#disinformation — Public Fediverse posts

NBC News: Deal or no deal? Trump’s social media posts add confusion to Iran conflict. “As President Donald Trump’s team was actively working behind the scenes over Memorial Day weekend to deliver an Iran peace deal, the president’s own social posts left a trail of confusion.”

bsky.app/starter-pack... #Accountability #CivicEmpowerment #ClimateEmergency #ClimateJustice #Corruption #Democracy #Crisis #DemocraticResilience #Disinformation #RightsOfNature #Truth

RE: https://bsky.brid.gy/convert/ap/at://did:plc:4sivhpo2an6off6obxrbc3oo/app.bsky.graph.starterpack/3mlvckwj3ev2w

Thursday, May 28, 2026

https://activitypub.writeworks.uk/2026/05/thursday-may-28-2026/

And now, The Donald J. #TrumpVirus #ReflectingPool / #ReflectivePond - with all the best #media #disinformation - like #Fox & the whole anti- #Truth #cult

No #reflection here... No #mindfulness. No #cognitive #ExecutiveFunctioning - like reasoning and understanding cause-effect, planning, sequencing, etc.)

No, Sir Stable Genius and his reflective pond are all about optics, not reflection

He has only concepts of concepts, his wants & fantasy, & all the best words. And #cognitive #WordSalad

@jmcrookston
Found a sovereign citizen leaflet in a cafe today.

In Australia, beside Western Australia, there is no significant separatist sentiment.

But luring people into a fantasy which has them outside the laws of the nation and states seems to be an alternative way to increase divisiveness.

#AusPol #SovCits #SovereignCitizens #Disinformation

@jmcrookston
Found a sovereign citizen leaflet in a cafe today.

In Australia, beside Western Australia, there is no significant separatist sentiment.

But luring people into a fantasy which has them outside the laws of the nation and states seems to be an alternative way to increase divisiveness.

#AusPol #SovCits #SovereignCitizens #Disinformation

@jmcrookston
Found a sovereign citizen leaflet in a cafe today.

In Australia, beside Western Australia, there is no significant separatist sentiment.

But luring people into a fantasy which has them outside the laws of the nation and states seems to be an alternative way to increase divisiveness.

#AusPol #SovCits #SovereignCitizens #Disinformation

@jmcrookston
Found a sovereign citizen leaflet in a cafe today.

In Australia, beside Western Australia, there is no significant separatist sentiment.

But luring people into a fantasy which has them outside the laws of the nation and states seems to be an alternative way to increase divisiveness.

#AusPol #SovCits #SovereignCitizens #Disinformation

@jmcrookston
Found a sovereign citizen leaflet in a cafe today.

In Australia, beside Western Australia, there is no significant separatist sentiment.

But luring people into a fantasy which has them outside the laws of the nation and states seems to be an alternative way to increase divisiveness.

#AusPol #SovCits #SovereignCitizens #Disinformation

Jakub Kalensky co-founded EUvsDisinfo in 2015 and is now a Visiting Professor at the College of Europe in Natolin.

At the Global Forum: FIMI & Hybrid Threats, Jakub argued that measuring disinformation by isolated incidents is like weighing one grain of rice and concluding rice doesn’t feed anyone.

Have a look here: https://www.linkedin.com/feed/update/urn:li:activity:7464995074737590273

#FIMI #Disinformation #EU #HorizonEurope

#NovaraMedia #heatwave #ClimateChange #denialism #disinformation #FossilLobby

(Fred Clark’s Law - correction of Hanlon's Razor: “Sufficiently advanced incompetence is indistinguishable from malice.”
There’s a certain point at which ignorance becomes malice–at which there is simply no way to become that ignorant except deliberately and maliciously.)

https://www.youtube.com/watch?v=xZ2tKOIfHAQ

#climate #ClimateScience #ClimateBreakdown #ClimateDisruption #globalWarming #globalHeating #ExtremeWeather #polycrisis

luma.com/19l5xgr9 "In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide." #booksky #literature #solarpunk #hopepunk

Radical Possibility: How Clima...

luma.com/19l5xgr9 "In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide." #booksky #literature #solarpunk #hopepunk

Radical Possibility: How Clima...

luma.com/19l5xgr9 "In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide." #booksky #literature #solarpunk #hopepunk

Radical Possibility: How Clima...

luma.com/19l5xgr9 "In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide." #booksky #literature #solarpunk #hopepunk

Radical Possibility: How Clima...

luma.com/19l5xgr9 "In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide." #booksky #literature #solarpunk #hopepunk

Radical Possibility: How Clima...

https://luma.com/19l5xgr9

"In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide. Where #authoritarianism thrives on inevitability, fear, and historical amnesia, climate fiction can insist on adaptability, memory, and possibility."

