A post simply to capture a wonderful 2016 lecture by Doug Hofstadter on Gödel and the limits to logic.

Wonderful – in a wonderfully nerdy sense, like so much Hofstadter. (In fact I only stumbled upon it randomly in the side-bar of a discussion of Dave Edmonds on Schlick and the Vienna Circle.)

[*Still an important issue for my own theses whether or not Gödel consequences really can be extended to non-number / non-integer contexts in (say) epistemology- real world human epistemology that is. But that of course is the point – any arbitrarily complex formulae or system of formulae – expressible in any formal notation – can be represented in Gödel numbering and their decidability analysed … formally*.]

Wonderful – because right from the start Hof is highlighting syntactic *properties* of sentences or even of integer numbers, strings of digits, symbols or anything (like length or size, a visual aesthetic) when talking about what we can know semantically about them, their content. Syntactic well-formedness and the semantic decidability or truth value of content, etc … every time I see this relationship it still seems nuts. It runs through so much of his work from “Gödel Escher Bach” via strange-loopiness to “Fluid Concepts and Creative Analogies” the former being seminal, the latter being an absolute tour de nerdy force.

Wonderful – because he *talks and transforms* formal logical notation in real time in his head whilst lecturing.

Wonderful – for the Collatz conjecture which was a new one for me – but is key to seeing the unpredictable size / shape relationship to a *theorem*, the special semantic content of that kind of string. The rest is basically the clearest possible exposition of what Gödel is about. Did I mention – wonderful?

Encoding a long complex string in a large (but not necessarily long) integer? And what operations can you perform on it, what properties can you extract from it?

Even a string that might represent a Bach composition … ha! … or more to the point, the whole three volume text of Principia Mathematica. This is Gödel numbering – Any string (even a complex mathematical or logical formula or sequence of formulae – the whole of the Principia) can be encoded in a large integer and analysed formally

So, for the integer (Gödel number) form of any arbitrary symbolic string, the analysis of its well-formedness will always be monotonic, convergent on a result, but the decidability of the truth value, the *provability* of (some part of) the content, may not be and may not be predicted.

Gödel’s formula G, whose number is g,

States that g is not the number of a theorem.

The number g exists (and has complicated number-theoretical properties)

(And yet) G says that it is *not* a theorem.

Says that it is provable, yet that very statement is not provable.

Not provable *because* it is true!

All (complete) axiomatic systems have such a (paradoxical) statement.

Goodbye professor Russell.

[*Spot the great throw-away remark about the photo of Gödel with “an unidentified peasant”. Brilliant teacher*.]

[*Lots more 21st C Hofstadter talks on YouTube, covering most aspects of his work … **the self-referential loops and analogies*.]