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Gödel's Incompleteness
I’ve mentioned Gödel’s incompleteness theorems many times on this blog, but I’ve never actually written about them in detail. I was asking, on twitter, for topics that readers would be interested in,...
View ArticleG&oum;del Numbering: Encoding Logic as Numbers
The first step in Gödel’s incompleteness proof was finding a way of taking logical statements and encoding them numerically. Looking at this today, it seems sort-of obvious. I mean, I’m writing this...
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Defining Properties Arithmetically (part 1): Gödel and Primitive Recursion
When I left off, we’d seen how to take statements written in the logic of the Principia Mathematica, and convert them into numerical form. What we need to see now is how to get meta-mathematical. We...
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The Meta of Gödel
As you may be figuring out, there’s a reason why I resisted walking through Gödel’s proof of incompleteness for so long. Incompeteness isn’t a simple proof! To refresh your memory, here’s a sketch of...
View ArticleFinally: Gödel’s Proof of Incompleteness!
Finally, we’re at the end of our walkthrough of Gödel great incompleteness proof. As a refresher, the basic proof sketch is: Take a simple logic. We’ve been using a variant of the Principia...
View ArticleGodel (Reposts)
I’m going to be on vacation this week, which means that I won’t have time to write new posts. But my friend Dr. SkySkull was just talking about Gödel on twitter, and chatting with him, I realized that...
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Godel Numbering
The first step in Gödel’s incompleteness proof was finding a way of taking logical statements and encoding them numerically. Looking at this today, it seems sort-of obvious. I mean, I’m writing this...
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Godel part 2: Arithmetic and Logic
In the last post, we saw how to take statements written in the logic of the Principia Mathematica, and convert them into numerical form using Gödel numbering. For the next step in Gödel’s proof, we...
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Gödel part 3: Meta-Logic with Arithmetic
On to the next part of Gödel’s proof of incompleteness. To refresh your memory, here’s a sketch of the proof: Take a simple logic. We’ve been using a variant of the Principia Mathematica’s logic,...
View ArticleGÖDEL PART 4: The Payoff
After a bit of a technical delay, it’s time to finish the repost of incompleteness! Finally, we’re at the end of our walkthrough of Gödel great incompleteness proof. As a refresher, the basic proof...
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