Logic Matters » Gödel’s theorems

Gödel Without Tears, again

Peter Smith — Sat, 28 Nov 2009 16:40:21 +0000

I’ve updated all the episodes so far to give them what I hope are more useful headers (a bullet-pointed list of topics). The last two episodes have also been more significantly revised. You can get the latest, greatest, versions here.

Two more episodes to follow very soon.

Gödel Without Tears — 6

Peter Smith — Fri, 27 Nov 2009 15:22:39 +0000

The latest episode showing how a theory like Q can express/capture primitive recursive functions and relations is now online. [As always, I'll be very glad to hear about typos/thinkos.]

Earlier instalments can be found here.

Gödel Without Tears — 5

Peter Smith — Fri, 06 Nov 2009 15:16:00 +0000

Here now is the fifth episode on the idea of a primitive recursive function. The preamble explains why this matters and where this is going. [As always, I'll be very glad to hear about typos/thinkos.]

The previous episodes are available:

Episode 1, Incompleteness — the very idea (version of Oct. 16)
Episode 2. Incompleteness and undecidability (version of Oct. 26)
Episode 3. Two weak arithmetics (version of Nov. 1)
Episode 4. First-order Peano Arithmetic (version of Nov. 1)

Gödel Without Tears — 4

Peter Smith — Mon, 02 Nov 2009 13:20:00 +0000

Here now is the fourth episode [slightly corrected] which tells you — for those who don’t know — what first-order Peano Arithmetic is (and also what Sigma_1/Pi_1 wffs are). A thrill a minute, really. Done in a bit of a rush to get it out to students in time, so apologies if the proof-reading is bad!

Here are the previous episodes:

Episode 1, Incompleteness — the very idea (version of Oct. 16)
Episode 2. Incompleteness and undecidability (version of Oct. 26)
Episode 3. Two weak arithmetics (version of Nov. 1)

Gödel Without Tears — 3

Peter Smith — Mon, 26 Oct 2009 11:14:00 +0000

Here’s the third episode (slightly updated to take account of some initial comments). Not anywhere near so exciting as the first two — but after all that arm-waving generality, we do need to get our hands dirty looking at some actual formal theories of arithmetic, mildly tedious though that is! And you really ought to know, e.g., what Robinson Arithmetic is.

Gödel Without Tears — 2

Peter Smith — Sat, 17 Oct 2009 15:21:00 +0000

As promised, Episode 2 of Gödel Without Tears (in which we prove sufficiently strong theories are undecidable and incomplete — just like that!)

As explained, I’m writing these notes as just-after-the-event handouts for weekly lectures. And each week I’ll be checking through the previous handout (and no doubt finding small corrections to make) before I give the next lecture. So here’s the latest version of Episode 1, dated 16 October.

Gödel Without Tears — 1

Peter Smith — Mon, 12 Oct 2009 10:40:00 +0000

Here, as promised, is the first of a series of lecture handouts (roughly weekly, and about twelve in all) encouragingly titled Gödel Without Tears — 1. As is the way with lecture handouts, this was dashed off at great speed, and I don’t promise that this is free of either typos or thinkos. So do please let me know of any needed corrections, or indeed of any passage which is too unclear/could do with just a little amplification. Enjoy!

Later: I’ve already replaced the first version with a slightly better one …