This is a ramification of the contents of the thread: 0.999...= 1 started by @jorndoe. Thanks jorndoe

First off, the simplest infinity that we know of is the natural numbers: N = {1, 2, 3,...}. The infinite set N is generated by the simple iteration of adding one to the preceding number as so: 1; hen 1 + 1 = 2; then 2 + 1 = 3; and so on and so forth.

What I'm particularly concerned about is the ratio between consecutive elements in the set N. The ratios look like below:

1) 1; there is no ratio here as there is no natural number that precedes 1
2) 2; the ratio is 1 : 2 = 0.5
3) 3; the ratio is 2 : 3 = 0.666...
4) 4; the ratio is 3 : 4 = 0.75
5) 5; the ratio is 4 : 5 = 0.8
6) 6; the ratio is 5 : 6 = 0.833..
7) 7; the ratio is 6 : 7 = 0.851742...
8) 8; the ratio is 7 : 8 = 0.875
9) 9; the ratio is 8 : 9 = 0.888...
10 ) 10; the ratio is 9 : 10 = 0.9
11) 11; the ratio iss 10 : 11 = 0.90...
.
.
.
3000) 3000; the ratio is 2999 : 3000 = 0.9996...

37896544) 37896543; the ratio is 37896543 : 37896544 = 0.9999999736123695078896904169075

As you can see as the numbers get larger the ratio between a natural number x and its successor x + 1, given by x : (x + 1) approaches, in the limit, 0.999...

But 0.999... = 9 * (0.111...) = 9 * (1/9) = 1

In other words, there will come a point in the sequence of natural numbers where a natural number x and its successor will have the relationship x : (x + 1) = 0.999... but since 0.999... = 1, x : (x + 1) = 1 and that means x = x + 1 which basically means there's a natural number which will not increase in size when you add 1 to it. We can't say that x is infinite because if it is then x : ( x + 1) = infinite : infinite which is undefined and can't equal 0.999... Ergo x must be a finite natural number but since adding 1 doesn't get us a number larger by 1, it follows that there is a largest natural number.

Replies to this post are to be addressed to @TheMadFool

If you're asking me about real riots which ended up destroying real black-owned businesses then that harm has been very much documented. That harm is very much real and will likely persist for years to come. — BitconnectCarlos

And the oodles of evidence for systemic racism?

Because it's incredibly unproductive and some of the those businesses we burn might even belong with disabled people. Or maybe siblings or parents of the disabled. — BitconnectCarlos

Why is hypothetical unsourced harm more important to you than real harm done on a daily basis? Why does it weigh heavier in your considerations regarding the protests than the lived reality?

By the same token, you can think of all the hypothetical instances of police brutality agitating against police brutality and for police reform would do.

An attempt at a left criticism of the movements, in the spirit of the forum's old advocatus diaboli.

I will not deny the suffering of people in poor neighbourhoods. It is understandable that those within them sometimes feel the need to turn to crime. That holds regardless of the race of their inhabitants; the same held for the Irish in 1800's London, inhabiting the slums and demonised for the effects of the poverty in the media of the time. The colour of someone's skin only matters insofar as history has made it matter; and it has been made to matter a lot. If someone deprived of opportunities seeks to better their communities through politics or business, they should be applauded insofar as what they do is moral.

What is moral must be, minimally, permissible to do. I don't mean by the standards of law, I mean by the standards of morality that all people have; amplifying the agency of themselves, those around them and of humanity in general. In this regard, severe or violent action must be weighed in terms of its efficacy at addressing severe and violent conditions - and let's not be stupid about it, violence must be permissible as a bargaining tactic of last resort against an oppressor; appealing to the moral sense of an oppressor has never and will never work (pace Assata Shakur). The situation in poor communities in America is dire; powerlessness, lack of representation, racialised police violence; the same or structurally analogous issues since the early 1900's; the state has not and will not save anyone, that much is clear.

The same history that condemns the American state as continually racist condemns almost all acts of protest against it in terms of sheer efficacy; it's been the same shit for hundreds of years. Appealing to the moral conscience of an oppressor, however loudly, bloodily and on fire, so long as those actions remain merely symbolic, protest is begging a racist state to be less racist. Even the best likely outcome of the protests in their current form is the harm reduction of less police brutality inflicted upon the racialised; well fought, well won, not enough. Symbolic violence alone is a spectacle of waste.

