wektor

Well-known member
I'd be interesting in trying some, provided I have a comfortable understanding of the biochemical impact. Tried gingko biloba, verdict inconclusive.

Do you have any recommendations?
I've been on and off noopept for longer (5 month-ish) periods of time and would highly recommend it, GABA felt like nice clarity when I tried it a while ago, lately I'm only taking some ashwaganda extract and aged ginseng extract (tastes like coldbrew). cannot tell whether it actually wakes me up as I'm still quite bemused after coveed.
 

poetix

we murder to dissect
Cartesian join between sets looks like "for every set A, and for every set B, there exists a set AxB such that for all x in A, and for all y in B, (x, y) is in AxB
@poetix taught himself set theory a few years ago. hes now considered a luminary in the field so it can be done. not by me, but it can be done.
I am not of course considered a luminary in the field of set theory. But I do know a bit.

Ordering of sets is an interesting topic.

If you have a set containing only natural numbers (positive counting numbers: 0, 1, 2...), then there's always a "least" element of that set, no matter how many numbers are in it (including if there are infinitely many). If 0 is in your set, then it has to be 0; if not, then the next number after 0, (i.e. 1), or the next after that, etc. This is the Well-Ordering principle.

If you have a set of other things, like all the rational numbers (fractions) greater than 0, and a function that uniquely maps every element of this set to a natural number, then you can pick out a "least" element of this set just by picking out the element that maps to the smallest natural number. This may seem counterintuitive. There isn't a "smallest" fraction greater than 0 - if you give me one, I can always come up with one smaller just by adding 1 to the denominator, e.g. you give me 1/1,000,000, and I go "yes, but 1/1,000,001 is even smaller" ad infinitum. But there is a way to uniquely map the rational numbers to the natural numbers, and this means we can say there's an "nth" rational number for every natural number n, which means that for any set of rational numbers we can say which comes first in that ordering, even if it isn't the smallest number in the set.

Once we get into real numbers, things get gnarly. There's no unique mapping from the natural numbers to the real numbers (as Cantor proved), so we can't rely on that trick to pick a "least" real number. But we can well-order the reals using the axiom of choice, which says that for any set of sets, there's a set containing just one element from each set (example: I have a big bag containing three smaller bags - one of fruit, one of sex toys, and one of books. The axiom of choice says that there exists a "choice function" which will give me a bag containing one thing chosen from each of the smaller bags - an apple, a cock ring and a copy of 1984). How do we do it? We take the set containing only the set of real numbers, and use the choice function magically granted to us by the axiom of choice to give us a set containing just one element from that one set - picking out an arbitrary real. That's our "first" real. Then we take that number out of the set we started with, and do the "choice" thing again - that gets us our "second" real. And so on.

The axiom of choice is equivalent to the Well-Ordering Theorem, which says that every set can be ordered in this way (but doesn't say how).

If we restrict our universe of sets to those which can be constructed by taking the empty set and repeatedly doing things to it to get larger sets, then every set in this universe has a unique "name" which is the smallest sequence of things you have to do to the empty set to arrive at it, and these "names" are countable, which means that both the axiom of choice and the Well-Ordering Theorem are automatically true in the "constructible universe". It's a bit like saying a) you can put computer programs in order, and b) every set we're prepared to consider is one that could be generated by a computer program, so c) the least element of any set is the element that is generated by the lowest-ordered computer program.
 

wektor

Well-known member
If we restrict our universe of sets to those which can be constructed by taking the empty set and repeatedly doing things to it to get larger sets, then every set in this universe has a unique "name" which is the smallest sequence of things you have to do to the empty set to arrive at it, and these "names" are countable, which means that both the axiom of choice and the Well-Ordering Theorem are automatically true in the "constructible universe". It's a bit like saying a) you can put computer programs in order, and b) every set we're prepared to consider is one that could be generated by a computer program, so c) the least element of any set is the element that is generated by the lowest-ordered computer program.
I suppose that it intuitively makes way more sense to start from the empty set, at least we know how we got to each specific set.
That makes me think, how are numbers constructed in classical mathematics?