#bookstodon #books #book #literature #solarpunk #hopepunk

https://luma.com/19l5xgr9

"In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide. Where #authoritarianism thrives on inevitability, fear, and historical amnesia, climate fiction can insist on adaptability, memory, and possibility."

#bookstodon #books #book #literature #solarpunk #hopepunk

https://luma.com/19l5xgr9

"In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide. Where #authoritarianism thrives on inevitability, fear, and historical amnesia, climate fiction can insist on adaptability, memory, and possibility."

#bookstodon #books #book #literature #solarpunk #hopepunk

https://luma.com/19l5xgr9

"In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide. Where #authoritarianism thrives on inevitability, fear, and historical amnesia, climate fiction can insist on adaptability, memory, and possibility."

#bookstodon #books #book #literature #solarpunk #hopepunk

https://luma.com/19l5xgr9

"In an era defined by #democratic backsliding, #climate crisis, #disinformation, fear-mongering, and the normalization of cruelty, #speculative #climate #fiction plays an active and necessary role in countering #narratives that divide. Where #authoritarianism thrives on inevitability, fear, and historical amnesia, climate fiction can insist on adaptability, memory, and possibility."

#bookstodon #books #book #literature #solarpunk #hopepunk

Tuesday, May 26, 2026

https://activitypub.writeworks.uk/2026/05/tuesday-may-26-2026/

Tuesday, May 26, 2026

https://activitypub.writeworks.uk/2026/05/tuesday-may-26-2026/

Tuesday, May 26, 2026

https://activitypub.writeworks.uk/2026/05/tuesday-may-26-2026/

Tuesday, May 26, 2026

https://activitypub.writeworks.uk/2026/05/tuesday-may-26-2026/

@ebbot 2/2 to be organized by leadership and propaganda. #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

Pope Leo Warns AI Images Are a ‘Powerful Amplifier’ of Disinformation https://petapixel.com/2026/05/26/pope-leo-warns-ai-images-are-a-powerful-amplifier-of-disinformation/ #ArtificialIntelligence #digitalmanipulation #disinformation #aiimages #popeleo #genai #News #pope

Pope Leo Warns AI Images Are a ‘Powerful Amplifier’ of Disinformation https://petapixel.com/2026/05/26/pope-leo-warns-ai-images-are-a-powerful-amplifier-of-disinformation/ #ArtificialIntelligence #digitalmanipulation #disinformation #aiimages #popeleo #genai #News #pope

Pope Leo Warns AI Images Are a ‘Powerful Amplifier’ of Disinformation https://petapixel.com/2026/05/26/pope-leo-warns-ai-images-are-a-powerful-amplifier-of-disinformation/ #ArtificialIntelligence #digitalmanipulation #disinformation #aiimages #popeleo #genai #News #pope

Pope Leo Warns AI Images Are a ‘Powerful Amplifier’ of Disinformation https://petapixel.com/2026/05/26/pope-leo-warns-ai-images-are-a-powerful-amplifier-of-disinformation/ #ArtificialIntelligence #digitalmanipulation #disinformation #aiimages #popeleo #genai #News #pope

Pope Leo Warns AI Images Are a ‘Powerful Amplifier’ of Disinformation https://petapixel.com/2026/05/26/pope-leo-warns-ai-images-are-a-powerful-amplifier-of-disinformation/ #ArtificialIntelligence #digitalmanipulation #disinformation #aiimages #popeleo #genai #News #pope

@ebbot 2/2 choice narrowed to ideas and objects brought to it attention through propaganda of all kinds. There is consequently a vast and continuous effort going on to capture our minds in the interest of some policy or commodity or idea. #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

It is not usually realized how necessary these invisible governors are to the orderly functioning of our group life. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

@ebbot 2/2 dominated by the relatively small number of persons—a trifling fraction of our hundred and twenty million—who understand the mental processes and social patterns of the masses. It is they who pull the wires which control the public mind, who harness old social forces and contrive new ways to bind and guide the world. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

We are governed, our minds molded, our tastes formed, our ideas suggested, largely by men we have never heard of. This is a logical result of the way in which our democratic society is organized. Vast numbers of human beings must cooperate in this manner if they are to live together as a smoothly functioning society. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

The conscious and intelligent manipulation of the organized habits and opinions of the masses is an important element in democratic society. Those who manipulate this unseen mechanism of society constitute an invisible government which is the true ruling power of our country. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

@ebbot 2/2 change the world that Edward Bernays, among others, made for us. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

@ebbot 2/2 just lying, but betrayal. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

By 1928, the word%27s troubling connotations had not faded: on the contrary. Throughout the decade there had been a gradual, disorienting revelation of just how systematically, and how ingeniously, the Allied governments had fooled the peoples of two great democracies, Great Britain and, in particular, the USA. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

@ebbot 2/2 be documented and sustained against perversion and betrayal. It can be elaborated and developed steadily and widely without personal, local and sectional misunderstanding. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

Propaganda will never die out. Intelligent men must realize that propaganda is the modern instrument by which they can fight for productive ends and help to bring order out of chaos. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

Here we are again on #MathsMonday, looking at the questionable statements of SmartmanApps, #debunking the #disinformation and seeing what real #mathematics it accidentally uncovers.

