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    As a non-mathematician who enjoys reading about math, I read this and it sounds great–the definition of surreal numbers is clear, elegant, and appears to be strictly more powerful than the usual definitions of real and imaginary numbers. So I have to ask: what’s the catch? Is there some reason this definition of numbers is difficult to use, or insufficiently rigorous, that would explain why it’s not used more widely?

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      As mentioned in the article, surreal numbers are a proper class (collection of things that too big to be a set). This means you can’t rely on things like the axiom of choice. Depending on what you want to do with them, this might turn out to not be an issue though. For example, we can’t talk about “set of metric compacta”, but there is a set of representatives on some topologies.

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        In addition to what kowale said, having infinitesimals and infinities is also trickier to work with in some cases.

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        How on earth are you supposed to pronounce “≱”?

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          “not greater than or equal”?

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            In my head I was just calling it “blob”, given that it’s supposed to be an opaque symbol to begin with.

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            There once was a scholar at Trinity
            Who derived the cube root of infinity.
            But it gave him the fidgets
            To count all the digits,
            So he dropped math and took up Divinity.

            (Can’t remember the author. Bertrand Russell?)

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              I’m stuck on Rule 2 unfortunately:

              … 0≱0 which we already know is false …

              I don’t see anywhere where we concluded this in Rule 1?

              In fact, I can’t even parse the sentence in its entirety:

              As it is equivalent to 0≱0 which we already know is false: 0≤−1 by definition, means ∅≱−1 and 0≱0

              What does the first “it” refer to, for example? Does it refer to 0≤−1? If so, and it is false as claimed, how can we conclude the final two expressions from it? In fact, the first and final expressions are identical – so how can the first one being false imply that the final one is true?

              Edit: Ah wait, falsehood implies anything you like, so this is fine. But then why not just stop at saying 0≤−1 is equivalent to 0≱0, which we (apparently) know is false? Why include consequents at all?

              I’m so tangled up >.<