From my limited knowledge of constructivism, I recall a number would be constructed as a choice sequence in which every time we pick we constrain the amount of our possible future choices to one specific element, effectively making the same choice every time. Or something like that.
 

poetix

we murder to dissect
There are various possible encodings of the natural numbers into sets.

For example, start with the empty set {} = 0. For each n, let n + 1 be the set containing every number up to n. So 1 must be the set containing 0, i.e. the set containing the empty set, {{}}. 2 is the set containing 0 and 1, hence {{{}}, {}}. And so on.

We can also encode an ordered pair in various ways - for example in the Kuratowski encoding the pair (p, q) is the set {{p}, {p, q}}.

This means we can encode the rational numbers as ordered pairs: 1/2 becomes (1, 2), and so on. Written in full, that's {{{{}}}, {{{}}, {{{}}, {}}}}.
 

IdleRich

IdleRich
Cartesian join between sets looks like "for every set A, and for every set B, there exists a set AxB such that for all x in A, and for all y in B, (x, y) is in AxB

I am not of course considered a luminary in the field of set theory. But I do know a bit.

Ordering of sets is an interesting topic.

If you have a set containing only natural numbers (positive counting numbers: 0, 1, 2...), then there's always a "least" element of that set, no matter how many numbers are in it (including if there are infinitely many). If 0 is in your set, then it has to be 0; if not, then the next number after 0, (i.e. 1), or the next after that, etc. This is the Well-Ordering principle.

If you have a set of other things, like all the rational numbers (fractions) greater than 0, and a function that uniquely maps every element of this set to a natural number, then you can pick out a "least" element of this set just by picking out the element that maps to the smallest natural number. This may seem counterintuitive. There isn't a "smallest" fraction greater than 0 - if you give me one, I can always come up with one smaller just by adding 1 to the denominator, e.g. you give me 1/1,000,000, and I go "yes, but 1/1,000,001 is even smaller" ad infinitum. But there is a way to uniquely map the rational numbers to the natural numbers, and this means we can say there's an "nth" rational number for every natural number n, which means that for any set of rational numbers we can say which comes first in that ordering, even if it isn't the smallest number in the set.

Once we get into real numbers, things get gnarly. There's no unique mapping from the natural numbers to the real numbers (as Cantor proved), so we can't rely on that trick to pick a "least" real number. But we can well-order the reals using the axiom of choice, which says that for any set of sets, there's a set containing just one element from each set (example: I have a big bag containing three smaller bags - one of fruit, one of sex toys, and one of books. The axiom of choice says that there exists a "choice function" which will give me a bag containing one thing chosen from each of the smaller bags - an apple, a cock ring and a copy of 1984). How do we do it? We take the set containing only the set of real numbers, and use the choice function magically granted to us by the axiom of choice to give us a set containing just one element from that one set - picking out an arbitrary real. That's our "first" real. Then we take that number out of the set we started with, and do the "choice" thing again - that gets us our "second" real. And so on.

The axiom of choice is equivalent to the Well-Ordering Theorem, which says that every set can be ordered in this way (but doesn't say how).

If we restrict our universe of sets to those which can be constructed by taking the empty set and repeatedly doing things to it to get larger sets, then every set in this universe has a unique "name" which is the smallest sequence of things you have to do to the empty set to arrive at it, and these "names" are countable, which means that both the axiom of choice and the Well-Ordering Theorem are automatically true in the "constructible universe". It's a bit like saying a) you can put computer programs in order, and b) every set we're prepared to consider is one that could be generated by a computer program, so c) the least element of any set is the element that is generated by the lowest-ordered computer program.
Just a couple of points to add there. Firstly it's worth saying that the Axiom of Choice (even though to you, me and anyone I would have thought, it sounds pretty intuitive) is not universally accepted so any consequences that derive from it are kinda up for challenge. Sometimes a theorem will be stated but there will be a kinda proviso saying "Proof depends on the Axiom of Choice" or "only known proof depends on AOC" and it's common that if someone proves something using the AofC then they will then try and we find a proof that does not require it. Or demonstrate that no proof can be made without it.
Not sure if I'm explaining the difference well but the first case
a) Proof can only be done using AofC means that if we reject the axiom then we reject the result
b) The only extant proof depends on AofC means that if we reject the axiom then we are merely back to not knowing whether or not the result is correct.

