1490: "Atoms"

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Re: 1490: "Atoms"

Postby ilduri » Tue Feb 24, 2015 11:24 pm UTC

da Doctah wrote:"What the hell are all those phlogistons doing in there, anyway?"

Well, as any good alchemist knows, the human body naturally contains some elemental fire. That's why we exhale phlogisticated air, after all. :)
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Re: 1490: "Atoms"

Postby LockeZ » Tue Feb 24, 2015 11:37 pm UTC

Sudden realization: That was never actually his mom. That was a nuclear-powered robot maid that his dad was in a relationship with.

Quercus wrote:would you expect a wafer of monocrystalline silicon to necessarily contain any atoms of beryllium? Or a drop of ultrapure water?


Difficult to say for sure without doing some science. Find me a drop of ultrapure water the size of a baseball, and I'll test it for you.
Last edited by LockeZ on Tue Feb 24, 2015 11:42 pm UTC, edited 2 times in total.

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Re: 1490: "Atoms"

Postby Kit. » Tue Feb 24, 2015 11:39 pm UTC

Quercus wrote:I'll accept pretty readily that this is probably true for any unprocessed environmental sample, but would you expect a wafer of monocrystalline silicon to necessarily contain any atoms of beryllium?

I would expect one gram of pure 11N monocrystalline silicon to contain an order of million beryllium atoms.

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Re: 1490: "Atoms"

Postby Quercus » Tue Feb 24, 2015 11:51 pm UTC

Kit. wrote:
Quercus wrote:I'll accept pretty readily that this is probably true for any unprocessed environmental sample, but would you expect a wafer of monocrystalline silicon to necessarily contain any atoms of beryllium?

I would expect one gram of pure 11N monocrystalline silicon to contain an order of million beryllium atoms.


Oooh, awesome. I love it when big numbers fight.

I concur, assuming that impurities in silicon reflect the abundance of elements in the earth's crust (which seems just about reasonable as a first approximation) - I should have just run the numbers for myself at the start, but I was lazy.

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Re: 1490: "Atoms"

Postby rmsgrey » Wed Feb 25, 2015 12:05 am UTC

Quercus wrote:
Kit. wrote:
Quercus wrote:I'll accept pretty readily that this is probably true for any unprocessed environmental sample, but would you expect a wafer of monocrystalline silicon to necessarily contain any atoms of beryllium?

I would expect one gram of pure 11N monocrystalline silicon to contain an order of million beryllium atoms.


Oooh, awesome. I love it when big numbers fight.

I concur, assuming that impurities in silicon reflect the abundance of elements in the earth's crust (which seems just about reasonable as a first approximation) - I should have just run the numbers for myself at the start, but I was lazy.


The 11N makes me suspect that the "pure" doesn't refer to the "silicon" but to a specific doping of silicon...

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Re: 1490: "Atoms"

Postby Quercus » Wed Feb 25, 2015 12:12 am UTC

rmsgrey wrote:The 11N makes me suspect that the "pure" doesn't refer to the "silicon" but to a specific doping of silicon...


Huh? I thought 11N was just "11 nines" i.e. >99.999999999% purity. Are you saying that the reason it's "only" 11N is because most of the rest is a specific dopant?

Edit: It appears that's not the case - you can buy 11N undoped silicon, and it seems to be the highest grade available. I do wonder if it's 11N because that's how pure it is, or because that's the detection limit to which whatever they use to detect impurities is certified. If it's the latter our calculations could be off by several orders of magnitude (although they could also be off if most of the impurity is something specific encountered during the manufacturing process, so I doubt it really matters).... I have now reached the limit of my pedantry.

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Re: 1490: "Atoms"

Postby Klear » Wed Feb 25, 2015 10:44 am UTC

LockeZ wrote:Sudden realization: That was never actually his mom. That was a nuclear-powered robot maid that his dog was in a relationship with.


FTFY

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Re: 1490: "Atoms"

Postby Copper Bezel » Wed Feb 25, 2015 10:58 am UTC

Klear wrote:
LockeZ wrote:Sudden realization: That was never actually his mom. That was a nuclear-powered robot maid that his dog was in a relationship with.


