Two collections of identical objects in a one-to-one correspondence are, by mathematical induction, identical collections. — Devans99

So you mean they're identical collections, but their elements aren't equal? :S

Two collections of identical objects in a one-to-one correspondence are, by mathematical induction, identical collections. — Devans99

Assume what you're saying is true. Then {1} = {2}, then 1=2, contradiction, then what you're saying is false.

There are two miscomprehensions in your post. The first is based on how sets are defined. The second is based on a conflation of an equivalence (of sets) with strict identity (of sets).

The first one:

We have two rows of identical bananas and each row stretches out to actual infinity. The two rows are lined up so that there is a one-to-one correspondence between them. According to Cantor, the two sets of bananas are therefore identical sets. — Devans99

Firstly, if all the elements of the sets are identical, then they just have one element. Sets are defined by what distinct elements belong to them; a set is a collection of distinct objects. If x is in X, there's only one copy of it. If you want to consider set like objects that allow multiple copies of identical elements in them, that's a multiset.



The second one:

3. So all bananas are in one-to-one correspondence
4. So the sets are identical — Devans99

Two sets being in a one to one correspondence says nothing about whether they are identical sets. The odds are in a one to one correspondence with the evens, but even numbers are necessarily not odd. However, if you stipulate the definition:

(C) Two sets $X,Y$ are related in sense C when and only when there is a bijection between them. When a bijection exists between the two sets, write $X \cong Y$.

You'll see that (C) is an equivalence relation. The equivalence classes of (C) are sets of the same cardinality. IE, insofar as C is concerned:

$\{1,2\} \cong \{3,4\} \cong \{5,6\}$

But

$\{1,2\} \neq \{3,4\} \neq \{5,6\}$

!

You'll also see that if two sets are identical, a bijection exists between them (the identity function), so a set has the same cardinality as itself. That is $X = Y \implies X \cong Y$, but the reverse does not hold. The reverse implication is precisely what you require to go from 3 to 4 in your argument, and you're obtaining it by equivocating between two sets being identical and a bijection existing between two sets.

homosexuals reproducing...:chin: — TheMadFool

Sexual preference, no matter strength, no make babby, no exclude babby. Sometimes babby come from gay man or woman and partner because like partner and want child. Sometimes drama. Sometimes artificial insemination. Sometimes woman have child for close gay friend. Sometimes person gay for loooong time and fuck other sex for loooong time until reaches personal epiphany, or until moves away from restrictive social context.

This has been your yearly dose of sex ed.

Bijections between sets only show they're of equivalent cardinality, not that they're identical sets. Sets are identical when and only when they share all and only the same members. So - the odd numbers:

{1,3,5,...}

And the odd numbers without 1

{3,5,7,...}

are not the same set, the second is a proper subset of the first, but they're the same cardinality (the bijection is f(x) = x+2 where x is an odd number).

Imagine if everyone was homosexual. Birthrate = 0. End of humanity or whatever species became completely homosexual. In other words, selection pressure would work against the homosexual gene. — TheMadFool

Even if you make the assumption that individuals which are homosexual are exclusively homosexual in sexual interest, that doesn't mean they don't reproduce. Moreover, even if they do not reproduce, it doesn't follow that any particular genotype that may result in homosexuality is not adaptive (or deleterious). It could be that a tendency towards exclusive sexual interest in individuals of one's own natal sex comes along with other traits that are adaptive (or just non-deleterious); it could be that the development of homosexuality is down to gene expression (perhaps due to natal hormonal environment).

Moreover, twin studies reveal that the development of sexuality in humans can't be reduced solely to genetic causes; like often hypothesised site mutations (as in ) or "the gay allele"; rather it seems that one's genotype plays only a facilitatory role - providing a genetic predisposition or consistency - with developing exclusive homosexuality.

A rough guideline is that your posts should stand alone without their references. Certainly, central parts of arguments or posts in general should usually not be deferred to a reference; it should at least be summarised and how it relates to the subject matter spelled out.

From the point of view of the greenlanders, the red people are monstrous. They're really not. If you could pick them up and move them around, you'd see that they're all the same. — frank

You don't need to demonise the Orange staff to demonstrate that what they did was wrong.

