Gurkenglas

I operate by Crocker's rules.

I try to not make people regret telling me things. So in particular:
- I expect to be safe to ask if your post would give AI labs dangerous ideas.
- If you worry I'll produce such posts, I'll try to keep your worry from making them more likely even if I disagree. Not thinking there will be easier if you don't spell it out in the initial contact.

Wiki Contributions

Comments

If you want to transfer definitions into another context (constructive, in this case), you should treat such concrete, intuitive properties as theorems, not axioms, because the abstract formulation will generalize further. (remark: "close" is about distances, not order.)

If constructivism adds a degree of freedom in the definition of convergence, I'd try to use it to rescue the theorem that the Dedekindorder and Cauchydistance structures on ℚ agree about the completion. Potential rewards include survival of the theory built on top and evidence about the ideal definition of convergence. (I bet it's not epsilon/N, because why would a natural property of maps from ℕ to ℚ introduce the variable of type ℚ before the variable of type ℕ?)

rename your "logs" directory to "sources"

The fair value input should be "what you expect to pay/get for this if this negotiation falls through", right? To serve as a BATNA.

Hmmmm. What if I said "an enumeration of the first-order theory of (union(Q,{our number}),<)"? Then any number can claim to be equal to one of the constants.

If Earth had intelligent species with different minds, an LLM could end up identical to a member of at most one of them.

Is the idea that "they seceded because we broke their veto" is more of a casus belli than "we can't break their veto"?

Sure! Fortunately, while you can use this to prove any rational real innocent of being irrational, you can't use this to prove any irrational real guilty of being irrational, since every first-order formula can only check against finitely many constants.

Chaitin's constant, right. I should have taken my own advice and said "an enumeration of all properties of our number that can be written in the first-order logic (Q,<)".

Oh, I misunderstood the point of your first paragraph. What if we require an enumeration of all rationals our number is greater than?

I claim Dedekind cuts should be defined in a less hardcoded manner. Galaxy brain meme:

  • An irrational number is something that can sneak into (Q,<), such as sqrt(2)="the number which is greater than all rational numbers whose square is less than 2". So infinity is not a real number because there is no greatest rational number, and epsilon is not a real number because there is no smallest rational number greater than zero.
  • An irrational number is a one-element elementary extension of (Q,<). (Of course, the proper definition would waive the constraint that the new element be original, instead of treating rationals and irrationals separately.)
  • The real numbers are the colimit of the finite elementary extensions of (Q,<).

I claim Cauchy sequences should be defined in a less hardcoded manner, too: A sequence is Cauchy (e.g. in (Q,Euclidean distance)) iff it converges in some (wlog one-element) extension of the space.

Load More