Our now-familiar friend says that infinitesimals are a standard thing taught in schools. This is false, with the possible exception of wherever he teaches, because infinitesimals have by and large been ejected from mathematics due to being unnecessary and confusing.

But what is true is that infinitesimals *used* to be taught, and he has the very old textbooks to prove it. Let’s find one of those textbooks, and its definition of infinitesimal:

> lim a = 0 means that a is an infinitesimal quantity

(Advanced Algebra, p226 by Collins, J. 1911. American Book Company. Symbols altered for typographical necessity)

So we need to know what “lim x = a” means, which is furnished to us on page 297 (it is no credit to this textbook that it uses undefined terms without even an apology):

> If a variable x assumes a given sequence of values such that the numerical difference between a constant a, and the variable x becomes and remains less than any assignable quantity, however small, then x is said to approach a as its limit [written] lim x = a

This is essentially the way Cauchy defined limits, and is the same as the definition of a limit I gave last time, with two changes:

1. It’s less easily translated into the symbolic version due to not using directly equivalent language like “for every”.
2. Instead of a sequence, we talk of a “variable”.

It is the second difference which is more consequential. In modern mathematics, in any given context, a variable only takes on a single value. If you think of x as possibly taking on multiple values, say 2 and 0 in alternation, then the following argument no longer works:

“x is not less than 1, and x is not greater than 1, therefore x is equal to 1”.

Imagine asking “is x less than 1?” and finding that x could be 2, so answering, “no”; then we may ask “is x greater than 1?” and find that x could be 0, so answer “no” again. But concluding that x is equal to 1 is not valid. So, we must be smarter about such arguments. Maybe our Smart friend is that smart, but I would prefer not to depend on it.

The solution modern mathematics has taken to this is that if we want to consider a changing quantity, we encode it into a different kind of object: a function. The function accepts a parameter, and that parameter determines what value it returns. Thus the dependence of that value on the parameter is made explicit; we save ourselves from the possibility of getting confused, because “x(0) is not less than 1, and x(1) is not greater than 1, therefore x(2) is equal to 1” is obviously rubbish!

In school it is typical still to describe functions in the old-fashioned language of variables, so you probably remember seeing a linear function described by “y = mx + c”, where x is a variable, and y another variable which *depends* on it. Further on in school it’s typical to replace “y” with “f(x)” which makes this dependence explicit.

Because functions are an entirely different type of thing, we also never compare them to numbers without first describing exactly how that comparison should happen. We don’t ask, “is f equal to 1”? Because the answer is clearly “no”, and similarly f is not less than or greater than 1; f is a function, not a number so these questions don’t even really make sense. This is not the case with variables; it is perfectly reasonable to ask whether x is equal to 1 - this kind of question is embedded in every single equation a school pupil is asked to solve.

Clearly, mathematicians and pupils of yesteryear were all able to work like this, but it is unnecessarily confusing and error-prone.

## Sequences

So, we will use functions instead of variables, and since it will not cause any extra difficulties, we will specifically use sequences which are functions whose inputs are natural numbers. This turns Cauchy’s definition of the limit into the modern one, and it turns his definition of an infinitesimal into “a sequence with a limit of zero”.

We have some questions to answer: while it is natural to compare a variable to numbers (such as when solving a system of inequalities) comparing a function (or sequence) to a number is less natural. What does “sin < ½” mean? Is it true, or false? Clearly sin(x) < ½ for some x but not for others.

Nonetheless, if we want sequences to occupy the role of infinitesimals, we need to be able to perform such comparisons, because the fundamental property of an infinitesimal e is that 0 < e < 1/n for all natural numbers n. Thus how we set up infinitesimals is fundamentally a matter of how we set up rules for comparing the sequences which represent them.

Let us first point out that instead of comparing sequences to *numbers* we can replace any fixed number with the constant sequence all of whose elements are that fixed number. And secondly, let’s establish that the *normal* equality of sequences is simply that each of their corresponding elements must be equal, that is, their first elements must be equal to each other, their second elements must be equal to each other, and so on. This means that there are many sequences whose limit is zero but which are not equal to one another. One of those sequences is the constant sequence all of whose elements are zero, i.e. the sequence which we are using instead of the number zero, written (0, 0, 0, …)

I should point out here that it is a standard fact about limits that the limit of a constant sequence is always that constant. This should be easy to see from the definition, but in a university course you would prove it rigorously. This means, also, that any sequence whose limit is zero must be different from any constant sequence, except that the zero sequence is equal to itself. This is good for our use of sequences which converge to zero (i.e. have limit zero) as infinitesimals, since they should be different from numbers larger than zero.

