Tag Archives: halting problem

Consequences of the Halting Problem: Gödel’s First Incompleteness Theorem

In my first post on this blog, I defined and proved my favorite fact: The halting problem is undecidable. You should probably read that post if you don’t know what that means, but in brief, it means that it is impossible to write a computer program that will always correctly tell you whether any given computer program will run forever or eventually stop.

I promised to talk about some of the far-reaching consequences of this fact in future posts, and here I am keeping my promise, if a little late. (Don’t get used to me keeping my promises on this blog, though…) I’ll start with arguably the biggest consequence of the halting problem’s undecidability: Gödel’s First Incompleteness Theorem.

As is my m.o., I’ll try to keep the discussion accessible to laypeople; I’ll provide lots of context; and I’ll just generally be extremely long-winded.

This post will be even more long-winded than usual because I feel the need to provide way too much context. To sort of acknowledge that this post is absurdly long, I’ve put a little (*) in the heading of sections that are less tangential. I’m incredibly biased, of course, but I suggest attempting to make it through the whole thing, and if you get tired somewhere around Euclid or Cantor, skip ahead to the next asterisk.

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My Favorite Fact: The Halting Problem

I figure the first topic on my blog should be my favorite. It concerns a question that is fundamental to computer science theory called the halting problem. It’s really not a fact about computers or computer programs at all; it’s just best formulated in that way. It was first solved in 1936 by Alan Turing, and one can argue that the basic question goes back at least to David Hilbert’s 23 problems in 1900 (although Hilbert initially assumed something that was false in his question). It was introduced to me by Professor Eugene Charniak a few years ago in his class on models of computation, and it’s a relatively standard problem that undergrads in computer science learn about at some point.

Most importantly, it’s really beautiful.

I’ll introduce the problem and prove the solution in this post. In a future post, I’ll discuss the wide-ranging consequences.

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