This bit... I expect that you know more about this than I ever knew, and certainly more than I remember but

If we restrict our universe of sets to those which can be constructed by taking the empty set and repeatedly doing things to it to get larger sets, then every set in this universe has a unique "name" which is the smallest sequence of things you have to do to the empty set to arrive at it, and these "names" are countable

What is this construction? It seems that if we restrict our universe of sets to a countable number of sets to a countable number then that is very.... restrictive.
Like if we have the set of real numbers (in fact any uncountable set) then that has an uncountable number of subsets doesn't it (in that every single element on its own can be a set)? So that countable list of sets generated from the empty set cannot contain any uncountable sets... is that right?
 

IdleRich

IdleRich
Just a couple of points to add there. Firstly it's worth saying that the Axiom of Choice (even though to you, me and anyone I would have thought, it sounds pretty intuitive) is not universally accepted
I suppose that I should say that it IS intuitive and obviously correct if the number of sets is finite, it's only (mildly) controversial if there are infinitely many sets to choose from.,
 

IdleRich

IdleRich
Yesterday clearing up the living room with the GF and trying to put records on the shelves led to me saying to her "There comes a point where you literally can't keep adding records to a shelf - it's not like Hilbert's Hotel". I'm sure I would not have mentioned Hilbert's Hotel if this thread had not somehow brought it to the top of my mind.

Hilbert's Hotel
 

poetix

we murder to dissect
Just a couple of points to add there. Firstly it's worth saying that the Axiom of Choice (even though to you, me and anyone I would have thought, it sounds pretty intuitive) is not universally accepted so any consequences that derive from it are kinda up for challenge. Sometimes a theorem will be stated but there will be a kinda proviso saying "Proof depends on the Axiom of Choice" or "only known proof depends on AOC" and it's common that if someone proves something using the AofC then they will then try and we find a proof that does not require it. Or demonstrate that no proof can be made without it.
Not sure if I'm explaining the difference well but the first case
a) Proof can only be done using AofC means that if we reject the axiom then we reject the result
b) The only extant proof depends on AofC means that if we reject the axiom then we are merely back to not knowing whether or not the result is correct.

This bit... I expect that you know more about this than I ever knew, and certainly more than I remember but



What is this construction? It seems that if we restrict our universe of sets to a countable number of sets to a countable number then that is very.... restrictive.
Like if we have the set of real numbers (in fact any uncountable set) then that has an uncountable number of subsets doesn't it (in that every single element on its own can be a set)? So that countable list of sets generated from the empty set cannot contain any uncountable sets... is that right?
I may have oversimplified (or been misled by my own computer program analogy) - the messy details are described here: https://en.m.wikipedia.org/wiki/Constructible_universe#L_can_be_well-ordered - it does seem that, within a given level of the constructible hierarchy, there's effectively a Godel-numbering for every formula that can "pick out" a set within that level, which if I'm not mistaken implies countability.
 

IdleRich

IdleRich
Yeah, I wasn't saying that you're wrong, I think I'm just saying that that countable number of sets within that construction must itself only contain countable sets. That is correct isn't it? Or am I missing something.
 

poetix

we murder to dissect
Yes, it's a very constrained universe - you're basically saying "sure, real numbers exist, but only the ones we can actually point to". But it does have the interesting property that every set within it is well-ordered, and the axiom of choice is just true within that universe by definition (your choice function in all cases is just "pick the 'first' element according to the well-ordering of this set"). I'm not quite sure about countability - perhaps it's true of any given level of the hierarchy, but not of the hierarchy in toto (since it extends upwards indefinitely).
 

