gottheUIblues

The normal term for the logic that has two truth values, the law of non-contradiction, and the law of the excluded middle is 'Classical Logic' which dates back centuries before George Boole. Non-classical logics which deny one or more of three properties that I've mentioned also exist

Rather COBOL is a living fossil? And today's Fortran is the FORTRAN family with horizontal gene transfer from the Algol lineage of programming languages.

Err? Peano Arithmetic is provably consistent in ZFC, but it is not in itself (if PA is consistent). Therefore if PA is consistent it is not equivalent to ZFC (regardless of whether ZFC is consistent or not)

"Embarrassingly" considered harmful?

To the contrary (as summarised by Gemini):

Gödel showed that arithmetic cannot prove everything about itself.

Turing showed that computers cannot predict everything about themselves.

Rice showed that we cannot automatically verify what programs will do.

Chaitin showed that mathematics is full of random, unprovable facts.

Lawvere showed that they are all failing for the exact same structural reason!

These are not fringe issues. They define the absolute boundaries of human and machine intelligence.

I don't think of 9 as really being a signal to the process at all, more of an instruction to the OS kernel to terminate the process

If that theory holds - have to ensure that the models have not been trained on any code that is licensed incompatibly with the GPL, in which case the models could not be distributed at all

So specified .. that it can actually prove it can't be completely specified by any single specification

Right so strictly speaking C++ could do anything here when passed a null pointer, because even though assert terminates the program, the C++ compiler cannot see that, and there is then undefined behaviour in that case

Shouldn't that be the "dam spelling" then?

Film score composers are quite famous for borrowing from 12 tone serialism - quite a bit of discussion on it available by Googling or using your favourite chatbot

Money is a social construct, not some kind of physical quantity subject to conservation laws, and can be and is introduced into the economic system all the time. The real question is really would introducing more money or a UBI cause social disruption by e.g. disrupting price signalling by high inflation or changing incentives to work so less goods and services that people actually value are produced.

A trit is log(3)/(8log(2))=0.19812031259014 of a byte

The verifier doesn't need to be deterministic, just to output a proof artifact that can be independently validated for correctness.

There are infinitely many different p-adic completions of the rationals for each prime p, so we have 2-adics, 3-adics, 5-adics etc, all different.

It's topography not topology! Would be funny if a country has a government department dedicated to the mathematical field of topology!

Except protium, deuterium and tritium

Netscape open sourced to try and head off Internet Explorer as a 'last throw of the dice'? Chrome based originally on WebKit from Safari which was forked off KHTML from KDE's konqueror, came along years later.

I wrote as my comment kind of as a rhetorical question, but perhaps in retrospect the connection between the string comparison problem the pigeonhole principle is not so surprising in the end.

I didn't know and not mentioned in the article is that there are two different programs in reverse mathematics, one relating to classical mathematics, the other relating to complexity questions. Claude says this: not sure about accuracy

Let me clarify the relationship between PV₁ and reverse mathematics systems, as there are actually two different research programs here: Two Different Programs 1. Classical Reverse Mathematics (over RCA₀) The standard reverse mathematics program, pioneered by Harvey Friedman and Stephen Simpson, works over second-order arithmetic and studies which set existence axioms are needed to prove theorems of ordinary mathematics. The main systems form the "Big Five": RCA₀ (Recursive Comprehension Axiom) WKL₀ (Weak König's Lemma) ACA₀ (Arithmetic Comprehension Axiom) ATR₀ (Arithmetic Transfinite Recursion) Π¹₁-CA₀ (Π¹₁ Comprehension Axiom) 2. Bounded Reverse Mathematics (over PV₁ or similar) This is a separate program studying computational complexity rather than computability. It analyzes which theorems require which computational resources.

And there's no clever trick you can do using eg hash functions or compression to shortcut the process? Of course the proofs that hash functions have collisions and there is no universal compression algorithm use the pigeonhole principle..