I feed crows on my nightly bike ride. Sometimes there are multiple crows, the proverbial "murder". If there are just a few, as far as I'm concerned it's just a manslaughter...
Show HN: I recently gave a talk on the incompleteness theorem, specifically expressed in the language of software. It starts with a bit of historical background and a discussion of some of the philosophical context in which he carried out his work. The second half of the talk is my attempt to show the beautiful essential idea at the core of Godel's idea, pitched to a technically knowledgeable general audience. These are the slides from the talk, not translated into web pages; YMMV.
Link: https://www.gregfjohnson.com/godel_incompleteness/
There is a beautiful application of the power set of the Naturals to denotational semantics. I assume Prof. Hamkins's book will cover that topic, but it is not mentioned in the linked article.
Dana Scott (and apparently Gordon Plotkin independently) came up with a clever way to create a model of the lambda calculus that employs the power set of the Naturals.
The problem is that in lambda calculus, the formal language permits every expression to appear in the left-hand slot of the "Function_Application" operator. I.e., every term is simultaneously permitted to be given as an argument to a function, and also to be used as a function.
So we have the conundrum of finding some set "S" where every element of S is a function (not a problem so far), BUT, those functions all take elements of S as inputs and produce elements of S as outputs. So we need a set S that is isomorphic to the set of functions "S -> S". Cardinality arguments show that this is not possible: the function space for any non-trivial set has greater cardinality than the set itself.
So, Scott and Plotkin devised a "computationally sensible" way to interpret an arbitrary set of integers as a function over sets of integers.
By standard encodings, interpret any integer "n" as an ordered pair "". Now, again via standard encoding, interpret M as a finite set of integers M_set.
The "function" defined by a singleton set { n } applied to some other set Q is:
{u} if M_set is a subset of Q, Empty_set otherwise.
Then for a set of more than one element, take the union of the outputs of each of the elements applied to Q as above.
One can define a topology over the power set of the Naturals, and the above functions turn out to be exactly the continuous functions relative to that topology. The continuous functions so defined have the same cardinality as the power set of the Naturals.
An interesting historical side-note: historically, mathematical structures were the starting point, and axiometized formal languages a la Frege, Russell & Whitehead, etc. were built later. In the case of lambda calculus, it was the other way around: the formal language came first, and it was a multi-decade riddle what actual mathematical structure (if any) this formal language actually described.
Been doing this for 20 minutes in the morning for a few decades now. It has had an impact. I am a rather “high gain” person emotionally. I now tend to notice emotions and observe them, which helps when they threaten to suck me into a self-perpetuating ruminative vortex. Also, it has made me gentler, and an easier person to be around. I’m now well aware that whatever disturbances occur, there will be echoes and resonances tomorrow morning during my sit! Now for a moment of honesty: this is a religious practice for me. A prayer that “landed” for me: “Lord, I ask to be a conduit of grace.”
How do you know whether that's not just you growing as a person? A few decades is quite a long time, a lot of personal growth could happen in that time with or without this practice.
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