## Equivalence and Order

But we still have to do more work, because to be infinitesimals, these sequences need to be smaller than 1/n, for every natural number n. We know the first step to making sense of this: replacing 1/n with the sequence (1/n, 1/n, 1/n, …). The next step is creating a notion of “smaller” and “larger” sequences.

In what follows, I will fix a sequence e representing an infinitesimal, e := (1, ½, ⅓, ¼, …), and x representing a small finite positive number, x:= (0.1, 0.1, 0.1, …)

### Global Domination

One very simple way of setting up an ordering is to say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1 and a2 < b2 and a3 < b3, and so on, for each index. This is no good for our purpose, because look at e and x. If we are doing things right, we should have e < x, but e1 = 1 is not less than x1 = 0.1. Even worse, if we go to the 11th element, x11 = 0.1 is not less than e11 = 1/11, so neither e < x nor x < e: this is not a **total** ordering, i.e. sometimes two objects just do not lie in any particular order.

### Lexicographic Ordering

The lexicographic (so called because it’s the ordering used by dictionaries - i.e. lexicons) ordering is perhaps the next most obvious way of ordering something that’s made out of multiple things which are themselves ordered.

Using this ordering we’d say that (a1, a2, a3, …) < (b1, b2, b3, …) if a1 < b1, or a1 = b1 and a2 < b2, or a1 = b1 and a2 = b2 and a3 < b3, or … and so on.

This *does* give us a total order; we can compare any two sequences and either one is less than the other, or they are exactly the same sequence. However, we still have a problem because we find that since 0.1 < 1, we have defined x < e, which is not what we wanted.

This ordering fails to capture the idea that it is the *eventual* behaviour of infinitesimals that we care about, and instead makes the early behaviour the most important.

### Eventual Domination

We can’t just do a “reverse lexicographic order” because we can’t start at the *last* element of a sequence composed of infinitely many elements.[^4] But we can do something else: we can say that (a1, a2, a3, …) ≤ (b1, b2, b3, …) if *from some point* all the a’s are smaller than all the b’s. You can hopefully see that, with this ordering, e ≤ x: from the 11th element onwards, all e’s are below 0.1. And indeed if y is *any* constant sequence consisting of rational numbers, eventually e will drop below that number and stay below.

It is not for no reason that I used ≤ in this definition instead of <. That is because this is again not a total order among all possible sequences: a sequence could oscillate above and below another sequence forever.

### Ordinary Real Numbers

What Cantor did in his development of real analysis was to notice that there is a certain kind of sequence that behaves very nicely, which nowadays we call a *Cauchy sequence*. It has the property that as the sequence continues, its values become arbitrarily close to all later values:

> A sequence a is *Cauchy* if, for every ε > 0, there is a number N such that for all n, m both greater than N, |a(n) - a(m)| < ε

The structure Cantor worked with was all of *these* sequences (of rational numbers) rather than arbitrary sequences. Then, he said that two sequences were equivalent if the difference between them becomes arbitrarily small. If not, then (one can prove) the difference between the values of a and b is either eventually positive, in which case we say a is larger, or eventually negative and we say b is larger.

We can go on to define arithmetic between such sequences and prove that everything works out wonderfully but there is one snag for us: there are *no* infinitesimals! What was the fate of our supposedly infinitesimal sequence e? Well, unfortunately the difference between it and the zero sequences becomes arbitrarily small, so they are rendered equivalent: any sequence which is “infinitesimal” in the Cauchy sense becomes merely zero.

[^4]: Unless the infinite order type is a *successor ordinal* which is a concept beyond the scope of this post, and is not the case for our sequences.

#math #maths

1/2

@ebbot 2/2 newspaper seeks to purvey news, it seeks to purvey entertainment. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

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@ebbot 2/2 be good housekeeping, or smart apparel, or beauty in home decoration, or debunking public opinion, or general enlightenment or liberalism or amusement. --Edward Bernays, Propaganda (1928) #propaganda #disinformation #misinformation #fakenews #uniparty #EconomicHappinessMachine

Von der Leyen to head to Lithuania for drone crisis talks – POLITICO

Lithuania issued an air alert on Wednesday after a stray drone was detected close to the country’s border…
#Lithuania #LT #Europe #Europa #EU #airdefense #airspace #AndriusKubilius #baltics #belarus #borders #crisis #défense #DISINFORMATION #drones #latvia #Lietuva #lithuania #military #missions #naujienos #oil #Procurement #Russia #safety #Ukraine #UrsulavonderLeyen #warinukraine
https://www.europesays.com/3015034/

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