FTFY

Win.
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Re: 1490: "Atoms"

Postby rmsgrey » Wed Feb 25, 2015 2:03 pm UTC

Quercus wrote:
rmsgrey wrote:The 11N makes me suspect that the "pure" doesn't refer to the "silicon" but to a specific doping of silicon...


Huh? I thought 11N was just "11 nines" i.e. >99.999999999% purity. Are you saying that the reason it's "only" 11N is because most of the rest is a specific dopant?

Edit: It appears that's not the case - you can buy 11N undoped silicon, and it seems to be the highest grade available. I do wonder if it's 11N because that's how pure it is, or because that's the detection limit to which whatever they use to detect impurities is certified. If it's the latter our calculations could be off by several orders of magnitude (although they could also be off if most of the impurity is something specific encountered during the manufacturing process, so I doubt it really matters).... I have now reached the limit of my pedantry.


Ah - I was confusing purity level with n-type and p-type semiconductors.

It's one of those bits of trade jargon that everyone seems to assume that you either know already, or have no reason to know, so Google is powerless to explain it...

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Re: 1490: "Atoms"

Postby orthogon » Wed Feb 25, 2015 2:14 pm UTC

rmsgrey wrote:It's one of those bits of trade jargon that everyone seems to assume that you either know already, or have no reason to know, so Google is powerless to explain it...

It's also one of those things that's quite hard to google anyway, because essentially the thing you're looking for is just "N". There's probably a term like "x-N" or "N-notation", but you need to know that term in order to find it in the first place. I had a colleague once who was trying to write a quiz that couldn't be solved easily using the web. He wasn't very successful: he apparently had little understanding of what was easy or hard to Google, since most of his attempts would yield to a single word search. But his idea was a very interesting one. It's probably been done now, but this was back in the late '90s. (Google was easier to outsmart in those days, too).
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Re: 1490: "Atoms"

Postby sevenperforce » Wed Feb 25, 2015 2:40 pm UTC

orthogon wrote:
rmsgrey wrote:It's one of those bits of trade jargon that everyone seems to assume that you either know already, or have no reason to know, so Google is powerless to explain it...

It's also one of those things that's quite hard to google anyway, because essentially the thing you're looking for is just "N". There's probably a term like "x-N" or "N-notation", but you need to know that term in order to find it in the first place. I had a colleague once who was trying to write a quiz that couldn't be solved easily using the web. He wasn't very successful: he apparently had little understanding of what was easy or hard to Google, since most of his attempts would yield to a single word search. But his idea was a very interesting one. It's probably been done now, but this was back in the late '90s. (Google was easier to outsmart in those days, too).

Google is not good at searching for a range of numbers, at searching for an uncommon name that is easily misspelled, at searching for certain types of trade jargon which have very close analogues outside their field, and so forth.

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Re: 1490: "Atoms"

Postby Quercus » Wed Feb 25, 2015 3:20 pm UTC

Yeh, that's a tough one to google without any context - I remembered vaguely that xN notation was in some way related to purity of stuff (I think the context was bottled gases, rather than silicon, though), so I could at least search for "silicon purity n numbers", which is a much more productive search than anything that doesn't include the word "purity".

In this case it's even worse than trade jargon being similar to something outside the field - because the N number means something totally different to N-type in the same field, and I'm not sure it's even possible to get google to tell those apart (does google index hyphens?).

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Re: 1490: "Atoms"

Postby Neil_Boekend » Wed Feb 25, 2015 3:32 pm UTC

Normal bottled gasses only go to 6 nines. For anything more you need to install purifiers (although some bottles have a built in purifier). With a purifier 7 through 9 nines is quite doable, although it does mean increasingly smooth and short supply lines with special connections and mechanics that have experience with ultrapure lines. A single fingerprint can destroy an installation if you need these kind of purities.

