Aatu Koskensilta — LessWrong

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Replying toWhy's equality in logic less flexible than in category theory?

Why's equality in logic less flexible than in category theory?

Right, but models of set theory without equality can contain many indistinguishable "copies" of the same set, and in this sense the "extensionally equivalent" predicate does not define equality (in the sense of the identity relation on the universe of the model).

Replying toWhy's equality in logic less flexible than in category theory?

Aatu Koskensilta4mo

Why's equality in logic less flexible than in category theory?

OK, maybe if we look at some other definitions of equality we can get a grip on it? In set theory, you say that two sets are equal if they've got the same elements. How do you know the elements are the same i.e. equal? You just know.

If we have the axiom of foundation -- or, more perspicuously maybe, the principle of epsilon-induction its equivalent to in presence of other usual axioms -- the picture is less mysterious.

Yes, in order to tell if A and B are equal, we need to see if they have the same elements, but in order to do that we need to already know how to tell... (read more)

Replying toWhy do the Sequences say that "Löb's Theorem shows that a mathematical system cannot assert its own soundness without becoming inconsistent."?

Aatu KoskensiltaApr 01, 2023

Why do the Sequences say that "Löb's Theorem shows that a mathematical system cannot assert its own soundness without becoming inconsistent."?

It's sometimes thought or said that the second incompleteness theorem shows that formal systems such as PA are in some sense "incapable of self-reflection", and their incapability to prove their own consistency reflects this lacuna in their "cognitive capabilities" or something along those lines.

Löb's theorem highlights how this is a much deeper phenomenon: it's not just that these formal systems lack abilities, or formalizations of relevant mathematical insight, or whatever; no, it's that it's formally impossible for any consistent system of the relevant sort to even baldly assert their own consistency, on pain of inconsistency!

That is, we can for instance construct, using the diagonal lemma, a sentence A such that A is,... (read more)

Replying toAre there specific books that it might slightly help alignment to have on the internet?

Aatu Koskensilta3y

Are there specific books that it might slightly help alignment to have on the internet?

It's heavily implied in the novels we only see the "disaffected" lot -- people who experience ennui, etc. and are drawn to find meaning out of a sense of meaninglesness even in somewhat inadvisable ways -- and the whole of Culture is mostly exploring the state space of consciousness and the nature of reality, sort of LARPing individual humanity as a mode of exploration -- you can for instance upgrade yourself from a humanoid into something resembling a Mind to a degree if you want to, it just seems this is not the path we mostly see mentioned. It's just that that sort of thing is not narratively exciting for most people,... (read 381 more words →)