housecarpenter

10 ms·

61.
by housecarpenter 4y ago
The reason sets are important is that they correspond to (Boolean-valued) properties. To each set, there corresponds the property of belonging to that set. To each property, there corresponds the set of all things with that property. I thin
62.
by housecarpenter 4y ago
What's sad about it?
63.
by housecarpenter 4y ago
I'm late replying to this, but---you need the axiom of choice once you have a family of non-empty sets rather than a single non-empty set. At some point you have to go into the formalism to really understand it. It's really about
64.
by housecarpenter 4y ago
I agree that infinity doesn't exist in the universe, but this isn't a problem for mathematics. Mathematics isn't about what exists in the universe, it's about what exists in the realm of concepts. Some concepts are more
65.
by housecarpenter 4y ago
I don't think Jech is using "naturally" as in "natural choice"; he's just using it as a synonym for "obviously". If X is non-empty set then there exists an x in X, because that's what it means fo
66.
by housecarpenter 4y ago
It's true that any real number can be written with a finite number of symbols. In fact, I can write any real number x with exactly one symbol, namely x, if I define x to mean that particular real number. Now if you fix a particular for
67.
by housecarpenter 4y ago
¬ is shift+` on the most common UK keyboard layout, but yeah, it would probably need to be popular in America .
68.
by housecarpenter 4y ago
I get your point that both of these are specialized notations/terminologies, but it's entirely possible for somebody to understand algorithmic complexity without knowing what "a rack, a host, a DC" is, there's no in
69.
by housecarpenter 4y ago
I'm a long time user of EditPad Pro. I use Sublime Text and VS Code as well, but one thing I find EditPad Pro particularly good for is modifying the syntax highlighting schemes, or writing my own, which seems to be a more involved proc
70.
by housecarpenter 4y ago
Yeah, televised election debates weren't a thing in the UK until 2010.
71.
by housecarpenter 4y ago
They state that only as a disclaimer, after a bunch of other statements that hint at the opposite. Just adding a disclaimer to misleading writing merely makes it confusing and unclear for those who are paying attention, and still misleading
72.
by housecarpenter 4y ago
In the RDBMSes I'm aware of, you can disable foreign key checks while still adding the constraints themselves---they won't do anything as far as the DBMS is concerned, but can presumably still be picked up by whatever tool you are
73.
by housecarpenter 4y ago
It depends what you mean by "simplification". The fact that phonics is something that can be taught and can be helpful for learning to read indicates that English does have phonetic spelling rules, inconsistently applied as they m
74.
by housecarpenter 4y ago
There are languages like that---the distinction is referred to as "proximative" vs. "obviative". (Though strictly speaking, it differentiates between "more topical" and "less topical" third persons, w
75.
by housecarpenter 4y ago
But I don't get paid on a piece rate; the amount of time I spend working is constant. Anything that increases my productivity just means I get more work done. (Others may differ, but I know from experience that I like to keep to a fixe
76.
by housecarpenter 4y ago
I'd be interested to hear why you think this, or if you can link to something that explains this perspective.
77.
by housecarpenter 4y ago
Since all three notions are closely interrelated, it's fairly trivial to take any one you choose as "fundamental" and define the others in terms of it: - Sets as fundamental: define ordered pairs via Kuratowski's definit
78.
by housecarpenter 5y ago
Any species could prove the same theorems given the same axioms, but (besides the fact that they might not choose the same axioms) I'm not sure if they would prove the same subset of theorems that we have proven/will prove. Pe
79.
by housecarpenter 5y ago
I am certainly not intending to intentionally spread FUD, though I guess you probably mean that I am unintentionally propagating FUD spread by other people. All I mean by saying it's a "logical error" is that it's not an
80.
by housecarpenter 5y ago
You can define closed sets positively as well. A closed set is one which includes its whole boundary. An open set is one which excludes its whole boundary. Clearly then it is possible for a set to contain only part of its boundary, in which
81.
by housecarpenter 5y ago
You might be interested to know that in intuitionistic logic, you still have a one-sided contrapositive law, i.e. (A > B) > (~B > ~A): [1] A > B(assumption) [2] ~B (assumption) [3] A (assumption) [4] B (modus ponens on
82.
by housecarpenter 5y ago
The issue is that some people believe in intuitionistic logic, so to them, there's a difference between assuming something is true and deriving absurdity in order to prove its negation (which they think is perfectly fine, and which is
83.
by housecarpenter 5y ago
I think when mathematicians call an operation denoted by juxtaposition a "product", that's usually because it's denoted by juxtaposition for some independent reason and they're drawing the syntactic analogy with the
84.
by housecarpenter 5y ago
It's entirely possible to trust something to work one way (e.g. keep a trivially restorable backup of previous state) when you do one thing with it and not trust it to do the same thing when you do another thing with it (especially if
85.
by housecarpenter 5y ago
This may be somewhat off-topic, but I think the point of "you lose debug stack traces!" is that with tail call elimination, you have more possibility for confusion from people who inadvertently write a tail call, or who write some
86.
by housecarpenter 5y ago
Python uses ARC plus a tracing GC for dealing with cycles, same as PHP: https://devguide.python.org/garbage_collector/ Perl uses ARC with no tracing GC, so you have to handle cycles yourself: https://www.per
87.
by housecarpenter 5y ago
& is also used for set intersection in Python. I think + for string concatenation isn't too bad, really. It fits in with the fact that length(s + t) = length(s) + length(t), the same way we write A × B for Cartesian product (since
88.
by housecarpenter 5y ago
It depends what you mean by tuple. In Python, tuples are basically just immutable lists. Just as lists with 1 element are useful, so are tuples with 1 element. You might be dealing with a tuple of unknown length, where the length could be 1
89.
by housecarpenter 5y ago
I use it, personally. The other two options I find too aesthetically displeasing: not indenting the string looks bad when it's within an indented block of code, and using join and putting the strings in a list is just too much boilerpl
90.
by housecarpenter 5y ago
It's not really an operator. It's part of the syntax of string literals. "foo" "bar" is an alternative way of writing the string literal "foobar". If foo is not a string literal, foo "bar" i

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