IdleRich

IdleRich
I think that I somehow decided arbitrarily that it wasn't supposed to be so constrained and then complained that it was. In a nutshell.
 

luka

Well-known member
what youre engaged in is a hunt for metaphor and homology and isomorphism rather than an earnest attempt to master any of these particular disciplines. which is cool. but also maybe a waste of time.
Prynne’s March 1982 letter to Crick, which goes on to offer structural isomorphism as the
condition for that deliberate sign left by the good scientist:
what might be required could be a recognisably non-random match (say, numerical or
morphological) between one part of the system and another which was functionally and
causationally distant from it. Such a match would be the message in a bottle, confirming
that we are survivors and not self-made.
As Katko helpfully points out, though not in reference to this particular letter, a ‘non-random match’
of this sort—what Needham describes as ‘a system of ratios and relations, which may be possibly the
same in all animals, in a word, a chemical ground-plan of animal growth’—is a key feature of the PTM’s
biochemical hinterland
 

Clinamenic

Binary & Tweed
Latest visualization format, in the interest of keeping it all within one grand abstraction. Y axis is the day's date in descending order (July in turquoise, August in gold, arbitrarily), and X axis is topic, with the 21 daily topics spaced out as the blue bars, in order of the focus of the day, as noted by the diagonal plotting of white cells:

Screen-Shot-2021-07-31-at-11-40-42-AM.png


Every cell allows for a note to be made and viewed as the cursor hovers over the cell. I have left a few rows of undated space between each cycle, to allow room for more formal experimentation, perhaps having to do with linking various topics together via arrows that move laterally across the gray space, in the manner of a piping infrastructure. The task, here, would be to further integrate the vast and otherwise unwieldy body of knowledge being accumulated.
 

Clinamenic

Binary & Tweed
As of now, each day has one primary topic and one complementary topic, but there are 21 topics total so each cycle is 21 days.
 

Clinamenic

Binary & Tweed
Monday, July 19, 2021: Economics / Finance
Supplement: Maths
Tuesday, July 20, 2021: Statecraft / Law
Supplement: Sociology
Wednesday, July 21, 2021: Geopolitics
Supplement: World History
Thursday, July 22, 2021: Philosophy
Supplement: Religion / Metaphysics
Friday, July 23, 2021: Business / Marketing
Supplement: Psychology
Saturday, July 24, 2021: Sociology
Supplement: Biology
Sunday, July 25, 2021: Literature
Supplement: Music

Monday, July 26, 2021: Maths
Supplement: Architecture
Tuesday, July 27, 2021: Physics
Supplement: Neuroscience
Wednesday, July 28, 2021: Chemistry
Supplement: Medicine
Thursday, July 29, 2021: Biology
Supplement: Statistics
Friday, July 30, 2021: Psychology
Supplement: Business / Marketing
Saturday, July 31, 2021: Engineering
Supplement: Physics
Sunday, August 1, 2021: Computer Science
Supplement: Geopolitics

Monday, August 2, 2021: Religion / Metaphysics
Supplement: Philosophy
Tuesday, August 3, 2021: Neuroscience
Supplement: Computer Science
Wednesday, August 4, 2021: Statistics
Supplement: Economics / Finance
Thursday, August 5, 2021: World History
Supplement: Statecraft / Law
Friday, August 6, 2021: Medicine
Supplement: Chemistry
Saturday, August 7, 2021: Music
Supplement: Literature
Sunday, August 8, 2021: Architecture
Supplement: Engineering
 

Clinamenic

Binary & Tweed
Now in color:

Screen-Shot-2021-07-31-at-4-19-51-PM.png


Might experiment with weaving the color of the day somehow into the daily stimulus, reinforcing the intra-day integration of information. The colors were chosen arbitrarily, based on what instinct the topic invoked, and are, as everything else, subject to revision.
 
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