11 nines is, well, let's call it really really interesting to design.
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Re: 1490: "Atoms"

Postby Kit. » Wed Feb 25, 2015 4:16 pm UTC

Semiconductor manufacturers use zone melting to achieve higher degrees of purity.

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Re: 1490: "Atoms"

Postby Quercus » Wed Feb 25, 2015 4:29 pm UTC

Kit. wrote:Semiconductor manufacturers use zone melting to achieve higher degrees of purity.


aaand down the Wiki-hole we go!

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Re: 1490: "Atoms"

Postby Quizatzhaderac » Fri Feb 27, 2015 4:54 pm UTC

Jackpot777 wrote:
Andries wrote:So we had fundamental forces on Friday, and atoms today.

What are we getting next? The alphabet?


Image

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|{{}}|, ha, ha, ha, ha.
|{{},{{}}}|, ha, ha, ha, ha.
|{ {}, {{}}, {{},{{}}} }|, ha, ha, ha, ha.
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Re: 1490: "Atoms"

Postby Pfhorrest » Fri Feb 27, 2015 6:23 pm UTC

Quizatzhaderac wrote:
Jackpot777 wrote:
Andries wrote:So we had fundamental forces on Friday, and atoms today.

What are we getting next? The alphabet?


Image

"Three! Three types of XKCD cartoons, ah ha ha ha!!!"
{{}}, ha, ha, ha, ha.
{{},{{}}}, ha, ha, ha, ha.
{ {}, {{}}, {{},{{}}} }, ha, ha, ha, ha.

It kind of bothers me how sloppy logic it is for mathematicians to treat the cardinality of the set as identical to the set itself. There needs to be some cardinality() function. {} is not zero, cardinality({}) is zero. { {}, {{}}, {{},{{}}} } is not three, cardinality({ {}, {{}}, {{},{{}}} }) is three. cardinality({5, blue, aardvark}) is also three, but {5, blue, aardvark} is clearly not identical to { {}, {{}}, {{},{{}}} }, they just have the same cardinality.
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Re: 1490: "Atoms"

Postby ucim » Fri Feb 27, 2015 6:41 pm UTC

Pfhorrest wrote: {} is not zero, cardinality({}) is zero.
You can't say the latter until you have defined what "zero" is. The former defines zero. In the latter, you are using a pre-existing definition of zero. So, the latter can't come before the former.

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Re: 1490: "Atoms"

Postby Quizatzhaderac » Fri Feb 27, 2015 7:05 pm UTC

There needs to be some cardinality() function.
Duly noted, I've edited my post.

If I wasn't joking I'd typically never work with a set of type:
<? extends Set<Void> || Set<? extends Set<Void> || Set<Set<Void>> || Set<? extends Set<Void> || Set<Set<Void>>>>
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Re: 1490: "Atoms"

Postby Pfhorrest » Fri Feb 27, 2015 7:11 pm UTC

ucim wrote:
Pfhorrest wrote: {} is not zero, cardinality({}) is zero.
You can't say the latter until you have defined what "zero" is. The former defines zero. In the latter, you are using a pre-existing definition of zero. So, the latter can't come before the former.

Not at all. Define zero as "the cardinality of the empty set". That's what it already is defined as, but then people speak as though "the empty set" simpliciter means the same thing as "the cardinality of the empty set". And that the cardinality of the set that contains only the empty set (which defines the number one) is identical to that set that contains only the empty set itself, and so on. When a set is clearly not identical to its cardinality, as different non-identical sets have the same cardinality. As it turns out there is only one set with the cardinality of the empty set, which is the empty set, which is why that's a useful place to start defining things, but the cardinality of that set is still not the same thing as the set itself.
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Re: 1490: "Atoms"

Postby Sizik » Fri Feb 27, 2015 7:23 pm UTC

Pfhorrest wrote:It kind of bothers me how sloppy logic it is for mathematicians to treat the cardinality of the set as identical to the set itself. There needs to be some cardinality() function. {} is not zero, cardinality({}) is zero. { {}, {{}}, {{},{{}}} } is not three, cardinality({ {}, {{}}, {{},{{}}} }) is three. cardinality({5, blue, aardvark}) is also three, but {5, blue, aardvark} is clearly not identical to { {}, {{}}, {{},{{}}} }, they just have the same cardinality.