Demonstration of empathy: I'm sure there were heated boardrooms meetings where managers protested against the poor management practices, that they had lots of consternation in their guts, but believed ultimately that what they were doing was for the greater good - the good of the company and all the workers, surely it's better for the company to succeed and have all those jobs than risk going bust from firing redundancy packages? It's for the good of the employees not to put undue risk to the company; which boils down to not jeopardising the bottom line; the profit rate; and doing what needs to be done to get the organisation in a competitive state of sustainable growth.

But - you need that myth to vindicate and justify their management's conduct. You don't need any myth to demonstrate that their conduct was intentionally abusive and substantially contributed to employee suicide and sickness.

Your concern for the workers is a tribute to either your kind heart or the fact that you direct most of your talent into sophistry. — frank

I had no idea that pointing out unsupported assumptions, equivocations and falsehoods was sophistry. Posting arguments in premise-entailment form as an initial response, that well known strategy of sophists.

It's better to work with nature and allow small adjustments (which might include 35 suicides), rather than prop up an artificial system that will eventually fail in a larger bloody adjustment. — frank

You attempt to portray Orange's provable mistreatment of workers as heroic, as necessary for adaptation for Orange's market success, as natural - as opposed to the 'artificial' contracts the targeted Orange employees had. All this does is portrays the management strategies, which were proved beyond reasonable doubt to lead to employee suicides, as beyond criticism by allying them with natural forces - despite being choices in management style (apparently nature flows from the management reorg). You claim that this is good in the aggregate despite there being very good evidence that such practices, even less extreme cases, cause huge productivity losses the world over and have demonstrably huge social costs. I substantiated both those claims and provided you with a source.

The fact that the job market was tight indicates that France is overpopulated and some portion of the workers need to move to where there are jobs — frank

"Orange's workplace management strategy of targeted harassment had to do with the fact that France was overpopulated and that the job market was tight, not to do with circumventing protections and compensation upon firing afforded to the workers at France Telecom from their previous contract"

These are just assertions with no argument.

It's a little harder see how to apply that in the case of psychological abuse because while asbestos has pretty much the same effect on everyone, moral harassment doesn't. Some people thrive on an emotionally charged environment that includes permission to be abusive (which is provided by an abusive executive.) — frank

"A work environment does not contribute to employee sickness if it does not result in that sickness for all employees" and "High pressure work environments are the same as intentionally abusive management strategies"

The first also contradicts:

In the case of depression and anxiety disorders the cause is likely to be genetic. You can't have a genetic predisposition to having your head ripped off in a combine accident, so agricultural accidents can be easily traced to a lack of safety precautions. Psychological disorders can't be. — frank

The idea that it's a genetic predisposition. That the worker suicides had anything to do with a genetic predisposition or genetic cause is also un-argued for. If you want to establish that the workers at Orange committed suicide predominantly because of their genetic predisposition, you need stronger evidence tying the workers at Orange who committed suicide's genetic profile to the environment, then you need to establish that the environment was not the main driving force regarding their development of the illness. As it stands, the management strategies were established as intentionally abusive.

You attempt to do this by simultaneously downplaying the demonstrated effects the workplace had on its workers and shifting the onus of responsibility to them for their harassment.

The world is full of assholes. Sooner or later we all have to learn how to deal with that. An asshole boss is an opportunity to either learn how to deal with abuse or grow a spine stiff enough to get yourself out of the situation. — frank

There's a big difference between an asshole boss with bad management (which, inevitably, leads to bad working conditions) and tailoring management practices to abuse people into quitting.

When the company became private, the executives struggled to keep the company afloat with their hands tied because the employees kept the protected status they had when the company was government owned. — frank

You continue to portray what happened as necessary or inevitable and leave it at that. Even if you grant that enforced redundancy was necessary for Orange to continue growing as a company, this does not establish that any particular way of enforced redundancy is good or bad. As it stands, their management practices were tailored to make people quit "out the door or out the window" (quote from Orange management), case reports were given in court of what the management did, it was established as abusive (not just "high pressure") and not necessary.

Your posts in this thread are full of unsubstantiated conjecture, rhetorical flourishes and reframing attempts, presented with conviction, you also "know" that what you're saying is true...

STRANGER: Then the Sophist has been shown to have a sort of conjectural or apparent knowledge only of all things, which is not the truth?

THEAETETUS: Exactly; no better description of him could be given.

