be compared by drawing but im not sure how i can prove that Jan 25, 2022 at 19:05
• Sorry, mistake! But why are you bringing continuity into this? It is not required that $U$ is continuous, is it? Jan 25, 2022 at 19:06
• @Giskard my lecture notes state that discontinuous preferences cannot be represented by a utility function, hence the continuity argument Jan 25, 2022 at 19:10

Take a strictly increasing mapping $$f:\mathbb{N} \to [0,1)$$, such as $$f(y) = 1 - \frac{1}{y+1}.$$ Then $$U(x,y) = x + f(y)$$ represents the Lexicographic preference in the $$\mathbb{N}^2$$ choice space.