Perhaps more organized force is required for enduring change on the level rightly desired by the movement. If you are of a more liberal persuasion, you will probably stop reading there; freely treat this as a disjunctive syllogism regarding stopping the merely symbolic violence of property destruction in the context of a recalcitrant system of racism; why cause undue harm when it merely appeals to the moral conscience of an oppressor, and thus falls almost entirely on deaf ears? It creates minor harms for the unlikely possibility of minor harm reduction.

Perhaps a reader may be thinking, "what you say is true, but the incremental gains of symbolic protest alone far outweigh the minor harm to people that destroying property does". Such a question however is flip-flopping; a standard argument that interlocutor makes goes like:

The American state is recalcitrantly racist and dealing with the same problems now as it was 100 years ago and thus violence against property is ok.

Really? When the same history finds at best minor gains, against countless murders and traumatisation of citizens by police, against the sheer weight of systemic discrimination done by both the state and the economy? And be under no illusions; without massive changes in state budgeting, a change in the American policing and legislative models, and a massive shift of the role of prison in society and industry... The same shit will be dealt with after the flames have settled.

How many more years of merely symbolic protest must be endured? All it creates is minor collateral damage, leaving the conscience of every oppressor fundamentally unswayed. The harm done so far is nothing but prayer.

It's the topic - mathematics is clear cut, and so it's harder to hide. — Banno

When it's so close to the usual definitions, yeah.

I don't think the article says very much. If you give a gigantic machine learning model lots of text, it's going to find patterns of all sorts in it. Something like poetry probably requires understanding of meter and rhyming structure; imagine that poem's lines are enumerated, a sophisticated text prediction algorithm working within an ABAB rhyming structure will likely have knowledge very similar to odd and even numbers, and that "adding 1" to the line number changes the required set of words to write to satisfy the rhyming pattern.

The stuff about arithmetic in it is about priming the model with explicitly arithmetical statements; even if it can add and subtract in terms of line number within a rhyming structure that doesn't mean it can generalise that addition and subtraction to the number symbols. It works in networks of connected symbols, rather than working in networks of connected concepts (though the distinction there is maybe just a matter of network complexity and scope of training data).

The rest is a discussion over whether increasing the number of parameters in the machine learning model is eventually going to stop leading to task improvements on various metrics.

There isn't much discussion over whether it's going to be "better poetry" or whatever, since that's a hazy thing to begin with.

After all, a society is judged by how they treat the least of them. — Outlander

Almost no one who should post in the introductory subforum would feel the need to post in the introductory subforum.

It means 1 is the limit. — EnPassant

Ok, I'll accept that. — EnPassant

You just accepted that 0.999... is the limit of {0.9,0.99,0.999,...}, and equal to 1.

Ok, I'll accept that. — EnPassant

Great. That means you accept $0.999...=1$!

:up:

to .

You can say 0.999...= 1 if by that you mean the limit of 0.999... — EnPassant

That is exactly how it is meant. That is what 0.999... means.



The infinite sum notation $\sum_{n=1}^{\infty}$ just means $\lim_{n\rightarrow \infty} \sum_{i=1}^{n}$, that $\lim$ means; for a sequence $\forall \epsilon \gt0 \exists N : \forall n \gt N |s(n)-L| \lt \epsilon$, the value $L$ is the limit. $0.999...$ is the limit of the sequence of partial sums $s(n)$, which is equal to 1.



Wrote a guide here. It's essentially LateX if you're familiar with it, though without lots of the standard packages.

Yes, but that is the limit which is different from equals. When you say 1 = you are saying 1 = the limit not simply 1 = — EnPassant

What is the limit of the series {0.9,0.99,0.999,...}? Call this x.
What does the symbol "0.999..." represent? Call this y.

Is x=y ?

$1 = \sum_{i=1}^{\infty} 9 \times 10^{-i} = \lim_{n \rightarrow \infty} s(n) =0.999...$

Tends towards the limit' — EnPassant

The sequence elements tend towards the limit. The limit is not a sequence element. 0.999... is the limit. It is equal to 1.

That only shows you didn't understand it. @jorndoe defines what a limit is in it! It doesn't even need infinity in the definition.

But if infinity is not a number how can you have an infinity "of"? — EnPassant

In one respect it is shorthand.

The sequence {0.9,0.99,0.999,..} is the sequence of partial sums $s(n)=\sum_{i=1}^{n}9 \times 10^{-i}$. IE {0.9, 0.99,0.999...}={s(1),s(2),s(3),...}. 0.999... is the limit of that sequence. It is equal to the limit of that sequence. Which is 1. 0.999... is equal to1.

