I can't offer much more insight (if I have at all this far). The above def is how ordinal numbers are built, and I think THERE it will prove more valuable (probably worth a quick Wikipedia read when you see the term used, as I don't think he went into them as much as you'll want).
It's an attempt to put the standard math systems we know and love into a unified set theory model, and then you can worry about what exactly set theory is I guess.
Wish I had better answers
Ha, no, I’m cool with figuring it out myself. Just trying to wrap my head around it.
FYI, can always do this:
Not bad, though a bit verbose. I think the question may be a bit off for getting the best response. One of the more important properties, and what distinguishes them from the rationals the most, is the properties of them as a linear order (every bounded subset contains a least upper bound inside the real set itself). And I think something like containing a dense countable subset (the rationals) yields their unique/categorical property. This is mentioned in the book, as you know.
So you have these desirable properties your intuition wants them to have, and are able to prove that the description given uniquely captures it. So if two people were to discuss them, you can be sure that their conceptions are identical. There's a little nuance im hesitant to even mention, but I will, and that's that these canonical model proofs rely on second order logic. Which is fine, but when you get to completeness theorem it might confuse a bit since there you are restricting to first order logic.
Ok, thanks. wil dig a little deeper in that case.
Enjoy discussing, helps me learn too.
Another way to phrase it, is you have a short set of rules: linear order, upper bounds, dense countable subset..., that you hone in on as you begin to formalize what you imagine reals to be, and then there are two interesting questions. Are these rules consistent (is there at least one mathematical model satisfying them, or do they somehow lead to contradictions), and are they narrow they don't allow for other interpretations. Ie, have you nailed down a set of rules such that you can declare they unambiguously define THE real numbers.
For instance, if you drop the countable dense subset requirement, you get nonstandard models that also satisfy the rest of the rules.
I'm blathering...
will stop posting these after this, but it’s amazing how you can drill down these things with AI when you don’t fully understand something.
Not bad. I've played with AI for math stuff before and it does ok. Been a few months since I have, so maybe it's gotten better, but early experience was that it quickly gets lost if you go too deep; mine started spewing incoherent sounds a few times lol
I had this problem too with very large numbers. Kept making errors, and once it “corrected”, I couldn’t be sure it was really right since it might still be making errors that just weren’t obvious to me. But for concepts with known explanations, think it’s pretty useful.
Yeah, same concerns lol. Been using it a little recent for coding (nothing fancy, just like helping me write a "fuzzy join" on some text fields), and it's damn useful there. Coding has never been so easy