In the case of depression and anxiety disorders the cause is likely to be genetic. You can't have a genetic predisposition to having your head ripped off in a combine accident, so agricultural accidents can be easily traced to a lack of safety precautions. Psychological disorders can't be. — frank

Equivocating between a cause and a predisposition doesn't help much. Let's grant that mental illnesses do not occur without genetic predisposition, genetic predisposition alone doesn't lead to the development of any mental illness - it also needs a facilitating environment for them to develop. The environment at Orange has been established as sufficiently terrible to cause mental illness; whether it caused it in only those with a genetic predisposition doesn't matter, it provably contributed to lots of death and even more sickness.

Moreover, why would a disease having a genetic predisposition absolve a company of their responsibility to their workers? They had contracts, Orange hated the contract, they abused their workers to make them fuck off.

The role genetic predisposition plays in your argument is to centre the legal (and moral) responsibility for Orange's harassment of workers onto the workers. For that, you need a better account of genetic predisposition and how it relates to workplace environments to be convincing.

"It isn't his fault he's drawing false analogies, he's genetically predisposed to!"

Source? — frank

Here ya go.

Has France started legally mandating good management practices? — frank

No, more people would go to the gulags if this happened. What happened was the work environment at Orange was judged as being sufficiently terrible to count as severe harassment, they had a responsibility for their workers' welfare, and these two things together make an enforceable claim that they contributed (bore responsibility) to workers' deaths.

I just pointed out that an abusive environment can produce efficiency gains. It's called bootcamp. It's you who wants to make the positive claim that abusive workplaces create costs for society in general. I'm not seeing it. As BC pointed out, every job has a downside. You get paid to put up with it. — frank

Why would an office environment be anything like a military boot camp in terms of what counts as a needless hazard?

And that argument is pretty easy to understand if we're talking about chemical exposure or the absence of safety equipment on machinery. — frank

Whatever causes the sickness and death doesn't really matter, does it? It just changes what should be done to address the issue, and the laws which may apply. Widespread workplace harassment and mental illness inducing work environments give business globally 1 trillion dollars in losses from productivity decrease alone, never mind the social costs and the weight that brings on social safety nets.

Some people thrive on an emotionally charged environment that includes permission to be abusive (which is provided by an abusive executive.) — frank

There's a big difference between a high pressure work environment and an abusive one. High pressure work environments can still have clear goals, efficient allocation of talent to tasks, and provide employees with downtime or appropriate compensation. The case with Orange wilfully stopped good management practices for the sole purpose of driving out employees they did not want to pay any more or provide a severance package for.

With moral harassment, it will be hard to quantify that cost. Less hard to quantify will be the revenues the government will lose for lack of companies like Uber in the economy, a company known for both ruthlessness and profitability. — frank

The WHO keeps statistics on it. It has the benefit of book-keeping on its side, whereas the reasons Orange treated their workers like they did was much more to do with, well, downsizing to avoid paying people (get those outflows off the books by any means necessary) and forcing them out without a severance package.

If it wasn't so clear cut that Orange's management practices caused the suicide of some of their workers, it wouldn't've been proved beyond reasonable doubt would it?

Edit: An unstated assumption in what you're saying is that such abuses actually produce efficiency gains or mitigate efficiency losses, which remains unargued for, and is implausible given the obvious effects of illness promoting environments on productivity and the costs they impose on societies.

Intellectually, it's hard to pinpoint it, but it's very clear emotionally. We're supposed to help each other. — frank

If you need a structural argument for why workplace safety should have laws attached, consider that sickness has huge social costs; it decreases workplace productivity and requires the use of social safety nets to care for those that are sick. In essence, workplaces harassing employees offloads the costs the business would have for dealing with the issue adequately onto whatever measures there are in their employees' social safety nets, the business loses productivity from the harassed workers, and the harassed employees' social relationships suffer too.

There are cases where workplaces stand to gain, or avoid loss, by harassing their employees or having an unsafe working environment (like with Orange), and in doing so their interests go against the public and the state. It makes sense to punish workplace harassment and unsafe working environments to impede the social and financial costs of this negligence from being offloaded onto the public and the state.

No, I'm not. The rightist argues that survival of the fittest generates healthy entities in human industry in very much the same way it does in the evolution of organisms. — frank

Can you spell out for me how that relates to workplace conditions that lead to unhealthy workers - exposed to conditions that are not necessary for the good functioning of an office?