http://en.wikipedia.org/wiki/Set-theore ... al_numbers
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Re: 1490: "Atoms"

Postby Pfhorrest » Fri Feb 27, 2015 10:31 pm UTC

Yes Sizik, I am obviously aware of that and nitpicking something about it.
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Re: 1490: "Atoms"

Postby ucim » Fri Feb 27, 2015 10:58 pm UTC

Pfhorrest wrote:Not at all. Define zero as "the cardinality of the empty set". That's what it already is defined as, but then people speak as though "the empty set" simpliciter means the same thing as "the cardinality of the empty set".
If you define zero as "the cardinality of the empty set" then you've defined zero. You're done.

However, you can instead define zero as "the color of mustard". This admittedly would not be a very useful definition, and not a lot of math would result from it. But you could do it; the caveat being that you must first throw out the prior ("cardinality of") definition, and everything that follows from it.

Likewise, you could (after throwing out all prior definitions of zero) define zero as "the empty set itself" and come up with some cockamamie rules for deriving the other integers. You might think that this is as silly as "the color of mustard", but it turns out that the system thus derived is actually rather interesting. It has useful mathematical properties. In fact, it seems to have the very same mathematical properties that we are used to with the conventional definition of zero, the integers, and arithmetic.
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It might be clearer if we had defined "glorph" as "the empty set itself", and come up with rules that describe "unf", "drobbie", "twix", etc. in terms of "glorph"; we would have discovered the same system under a different name.
Anyway, having done that (defined zero as "the empty set", etc.), we might then consider the question of what the cardinality of a set is. That is, having invented an abstract symbol-manipulating system (that happens to look a lot like the counting numbers and such), we would then need to explicitly associate these symbols and ideas with the idea of quantity.

You are taking this (association) as a given. It is not. That's the source of the discord (as I see it).

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Re: 1490: "Atoms"

Postby rmsgrey » Sat Feb 28, 2015 12:04 am UTC

ucim wrote:Likewise, you could (after throwing out all prior definitions of zero) define zero as "the empty set itself" and come up with some cockamamie rules for deriving the other integers. You might think that this is as silly as "the color of mustard", but it turns out that the system thus derived is actually rather interesting. It has useful mathematical properties. In fact, it seems to have the very same mathematical properties that we are used to with the conventional definition of zero, the integers, and arithmetic.


It would be more accurate to say that you can define conventional arithmetic on top of such a system, and, as a simpler system (though less immediately intuitive) there's less to assume in constructing it, so it's more likely that there's no gremlins lurking to bring the whole thing crashing down in a puff of inconvenient logic...

To do arithmetic, it turns out all you need is a structure for performing mathematical induction - a base entity and a rule which, given any entity in your list, lets you find the unique next entity in your list - with the added constraint that no two entities share the same next entity. In other words: if you can count without ever reaching a highest number, you can do arithmetic with those numbers.

The entities you end up with by following this approach may not look like anything we think of as numbers, but they work the same way, and they are constructed, literally, out of nothing, rather than requiring us to assume the existence of an infinite number of numbers with certain relations between them before we can even start. And the fact we can construct an actual system that allows arithmetic means that there isn't some subtle contradiction lurking that would make arithmetic logically inconsistent - you can attempt to define a system by listing its properties, but not every list of properties can be satisfied by a single system (for example, it turns out "powerful enough to do arithmetic in" and "free from undecidable propositions" are incompatible with each other) - being able to look at the list of properties we associate with numbers and arithmetic and show that there's a system that has all those properties means we can do arithmetic without worrying about what the underlying system is - we know there's at least one system for which everything we're proving holds true.

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Re: 1490: "Atoms"

Postby Pfhorrest » Sat Feb 28, 2015 12:15 am UTC

ucim wrote:You are taking this (association) as a given. It is not. That's the source of the discord (as I see it).

