The Sigma Club

The presentation by Professor Sir Richard Sorabji of Michael Redhead's paper entitled A SIMPLIFIED VERSION OF GODEL'S THEOREM given to the Sigma Club at The London School of Economics, May 2016

[michaelredhead]

I have made some comments about Godel by Michael Redhead:

1)True means agreement with fact or reality. Correspondence means agreement with statements of objective facts. There are many variations on this notion such as coherence theory, pragmatic theories etc. We follow here the Greek usage!

2) Proof of my result. Since for Godel true means that G is unprovable i.e. it means what it says, which is true, and then goes on to show that it is unprovable, the hard part. In my example unprovability is taken for granted so to speak
and we concentrate on playing about with truth!

3) For our example truth commutes with the universal quantifier, whereas provability does not. Since the theorems are analytically true we can replace proof with truth, in the sense that they express defining properties of the numbers 0, 1, 2 etc. If the theorems were false we would not be talking about numbers, ie if we are talking about numbers then the theorems are true.

4) The step from 3 to 4 is an instance of universal generalisation in quantification theory.

5) Recursion moves from n to n-1 to n-2 etc until it stops at the so-called base case.

6) For equal numbers the number of steps goes abruptly to zero! A suggested example shows how switching rows and columns one at a time increasing strictly from zero as the numbers get more different, which might be considered more satisfactory!

7) If we employ an infinitary logic incorporating the so-called omega-rule, then our argument fails. But such logics cannot be implemented on a machine which has a finite number of operations. This is what Lucas has in mind when he talks of digital machines.

8) General arguments. What do you make of my own argument for a simplified proof.

9) Strict finitism gives a let out for my own argument. What do you think?

10) Lucas conception of an informal semantics. Is this possible? As a way out of our basic conundrum the full force of the Lucas argument applies. But try my simplified example!

11) Do you agree that the syntactic version of Godel is the only reasonable way to go, despite our reservations about consistency.

12) Another point. We are dealing here with the semantic version of the induction axiom. From this perspective Peano's own treatment has, to borrow Russel's phrase, all the advantages of theft over honest toil!