In another respect, and you will not like this even more because the math is more advanced, the cardinality of the set of sequence elements {0.9,0.99,0.999,...} is aleph-null, the smallest infinity. That transfinite number is not a real number.

If you don't understand these issues, you should read through @jorndoe's document. If you have any questions regarding its content, ask in thread and I will try and address them for you.

n infinity of 9s. And we can't write that, whatever it means. — EnPassant

Which is exactly why you write 0.999"...". It is the limit.

But does anybody know? — EnPassant

Yes, many people do. Those who've taken the time to study it.

Intuitively yes, we can see that the limit is 1.

0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which IS 1.

0.999... is not an element of the sequence {0.9,0.99,0.999,...}, it is the limit of that sequence, which is 1.

There are always people that disbelieve in it. It isn't so surprising, since it involves infinity, limits, convergence and monotonicity. What I find especially frustrating about it is that people can be shown a formal proof of it with sources for everything and still refuse to read it and ask exploratory questions.

Then you don't know what the symbols mean and should read the OP's article!

.999... — EnPassant

That ... MEANS the thing on the left IS the limit. 0.999... IS the limit of the sequence {0.9,0.99,0.999,...}, which is 1.

I think we'd be remiss not to explore those possibilities. But maybe you consider them sufficiently explored and satisfactorily put to bed already. Certainly I feel like I'm repeating myself a bit, so maybe there's not much more fruit to be had from this branch. — Isaac

I'm not saying it is valueless to consider; I just think it's not the right time or movement to easily pivot into non-domestic issues. The focus seems to me clearly on domestic issues of systemic racism; the current lack of emphasis on the post-colonial and international trade aspects of systemic racism doesn't undermine the real chances of domestic gains.

It's amazing how many people will respond without reading through and understanding @jorndoe's well written proof. Go educate yourself! Someone's made it super easy for you!

Yes but 'limit' is not the same as 'equals'. — EnPassant

0.999... is equal to the limit of the sequence {0.9, 0.99, 0.999, ...} is equal to 1.

Very well said.

The police stuff, and social policy to address that mobility issue are domestic, and are what the momentum of this movement is towards in the US I think. By the looks of it the international movement is focused on domestic policies within the effected nations. I don't think it would be or should be repurposed towards addressing international systemic racism/colonialism, I simply hope that the movement gathers enough momentum and scope to "Yes, and" the international stuff; using that domestic levers can pull on international trade policies too.

The limit = 1 but not .999...= 1 — EnPassant

the ... means the limit is taken. IE, .999... = 1

More concretely; questions of systemic discrimination are also questions of resource distribution and social planning. To the extent those things are affected by economy managing policies, those impact systemic racism. To the extent economy managing policies are affected by the constraints imposed by current economic structure, it impacts systemic racism.

The same would be true of any economic system with systemic racism, just happens this one is capitalist and the current form is very finance skewed.

Destroying property and resources achieves nothing; it really just amounts to the mindless destruction of property and resources. Nothing will happen without the enlightenment of the masses, and a collective will to coordinated action against the financial elites. But this seems almost impossible given that the masses mostly have no desire to be educated, and people's capacity to genuinely care about others usually extends only to a relative small number of family and friends. — Janus

What stops you from reading all the collective knowledge gained through the protests as education?

Anyone with a problem they don't know how to learn to solve.

A man who's concerned about his woman's sexual needs, especially if she looks to greener pastures, will, in all likelihood, keep her on short leash lest he become cuckold or the like; this tendency of men probably spills over into other freedoms a woman can have. — TheMadFool

If someone thought that, it betrays
(1) A lack of trust
(2) A lot of insecurity
(3) A willingness to constrain their partner from activities they strongly desire doing to mitigate the insecurities.

In treating someone like that, they make their insecurities their partner's problem; and not in the intimacy expanding sense of mutual exploration - it's pure confinement. Doing so will only hasten the demise of the relationship if they're lucky, or perhaps lead to traumatising the person they (allegedly) love more than anyone else.