I'm not actually. I'm more calling for it to be made explicit. Starting with the empty set and then constructing a bunch of other sets in a systematic way that gets you a bunch of objects that you can manipulate exactly like we do numbers is great. Then we make explicit that when we are talking about quantity, that is to say cardinality, that is to say number as ordinarily numberstood, we do not say "the number of (members in the set) Stooges is identical to the number of (members in the set) Musketeers which is identical to { {}, {{}}, {{},{{}}} }", but rather "…which is identical to the number of (members in the set) { {}, {{}}, {{},{{}}} }".
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Re: 1490: "Atoms"

Postby ucim » Sat Feb 28, 2015 12:36 am UTC

We can talk about the cardinality of a set by doing a one-to-one comparison without ever counting. So it makes sense to say the cardinality of stooges is the same as the cardinality of three (as defined using {} for zero), because "three" is a set, just like stooges is.

The thing we can't say, without the extra step, is that the cardinality of stooges is three, any more than we can say that the cardinality of stooges is musketeers. We first have to show that three (the set) is a good way to count things... which establishes that three (the set) is the same as three (the integer we are familiar with).

Then we have the interesting result that, for this specific set of Sets (the integers as defined through sets), the cardinality of a member of this Set (an integer) is equal to the set itself (the common value of the integer it represents).

It's an interesting result, but not an inconsistent one.

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Re: 1490: "Atoms"

Postby rmsgrey » Sat Feb 28, 2015 12:39 am UTC

Pfhorrest wrote:
ucim wrote:You are taking this (association) as a given. It is not. That's the source of the discord (as I see it).

I'm not actually. I'm more calling for it to be made explicit. Starting with the empty set and then constructing a bunch of other sets in a systematic way that gets you a bunch of objects that you can manipulate exactly like we do numbers is great. Then we make explicit that when we are talking about quantity, that is to say cardinality, that is to say number as ordinarily numberstood, we do not say "the number of (members in the set) Stooges is identical to the number of (members in the set) Musketeers which is identical to { {}, {{}}, {{},{{}}} }", but rather "…which is identical to the number of (members in the set) { {}, {{}}, {{},{{}}} }".


The trouble with talking about cardinality here is that to add |{ {}, {{}}, {{},{{}}} }| and |{ {}, {{}}, {{},{{}}}, {{},{{}},{{},{{}}}} }| and get the right answer, you have to talk about the cardinality of a set I'm not going to try writing out, which is not obviously related to the two sets you're adding (apart from having them both as subsets, and members, and subsets of members and members of members and...) - to get from sets A and B to set C without invoking a prior concept of addition - obviously, blind mice plus seasons gives days of the week, but only because we already know the answer - you need to back off from cardinality and go back to the process by which the sets were constructed in the first place - 0++++0++++ = (0++++0+++)+ = (0++++0++)++ = (0++++0+)+++ = (0++++0)++++ = (0+++)++++ = 0+++++++.

Having to process the concept of cardinality for each number in the process is an added step - if you're invoking the sets at all, then invoke them directly; if you're not invoking them, then you don't need to worry about cardinality either...

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Re: 1490: "Atoms"

Postby Pfhorrest » Sat Feb 28, 2015 12:53 am UTC

ucim wrote:We can talk about the cardinality of a set by doing a one-to-one comparison without ever counting. So it makes sense to say the cardinality of stooges is the same as the cardinality of three (as defined using {} for zero), because "three" is a set, just like stooges is.

The thing we can't say, without the extra step, is that the cardinality of stooges is three, any more than we can say that the cardinality of stooges is musketeers.

