Patrick Stevens

Karma: 24

Patrick Stevens Aug 28, 2016, 8:50 PM
2 points
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on: Empty set
Ilia Zaichuk Thanks for the edit! I made a couple of linguistic changes, and made the “uniqueness of $\emptyset$ ” a bit less compact.

Patrick Stevens Aug 17, 2016, 1:01 AM
2 points
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on: Project proposal: Intro to numbers
The non-existence of a total order on $C$ is fun and interesting, I think, and also not very difficult. An excellent exercise in proof by contradiction.

Patrick Stevens Aug 15, 2016, 3:05 AM
1 point
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in reply to: Patrick Stevens’s comment on: Rice’s Theorem
I think the answer is no. Indeed, there are uncountably many $S$ , but only countably many machines which can access oracles.

Patrick Stevens Aug 14, 2016, 11:41 PM
1 point
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on: Rice’s Theorem
Surely they are equivalent. Given a Rice-deciding oracle, we can ask the oracle, “Does the partial function defined by machine $[n]$ specify where input $k$ should go?”; that determines whether $[n]$ halts on input $k$ or not.

Patrick Stevens Jul 31, 2016, 2:16 AM
3 points
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on: Rice’s Theorem
I think the halting problem probably should have its own page, rather than being linked to the umbrella uncomputability page.

Patrick Stevens Jul 23, 2016, 1:17 PM
1 point
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in reply to: Eric Rogstad’s comment on: Freely reduced word
Are you otherwise broadly Math 3? It would be good to have a guinea pig for group theory.

Patrick Stevens Jul 23, 2016, 1:15 PM
1 point
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in reply to: Kevin Clancy’s comment on: Division of rational numbers (Math 0)
I think this actually belongs in the Multiplication article, but you’re quite right that I’ve not been explicit enough. I intend to have a meditation on the various ways that the notation is consistent, but this one doesn’t need any division at all so it should appear earlier.

Patrick Stevens Jul 23, 2016, 5:12 AM
1 point
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in reply to: Eric Rogstad’s comment on: Freely reduced word
It sounds like you didn’t already know what the free group is; in that case (and even if you did already know), it’s very gratifying to know that someone is actually reading this carefully!

Patrick Stevens Jul 23, 2016, 5:09 AM
2 points
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in reply to: Eric Rogstad’s comment on: Freely reduced word
You’re quite right to flag this up; I was being sloppy. There are three main ways to construct the free group, and I’ve kind of mixed together the two of them which are most intuitive. I’m trying not to simply define the free group here, but you’re right that I’ve done it confusingly. I’ll fix it.

Patrick Stevens Jul 10, 2016, 1:11 AM
6 points
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in reply to: alexei’s comment on: Uncountability
A summary of the relevant cardinal arithmetic, by the way (in the presence of choice): $ℵ_{α} + ℵ_{α} = ℵ_{α} = ℵ_{α} ℵ_{α}$ while $2^{ℵ_{α}} > ℵ_{α}$

Patrick Stevens Jul 7, 2016, 4:16 AM
1 point
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on: Löb’s theorem
Something I learnt from Mietek Bak is that Löb’s Theorem is kind of more subtle than this. In provability theory, it’s fine to have a “box” operator that we informally read as “is provable”; but what Löb’s theorem tells us that we can’t simply interpret it literally as “is provable” without difficulties. One should define the “provability” predicate formally, to avoid getting confused (or one should specify that it is simply a formal symbol to which we have not assigned any semantic meaning, although that is somewhat against the point of the angle taken by the parent article); for example, the provability predicate could be defined by a certain first-order formula which unpacks a Gödel number, checks it’s encoding a proof and verifies each step of the encoded proof.

Patrick Stevens Jul 4, 2016, 5:18 PM
1 point
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on: Report likelihoods, not p-values
Have I gone mad, or do you mean “L(H|e) is simply the probability of H given that the the actual data e occurred”?

Patrick Stevens Jul 3, 2016, 3:03 PM
1 point
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in reply to: Kevin Clancy’s comment on: Real number
You’re right; I was sloppy. I’ll fix it, thanks.

Patrick Stevens Jun 29, 2016, 3:45 PM
1 point
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in reply to: Joe Zeng’s comment on: Injective function
I’ve edited something about that into the text. Basically I think it’s to do with the symmetry of the words in “one-to-one”: it looks like it should go both ways, as “one thing in the domain hits one thing in the range, and vice versa”.