Edit: Maybe a starting point for a discussion on monogamy as a social norm, I'll just gesture in that direction with song lyrics -

I want you, and I want you to want me to want you
But I don't need you
Don't need you to need me to need you
That's just me
So take me or leave me
But please don't need me
Don't need me to need you to need me
Cos we're here one minute, the next we're dead
So love me and leave me
But try not to need me
Enough said

I agree - casual sex is an unspoken agreement that the ‘relationship’ is only the sexual encounter - but this is not a suspension of agency. I think there’s a difference between a fuck that denies agency and one which respects that you have preferences and choices during the sexual encounter, even if we never speak again after. — Possibility

But I get your point - as long as everyone's on the same page, and both (or more!) parties are OK that situation - that one has permission, as it were, to be treated like that, then that's cool. I dunno how to put it - like an agental suspension of agency maybe. — StreetlightX

If I could interject, maybe a good vocabulary to describe it is the distinction between a subjectivity and agency. In one of @StreetlightX's old threads I posted the following characterisation of what a subjectivity is.

• Subjectivities are more than roles, they become integrated capacities of a person which are exercised in how they live their life.
• Subjectivities are more than the application of an on-off property to a subject, like 'disabled' or 'traumatised'; they can inform and transform people in different degrees of similar ways; like episodic flashbacks vs more mundane intrusive memories; or in much different ways; like generalised anxiety vs dissociative disorder as comorbidities of PTSD.
• Subjectivities are to a large degree impersonal; they are composite patterns of behaviours, feelings and events which constrain individuals along a mode of variation. A person can be said to 'inhabit' the unfolding of PTSD just the same as they can inhabit walking; being a sufferer of PTSD or a walker respectively.

A subjectivity is close to a role a person embodies; a person might be a window shopper, a person might be a wheelchair user, a person might have depression. A role understood as instantiated with possible misfit in a person through their conduct. Every person embodies a spectrum of subjectivities.

I think perhaps the idea of a sexual subjectivity is relevant here; the dance of fantasy and social ritual ascribes a subjectivity to one's partner in the encounter, and one adopts a subjectivity to play a part in the dance. What role each plays may be found more or less agreeable, more or less erogenous, more or less freeing or transgressive, with the moving composition of their partner's desires.

What seems relevant here is the extent to which those subjectivities are negotiated in the encounter, whether it is an agent-agent relation manifesting as a (series of) subjectivity-subjectivity relation( s ), or whether one of the agents is nothing more than the desired subjectivity they could embody.

But what about the correlationist??? What is his view on objects when no minds exist??? Do objects still exist but in a different form than what we measure them to be?? Or do they not exist? But then this is solipsism. — francis20520

That place of ambiguity is where the arche-fossil argument works.

This argument isn’t actually valid, because it could arguably be the case (if not for other, valid proofs that 0.999... = 1) that 0.999... is the very last number before 1, so there is nothing between them and even though they’re not (this hypothetical person would argue) the exact same. — Pfhorrest

It would be valid when restricted to the real numbers as originally intended, I think. There does not exist an x between the limit of that sequence and 1; it would simultaneously have to be greater than every sequence element of 0.9 0.99, 0.999... and less than 1, but since that sequence is monotonic, the "least" such strictly positive real number strictly between the sequence limit and 1 doesn't exist as you can always find one less than it. It's another way of reading the epsilon N proof.

Looks fine to me!

I think there's a common source of confusion related to the one you pointed out, 1 isn't in the infinite series, but the infinite series converges to its supremum, which is 1. People confuse maximum and supremum a lot.

I'd like to see how the legalization of outsourcing from places with illegal(in the States) labor practices ties into the systemic racism in the US government. — creativesoul

Wrote something related to it here. The history of systemic racism is colonial history; and the majority of current systemic racism is done through international business.

Topic is systemic racism... while capitalism overlaps with racist issues, it's coincidental. — creativesoul

If systemic racism didn't work through economic means, you would not expect minorities to be economically disadvantaged. The current economic system over the world is capitalism; so studying how capitalist economies deploy or manifest racism is necessary intellectual labour here.

1.Feeling attractive and sexy feels good, and it feels good for the same reason that feeling unattractive and unsexy feels so bad: our self-worth is wrapped up in it. — 3017amen

Self concept is much different from treating oneself as an object; a person enacts a self concept in how they evaluate themselves - and hows are not whats. Unless you're making the previous metaphysical move that any way of interacting with a distinct entity is an objectification, in which case why bother, if you already "get that"? The metaphysical baggage you're bringing to the discussion is obfuscating the issue.