And I'm saying that the only reason why we couldn't do that is if we foolishly defined the integers as identical to sets, rather than as identical to the cardinalities of those same sets. "Three" shouldn't be the name of a set, that goes completely against the ordinary usage of numbers. "Three" should be a size of (many) sets, which size exactly is specified by reference to one particular set in a series constructed systematically from {}.
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Re: 1490: "Atoms"

Postby ucim » Sat Feb 28, 2015 4:45 am UTC

Pfhorrest wrote:"Three" shouldn't be the name of a set, that goes completely against the ordinary usage of numbers.
...but there's no such thing as "numbers" at this point. Not if we start with {} and some set operations. All these set operations can be done without numbers. We happened to give them the names we use for numbers, because we know how it will turn out. But even if we didn't, and gave them different names, we'd still end up with a system that is isomorphic to arithmetic. We just wouldn't know it, until some genius came out and said:

Lookie here - if we call "mook" 0, and "ourim" 1, and "tavek" 2, and "twine" 3... and we gavok ourim and tavek, we get twine. That's the same as if we add 1 and 2; we get 3. And it works for all of them! I've discovered an isomporphism! (And that genius would publish a paper and win the Fields medal.)

And that would accomplish the same thing as the "extra step" referred to upthread.

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Re: 1490: "Atoms"

Postby Minerva » Sat Feb 28, 2015 2:29 pm UTC

The rated life for the electronics in a pacemaker in the early 1970s was only 5 years.


I'm impressed with that NEJM letter... if the pacing configuration is still working well for the patient, and our Voyager-lady is still ticking along happily 40 years later, then everyone wins. No further surgery is required to change the pacemaker, and as the paper notes, it's extremely cost effective healthcare over the lifetime.

So far, there have been no local or systemic consequences of prolonged exposure to ionizing radiation


The paper notes that, but it seems like it wouldn't be significant at all - external radiation emission from the case of the 238Pu RTG would be essentially nonexistent.
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Re: 1490: "Atoms"

Postby Kit. » Mon Mar 02, 2015 1:22 pm UTC

Pfhorrest wrote:
{{}}, ha, ha, ha, ha.
{{},{{}}}, ha, ha, ha, ha.
{ {}, {{}}, {{},{{}}} }, ha, ha, ha, ha.

It kind of bothers me how sloppy logic it is for mathematicians to treat the cardinality of the set as identical to the set itself. There needs to be some cardinality() function. {} is not zero, cardinality({}) is zero. { {}, {{}}, {{},{{}}} } is not three, cardinality({ {}, {{}}, {{},{{}}} }) is three. cardinality({5, blue, aardvark}) is also three, but {5, blue, aardvark} is clearly not identical to { {}, {{}}, {{},{{}}} }, they just have the same cardinality.

It's not like "three" or "3" or "III" or "1+1+1" were logically any different from "{ {}, {{}}, {{},{{}}} }". They are all just labels of different degrees and ways of convenience. The "braced" one is rarely convenient, though.

We can have several isomorphisms between the ordinals and the symbols to express them. Some of these isomorphisms can also be homomorphisms preserving a structure that might be of interest to us.

ucim wrote:because "three" is a set,

Is square root of 3 a set?
Is square root of 3 squared a set? How would you know?

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Re: 1490: "Atoms"

Postby RobinEJ » Mon Mar 02, 2015 1:51 pm UTC

Kit. wrote:
ucim wrote:because "three" is a set,

Is square root of 3 a set?
Is square root of 3 squared a set? How would you know?

TypeError: sqrt expected a Real, but got a Nat.

This series of sets is isomorphic to the natural numbers, not the real numbers, so your question doesn't make sense.

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Re: 1490: "Atoms"

Postby Kit. » Mon Mar 02, 2015 2:09 pm UTC

RobinEJ wrote:
Kit. wrote:
ucim wrote:because "three" is a set,

Is square root of 3 a set?
Is square root of 3 squared a set? How would you know?

TypeError: sqrt expected a Real, but got a Nat.

Throw it away and try google.

RobinEJ wrote:This series of sets is isomorphic to the natural numbers,

This statement is quite different from '"three" is a set'.

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Re: 1490: "Atoms"

Postby Pfhorrest » Mon Mar 02, 2015 9:54 pm UTC

Kit. wrote:
RobinEJ wrote:This series of sets is isomorphic to the natural numbers,

This statement is quite different from '"three" is a set'.

Yes, and that is really my point.

