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To my knowledge, the idea of representing weak-order with a function dates back to Cantor. So my questions are:

1) Was Cantor the first person to rigorize these kinds of representation?

2) Were there anyone in the ancient world who (though not rigorously) developed the idea of representing social choice or preference by a function?

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Not to be too obvious about it, but couldn't you just say Jeremy Bentham? The calculus of choosing bundles that maximizes pleasure specified a very simple utility function in words.

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Gérard Debreu precisely defined the conditions required for a preference ordering to be representable by a utility function.

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  • $\begingroup$ Check out Cantor and Birkhoff representation $\endgroup$
    – High GPA
    Commented Mar 8, 2019 at 18:31
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Swiss mathematician Gabriel Cramer suggested taking the expected utility of money to resolve the St Petersburg paradox in 1728, and Daniel Bernoulli did the same 10 years later. Cramer took $u(x)=\sqrt x$ and Bernoulli took $u(x)=\log(x)$

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