The ideal object is that which is being perceived. How important is the object being perceived? — 3017amen

So yes, it seems you are making the move that any way that a subject relates to anything besides itself contains an aspect of objectification. Entirely irrelevant to objectification as a mode of human conduct; even non-objectifying ethical conduct could be objectifying in terms of your subject-object stuff; which is really a sign that you're talking about something much different.

If any relationship towards another human, or even oneself, has aspects of objectification (relating to something as an object (the metaphysical category)), what you're saying is quite irrelevant to the notion of objectification under discussion in the thread. The only purchase you've gained in the argument is by muddying the terms.

I'm arguing one cannot escape pure objectification in a world of physical appearances. — 3017amen

Insofar as anything can be treated as an object, that is an objectification. But it doesn't mean the same thing as objectification in the way other people besides you are using it. SEP lists the following aspects of objectification:

instrumentality: the treatment of a person as a tool for the objectifier’s purposes;
denial of autonomy: the treatment of a person as lacking in autonomy and self-determination;
inertness: the treatment of a person as lacking in agency, and perhaps also in activity;
fungibility: the treatment of a person as interchangeable with other objects;
violability: the treatment of a person as lacking in boundary-integrity;
ownership: the treatment of a person as something that is owned by another (can be bought or sold);
denial of subjectivity: the treatment of a person as something whose experiences and feelings (if any) need not be taken into account.
Rae Langton (2009, 228–229) has added three more features to Nussbaum’s list:
reduction to body: the treatment of a person as identified with their body, or body parts;
reduction to appearance: the treatment of a person primarily in terms of how they look, or how they appear to the senses;
silencing: the treatment of a person as if they are silent, lacking the capacity to speak.

If you really can't see the difference between what you're talking about and the intended topic after reading that list, I dunno what to tell you. If you want to treat them both as the same thing - why? What purpose could it possibly serve? :chin:

Let me ask you: do you believe that there’s a point (which may vary between men) at which a sexual act is a foregone conclusion? — Possibility

Nah. I think I felt more like that when I was younger though - not that I was "entitled to sex", just that I got more pissed off and bamboozled with rejections.

It’s actually a good question. Casual sex for the most part is a mutual fantasy - it’s a dance of cognitive dissonance. He pretends he’s interested in a relationship beyond sex, and I pretend not to notice the pretense. When I stretch and test his performance, though, his true colours will show through under the surface. From this I can decide whether I think the game is worth playing out, whether I buys into the fantasy, or not. — Possibility

That's interesting. So for you it actually doesn't matter whether the connection is "genuine" or not, just whether you're in the mood, physically attracted enough and whether he's performed/improvised/responded in a manner that suits your moment to moment expectations/desires?

Their aim is to gradually eliminate or obscure her options. That’s the transgression. It’s a danger that women are not always prepared for, and that men may not even realise is a problem - until they’re accused of rape. — Possibility

:up:

It's very easy to reduce someone to the role they are subscribed/expected to perform.

A more pernicious (and all too common) form of it seems to be doing whatever you can to erode your target's personal boundaries.

So, for me at least, there’s a difference between not giving a damn about an extended relationship and not giving a damn about your sexual partner as a fellow human being with agency. I enjoy the pretense of a ‘possible’ romantic connection as much as the next girl, but underneath that is the real question: Is he respecting my freedom to choose? — Possibility

That also makes a lot of sense to me. Any form of intimacy shouldn't have to be linked to a romantic relationship for its validation, conversely; cherish transient instances of intimacy, they are all too rare.

Thank you for indulging my question.

For you it might be. For others, casual sex is most often a much more complex and interesting connection between two people than mere masturbation, no matter how brief the encounter. — jamalrob

:up:

But I don't even think it's clear how to distinguish genuine from fake interest. In performing a connection, a connection is made, I think. Often. — jamalrob

I don't wanna draw a line for other people regarding what consentual games are played. They seem to enjoy it. There's even some nice moments of erogenous surprise in flirting with strangers - I miss those, all too rare to be surprised by joy (hashtag CS Lewis) and genital sensation. Creating desire and having desire created in yourself is a lot of fun.

If both parties are playing the game, it's a form of relating to someone as a person, not merely using someone as a means or treating them as an object. — jamalrob

I think that is true, what makes me uncomfortable are situations where one person is signalling that they are "connected" for the sole purposes of getting sex, and the other is unaware that their game partner is using it simply as a strategy to obtain casual sex. I believe that's a way of not playing the same game.

I don't know how common it really is, or whether the Connection Flirting Script is just seen as part of the game.