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Ilia Zaichuk Thanks for the edit! I made a couple of linguistic changes, and made the "uniqueness of " a bit less compact.
Ilia Zaichuk: I've made the appropriate changes to the markup to make text display in MathJax (which is the LaTeX-syntax markup language used for maths on this site and on Stack Exchange). However, I think it's a bug in Arbital (which I've just pointed out to the developers) that it's not rendering correctly. (EDIT: I've altered the markup into a form that works around the bug. It just makes the markup look a bit less nice.)
In general, you can use \text{text here}; if you want to put maths inline with the text here, you can use dollar signs:
\text{Heinz $57$ Varieties}
Additionally, if you have a mathematical string you want to typeset, like "sin", you can use \mathrm{sin} which shows up as . (It so happens that there is already a built-in symbol that does that for sin: \sin.)
This is not universally agreed-upon, but I use " decides whether or not holds" to mean " outputs if holds, and outputs otherwise".
If I said " decides if holds", I would consider that ambiguous: it might mean " outputs if holds" without the requirement on 's behaviour if doesn't hold.
Looks good to me!
The non-existence of a total order on is fun and interesting, I think, and also not very difficult. An excellent exercise in proof by contradiction.
I think the answer is no. Indeed, there are uncountably many , but only countably many machines which can access oracles.
Surely they are equivalent. Given a Rice-deciding oracle, we can ask the oracle, "Does the partial function defined by machine specify where input should go?"; that determines whether halts on input or not.
I think the halting problem probably should have its own page, rather than being linked to the umbrella uncomputability page.
Simply that I didn't know the name :) I'll edit it in.
Thanks: quite correct.
I don't think this is what you mean, is it?
Are you otherwise broadly Math 3? It would be good to have a guinea pig for group theory.
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.
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!
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.
A summary of the relevant cardinal arithmetic, by the way (in the presence of choice): while
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.
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"?
You're right; I was sloppy. I'll fix it, thanks.
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".
To the original author: xkcd images are CC BY-NC (2.5), and as such require attribution.
This page doesn't disambiguate between "left inverse" and "inverse". Strictly an "inverse" is a two-sided inverse, so gf = 1 and fg = 1.
A question about the requisites for this page: should the alternating group on five elements is simple be a requisite? It's necessary for the base case of the induction, but one can probably understand the proof without it, simply referring to it as a known fact.
I think this probably wants a diagram of the two graphs, being differently laid out in the plane but isomorphic.
This is definitely a page which admits two lenses: the "easy" proof and the "theory-heavy" proof. What kind of lens design might people use?
"identity" is probably not a sufficiently specific link; I'd go for math_identity, probably.
I feel like symmetric_group should be a requisite for this page. However, this page is linked in the body of symmetric_group, so it seems a bit circular to link it as a requisite. I think this situation probably comes up for most child pages; what's good practice in such cases?
None that I'm aware of, but I've found it convenient to know when I was doing exercises in a first course in group theory.
I took the plunge and put it on its own page.
Request for comment: is the definition of "cycle" something that should be on its own page? They're not about the symmetric group per se, but I've only heard of cycles being used in the context of symmetric groups.
I have a question about general Arbital practice here. A mathematician will probably already know what a group homomorphism is, but they probably also don't need the proofs of the Properties, for instance, and they don't need the explanation of the trivial group. Should I have split this up into different lenses in some way?