We can construct a series of sets that are isomorphic to the natural numbers, but that doesn't mean that they are the numbers. The reason they are isomorphic to the natural numbers is because they start at size zero and add one member at each step in the series, so that their sizes are the natural numbers. Which makes them a great reference for defining the natural numbers: "the natural numbers {0,1,2,3,…} are defined as the sizes of the sets { {}, {{}}, {{},{{}}}, {{},{{},{{}}}}, ...}". And you can do whatever you can do to those sets and the size of the resulting set will be the number you would have gotten if you did those things to the numbers whose sizes the input sets were. That's wonderful! But it doesn't mean that the sets are the numbers.
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Re: 1490: "Atoms"

Postby Kit. » Mon Mar 02, 2015 10:39 pm UTC

Pfhorrest wrote:Which makes them a great reference for defining the natural numbers: "the natural numbers {0,1,2,3,…} are defined as the sizes of the sets { {}, {{}}, {{},{{}}}, {{},{{},{{}}}}, ...}".

Not really, as that:

1. gets it backward (the natural numbers are not just the sizes of "braced-only" sets. 3 is not only the size of { {}, {{}}, {{}, {{}}} }, but also the size of { 0, 1, 2 }, and the size of { {}, {{{{8}}}}, {1,2,{3,4},5,6} } as well);

2. makes ω and other infinite ordinals(*) "natural numbers" by definition (as there is nothing in the set theory itself that would stop transfinite induction over "braced-only" sets... which was exactly the point of introducing the "braced-only" induction).

Edit: *) well, maybe "cardinals", depending on how we define "sizes".

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Re: 1490: "Atoms"

Postby rmsgrey » Tue Mar 03, 2015 1:46 am UTC

Kit. wrote:
ucim wrote:because "three" is a set,

Is square root of 3 a set?
Is square root of 3 squared a set? How would you know?


" 'three' is a set" refers to "three" in a specific context.

"Three" is a mobile phone network. That doesn't mean it makes sense to ask about the squares and square roots of mobile phone networks...

The label "three" is used to refer to many different things - just within mathematics, it can refer to a natural number, an integer, a rational number, a real number, a complex number, an element of any of an infinite number of sets, groups, rings, fields, etc of various sizes, and so on. In most cases, "three" in mathematics refers to something analogous to the natural number three, and if "one", "two", and "plus" all make sense in that context, "three" is usually going to be "one" "plus" "two".

So, when you ask whether the square root of three is a set, the question that needs to be answered first is what the context of "three" and "square root" is. If you're dealing with "three" in a context where you've derived the natural numbers from the empty set, and haven't extended your list of numbers to anything else, but have defined operations for addition, multiplication and exponentiation, as well as their inverses where the answers exist, then "the square root of three" is nonsense - in that context, the square root operation can only be applied to square numbers.

You can extend your concept of number by defining positive rationals as equivalence classes of ordered pairs of sets, where the second set in any pair cannot be the empty set, and the equivalence relation is derived from multiplication ([a,b]=[c,d] if, for some r, either a*r=c and b*r=d or a=c*r and b=d*r) and again by defining positive reals as the limits of convergent sequences of equivalence classes of ordered pairs of sets, with the appropriate conditions. In that context, with a suitable definition of "square root", "the square root of three" refers to the shared limit of any of a number of convergent sequences of equivalence classes of ordered pairs of sets. You could, in principle, if you were sufficiently bloody minded, write out an expression in terms of nested braces representing each of the sets in a suitable member of the nth equivalence class of a suitable convergent sequence of them.

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Re: 1490: "Atoms"

Postby PM 2Ring » Tue Mar 03, 2015 5:52 am UTC

I find those Von Neumann ordinal sets hard to read in monochrome - I prefer them in colour...

0 : {}
1 : {{}}
2 : {{}, {{}}}
3 : {{}, {{}}, {{}, {{}}}}
4 : {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}
5 : {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}}

If you'd like to see them as Venn diagrams, see this post from a few years ago: Are integers arbitrary?.

