This is the issue I tried unsuccessfully to raise earlier. We are getting lost in what may (or may not) be interesting mathematical properties but are confusing the idea of value. The problem is teasing apart whether holding the ticket has any value prior to the redemption itself. Can the owner redeem the ticket at a value of his choice less than $100 with certainty at any time he wants? Is the person on the other side of the transaction have any agency? Can he accept or reject an offered trade?
I’m not trying to fight the hypothetical here. These things are critical to the idea of value, especially mark-to-market valuation. This makes it a different question from “what’s the highest number less than 100.”
It’s worth $99.99. Any work dividing it into mills takes more energy than it is worth, and nobody will give you a $100 for it, because the Benjamin in their hand is worth just as much without the headache. The answer is $99.99
Everyone has complete free agency to accept or reject whatever trades they like. However, the government, in all its wisdom or folly, has set up centers at which one can exchange the ticket for any desired amount less than $100.
This is how I would handle the semantics.
It’s worth is whatever is the maximim that someone will offer me for it in money or product. While money may be continuous, the number of people in the world is not and someone would offer a maximum amount like $99.9999999999962 The same can be said for product in that there is an object that is the maximum price still under $100 like a really big meatball sandwich for $99.99999999999999999915125. The larger of these two numbers is the worth.
It’s value is $100 because of the supremum argument that I can get as close to $100 as I want. But of course the value is theoretical while worth is actual.
Think of it this way. I own a painting that I have to sell and it is appraised at $1,000,000 but the highest offer I get is $900,000. What is the value, what is the worth and which one do I get when I sell?
In the sense of being willing to exchange the same amount of ordinary cash for them? Not necessarily, but one might suppose that hypothetical omnirational agents would make the same value judgements, as (as opposed to, say, ice cream or bus rides or other products with intrinsic desirability for some market of consumers) the ticket is merely a means to other ends.
Also, let us suppose we live in a philosophers’ paradise where any opportunities for arbitrage are seized mercilessly.
They won’t take you up. The government center has one job and one job only: to redeem tickets for a value less than $100, as chosen by the ticket-holder.
There doesn’t really need to be a government center. It could be a button on the ticket that turns it into a suitable amount of cash (legally sanctioned and indistinguishable from cash obtained by any other means), or deposits the amount directly into your bank account. Whatever story for the setup of “The ticket is directly redeemable for any value less than $100, but not (directly) for any other value.”