Kurt Gödel proved the first incompleteness theorem in 1931. He published the proof in his famous paper, “Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme” ("On Formally Undecidable Propositions of Principia Mathematica and Related Systems"). (Syntactic) completeness means that for any statement within an axiom system, either the statement or its negation is provable within the axiom system. Gödel's (first) incompleteness theorem says that every consistent (= free of contradictions) axiom system rich enough to contain "elementary arithmetic" (i.e. addition and multiplication of integers) always contains statements that are undecidable (neither the statement nor its negation can be proven from the axiom system). Hence those systems are incomplete, no matter how many axioms we add to them. https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems Other simpler systems have been proven to be complete, such as propositional logic (Hilbert, 1918), first order predicate logic (Gödel's completeness theorem, 1929) and Euclidean geometry (Tarski, 1930). However, Principia Mathematica (PM) by Russel and Whitehead was published in 1910. So - did PM contradict Gödel's incompleteness theorem? This was before Hilbert's program in the 1920's, that aimed to prove the completeness of mathematics. However, in PM, no claim is made to either prove or disprove the completeness of mathematics, so it doesn't contradict Gödel's incompleteness theorem. #mathematics, #gödel, #logic, #incompleteness

I just tried out #amethyst on #grapheneos via #zapstore. It works great, and might be one of the best nostr apps on mobile. You can also easily access your on-chain, nostr-key based #bitcoin wallet on amethyst via settings/wallet. That on-chain BTC wallet is derived from your private nsec nostr key, so you always have it if you use #nostr. Links: ------ Zapstore https://zapstore.dev/apps/dev.zapstore.app Amethyst https://opensats.org/projects/amethyst GrapheneOS https://grapheneos.org/

Also, here is a completely transparent way to generate an electrum (legacy) seed phrase from dice without trusting any electronic device to process the dice rools: https://bitcointalk.org/index.php?topic=878614.0 WARNING! Use at your own risk!! #dicerolls, #bitcoin, #seed

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Wow - where can I get one of those dice boxes?

#eu, #chatcontrol, #privacy, #masssurveillance, #ursula

Yes, the private key for larger funds should be air-gapped and be created with maximum randomness.

There is a book that I viewed, when studying mathematical logic, as "that legendary brutal book (in three volumes) that nobody ever reads" - Principia Mathematica by Alfred North Whitehead and Bertrand Russell. I have now finally bought volume one of this book. It was written between 1900 and 1910, and its goal is to create a foundation of mathematics based merely on logic, a project called logicism. Previous work by Gottlob Frege, who employed what could be called naive set theory, contained a contradiction, discovered by Russell, known as Russell's paradox: M = {the set of all sets} S = {all sets X such that X is not a member of itself} then: if S is a member of itself => S is not a member of itself and: if S is not a member of itself => S is a member of itself, hence the contradiction. Instead, Russell and Whiteheads project avoids this paradox by using the first type theory that only allows construction of certain sets (called classes). You are allowed to construct sets of individuals, sets of sets of individuals and so on, but you are not even allowed to construct the combined class: {a, {a}} hence a very restrictive type theory. But it is sufficient to construct elementary arithmetic, ordinals, cardinals and many other things. Volume one is 666 pages long. On page 379 they finally prove that 1+1=2: I have now reached page 29, getting through most of the notation stuff: The notation can be a bit awkward, since they use dots instead of parentheses, and also at the same time use a dot for the "and" operator. However, you kind of get used to it, and it sometimes looks better that loads of parentheses that you have to count. #books, #mathematics, #logic, #bertrandrussell, #typetheory, #settheory

I just noticed that #darkreader (https://darkreader.org/) for #darkmode now requires a one-off payment to work, but it seems it still works on Linux. So they really went Dark Vader. I started using this instead (https://chromewebstore.google.com/detail/dark-theme-dark-mode-for/gjjbmfigjpgnehjioicaalopaikcnheo): However, not sure if its open source. Any other tips? Also, I've noticed that most of the internet is #darkmode by default anyway now.

Hello world!