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- scott on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943
- Peter Smith on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943
- scott on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943
- David Auerbach on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943
- Peter Smith on Gödel’s First Theorem, from Gödel 1931 to Kleene 1943
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Category Archives: Logic
Gödel’s First Theorem, from Gödel 1931 to Kleene 1943
As ‘homework’, before writing more of the second edition of my Gödel book, I’m reading through the literature to see how others have handled the First Incompleteness Theorem, both in the early papers from Gödel on, and then in the … Continue reading
Posted in Gödel's theorems
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Carnap and the Diagonalization Lemma (Continued)
Let’s distinguish what I’ll call the Diagonalization Equivalence from the familiar Diagonalization Lemma. The former is a semantic claim: in the right conditions, for any one-place predicate of theory T there is a corresponding sentence such that is true if … Continue reading
Posted in Gödel's theorems
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Carnap and the Diagonalization Lemma
Carnap is often credited with proving the Diagonalization Lemma in Logische Syntax der Sprache. But where does he do it? Well, in §35 Carnap notes the general recipe for taking a one-place predicate and constructing a sentence such that is … Continue reading
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Somewhat gappy Gödel
When planning and actually writing my Introduction to Gödel’s Theorems, I intentionally consulted other books as little as possible, trying to reconstruct strategies and proofs from memory as far as I could. I thought that would be a good discipline, … Continue reading
Posted in Gödel's theorems
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IGT2 A first instalment of the second edition of my Gödel book!
As I’ve said before, CUP have agreed to publish a second edition of An Introduction to Gödel’s Theorems. Camera-ready copy is due to be sent to them rather implausibly soon, by the end of July 2012, with publication six months … Continue reading
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Next up: Truth, Gödel, and other delights
OK: a review of Maddy’s very engaging recent book (written with Luca Incurvati, which was fun to do) has gone off to Mind. And in the next day or two, I must also put together a review for Phil. Math. … Continue reading
Posted in Books, Gödel's theorems
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Maddy on mathematical depth
This is a very belated follow-up to an earlier post on Penelope Maddy’s short but intriguing Defending the Axioms. In my previous comments I was talking about Maddy’s discussion of Thin Realism vs Arealism, and her claim that there in … Continue reading
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And who is for 0-ary function expressions?
In defining a first order syntax, there’s a choice-point at which we can go two ways. Option (A): we introduce a class of sentence letters (as it might be, ) together with a class of predicate letters for different arities … Continue reading
Posted in Logic
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Two-place functions aren’t one-place functions, are they?
Here’s a small niggle, that’s arisen rewriting a very early chapter of my Gödel book, and also in reading a couple of terrific blog posts by Tim Gowers (here and here). We can explicitly indicate that we are dealing with … Continue reading
Posted in Logic
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Tennenbaum’s Theorem Workshop
Sean Walsh organized a one-day workshop on the philosophical significance of Tennenbaum’s Theorem on Saturday. It kicked off with me presenting a short piece that Tim Button and I have forthcoming in Philosophia Mathematica: here’s a preprint of our paper. … Continue reading