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Re: 1490: "Atoms"

Postby Kit. » Tue Mar 03, 2015 10:09 am UTC

rmsgrey wrote:" 'three' is a set" refers to "three" in a specific context.

"Three" is a mobile phone network. That doesn't mean it makes sense to ask about the squares and square roots of mobile phone networks...

The label "three" is used to refer to many different things - just within mathematics, it can refer to a natural number, an integer, a rational number, a real number, a complex number, an element of any of an infinite number of sets, groups, rings, fields, etc of various sizes, and so on. In most cases, "three" in mathematics refers to something analogous to the natural number three, and if "one", "two", and "plus" all make sense in that context, "three" is usually going to be "one" "plus" "two".

Sorry, then it's probably not ucim who I should have quoted. The argument still stays.

There is no point in wasting a label "three" on such things as { {}, {{}}, {{}, {{}}} }. There is nothing interesting enough in this particular set to give it any particular label (as opposed to {} and ω, for example). There's also nothing interesting that relates this particular set to the same-named item in the ring of integer (Z is an initial object in the category of rings, that's why the names for objects in Z are so well-represented in other areas of mathematics). There's also no reason why, while trying to reuse "integer" names for sets, we wouldn't call {{{{}}}} "three" as well - after all, it's even a better fit, because {{{{}}}} was created from {} using arithmetic, not transfinite, induction.

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Re: 1490: "Atoms"

Postby rmsgrey » Tue Mar 03, 2015 1:55 pm UTC

Kit. wrote:
rmsgrey wrote:" 'three' is a set" refers to "three" in a specific context.

"Three" is a mobile phone network. That doesn't mean it makes sense to ask about the squares and square roots of mobile phone networks...

The label "three" is used to refer to many different things - just within mathematics, it can refer to a natural number, an integer, a rational number, a real number, a complex number, an element of any of an infinite number of sets, groups, rings, fields, etc of various sizes, and so on. In most cases, "three" in mathematics refers to something analogous to the natural number three, and if "one", "two", and "plus" all make sense in that context, "three" is usually going to be "one" "plus" "two".

Sorry, then it's probably not ucim who I should have quoted. The argument still stays.

There is no point in wasting a label "three" on such things as { {}, {{}}, {{}, {{}}} }. There is nothing interesting enough in this particular set to give it any particular label (as opposed to {} and ω, for example). There's also nothing interesting that relates this particular set to the same-named item in the ring of integer (Z is an initial object in the category of rings, that's why the names for objects in Z are so well-represented in other areas of mathematics). There's also no reason why, while trying to reuse "integer" names for sets, we wouldn't call {{{{}}}} "three" as well - after all, it's even a better fit, because {{{{}}}} was created from {} using arithmetic, not transfinite, induction.


You still need to define addition (and other operations) in a way that makes sense for {{{{}}}} + {{{}}} if you're going to get {{{{{{}}}}}} out of it. And personally, I find "three = {zero, one, two}" to be more appealing than "three = {two}" - in the former case, you're counting your list of numbers you already have to get your new number; in the latter, you're counting layers of recursion, and the "threeness" of the object is only apparent when it's expressed as "{{{zero}}}".

Like I said before, the point here is not to come up with a practical notation, but to shore up the theoretical underpinnings of the common notation by proving that there is some set which has the properties we say the natural numbers have. Once you've established that some such set exists, you can then do arithmetic without worrying about what "three" might be in this new context.

Once you have natural numbers with defined labels, you can do whatever maths you like with the labels, knowing that your results all apply to any set which can be built up recursively from a base element and an injective successor operation, so long as your arithmetic operations, etc, are based on the base element and successor operation of your set - for example, if you want your set to be the integers starting at 42, then 42+43 would be 43 not 85, and so on.

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Re: 1490: "Atoms"

Postby Neil_Boekend » Tue Mar 03, 2015 2:08 pm UTC

Somehow, every time I see this comic tread I think "You got some nice atoms. It would be a shame if a vacuum instability were to happen to it. Better pay your 'insurance'. "
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