The fact that the job market was tight indicates that France is overpopulated and some portion of the workers need to move to where there are jobs. By forcing French companies to maintain happy work environments, they're effectively making the problem worse. French people will stay in France and produce more French people into an overloaded system. Companies will struggle to maintain the happiness quotient until they finally go out of business and the crowds of hungry French people will start executing people by guillotine. — frank

Are you arguing that it is necessary for people to be harassed into suicide in every workplace because if it doesn't happen civilisation will collapse quicker?

Maybe this will help you rustle people's jimmies.

(1) Countries' legal systems should be able to punish those responsible for working conditions that provably and significantly impede health.

(2) Working conditions that provably and significantly contribute to death and sickness impede health. (1, consequence)

(3) There are many independent reports of working conditions at Orange significantly contributing to deaths and sickness and they should be trusted and treated as evidence. (Premise)

(4a) It is reasonable to believe that working conditions at Orange contributed significantly to deaths and sickness. (3, using the evidence).

(4b) It is unreasonable to believe that working conditions at Orange did not contribute significantly to deaths and sickness (3, using the evidence).

(5) France's legal system should be able to punish those responsible at Orange for the working conditions at their company as they provably and significantly impeded health.

I was using "acceptable" as a substitute for "satisfying," which I think eliminates the circularity you pointed out. — quickly

Highlighting a circularity wasn't really my intention. Noting that you used 'acceptable' there was a rhetorical move to highlight the territory knowledge required to set up a modal logic. The status of this knowledge is problematic. If it is imagined as a system of established propositions (which can be encoded as axioms) expressing the metaphysical character of the domain, we have to ask what process of reasoning, discovery and description allows the formation of that system of propositions and allows us to tailor it to the domain in question. This, precisely, is practicing metaphysics as it is relevant to the domain in question.

For example, an engineer might be satisfied with a description of the screen's material properties. This might not satisfy the philosopher, but it will satisfy anyone asking whether actual punches will break actual screens. — quickly

Maybe the engineer and the philosopher might have a common interest here. The engineering description of the compressive strength of the laptop screen engenders a series of metaphysical questions and is accompanied with its own metaphysical insights. Maybe the engineer and the philosopher would think, in an Aristotelian vein, that the compressive strength of the laptop was a potential of it - characteristic properties that engender and constrain what laptop screen will do and what it can do. The only way the screen can break is if the thrown punch can exceed the compressive strength. In context, these are "territory" considerations.

The "possible worlds" in which the laptop screen breaks develop out of the ones in which it is punched through, maybe, being subjected to a compressive force sufficiently greater than its compressive strength. Spelling out the transition mechanism of "worlds where the laptop is punched" to "worlds where the laptop breaks given a punch x": these are those worlds in which the laptop screen's compressive strength is sufficiently less than that of the punch. You then check to see if the domain of possible thrown punches (by a particular agent) is capable of applying enough force to the laptop screen to break it. Then the worlds where the laptop screen breaks given a sufficiently strong punch are connected to the worlds where that punch is thrown. These are mere "map" considerations (albeit given a rather stultifying framework for making such a map, in which we already have (weak) existential commitments to possible worlds, more than the bodies in play).

Neither the philosopher nor the engineer should be particularly happy with "it is possible that the laptop screen breaks when it is punched because it is a physical possibility", or "the laptop screen might break when struck because the possible world where the laptop screen breaks when struck is connected to the ones where the laptop is struck"...How, why does it have that status? In context? There's metaphysics elsewhere, prior to it, setting up that logic is a lot about rearranging representations when we already know how to move them about.

My specific contention is that once the set of acceptable answers is determined, the metaphysical and logical questions are mostly settled and usually irrelevant. The "phenomenology" and "intuitions" are mostly determined by asking someone what they're looking for in an answer. — quickly

I agree that once you "determine a set of acceptable answers" the "metaphysical and logical questions are mostly settled", but that process of "determining the set of acceptable answers" is precisely part of settling the metaphysical and logical questions.

The only reason, I think, that metaphysical questions concerning modality seem largely irrelevant to you (in this context) is that you (seem to) think that "determining the set of acceptable answers" has nothing to do with reasoning metaphysically about modality. Note that you expressed this using the word "acceptable"; itself a modality. Reasoning about the sense of "acceptable" there; what is appropriate for the situation is in part extra-logical; you have to think about the logic from the outside while negotiating its axiomatisation to ensure it works well for the sense of modality in question.

This isn't so much a weakness of a modal logic, it's a recognition of its limitation in scope. It's a map, there's a territory, and we negotiate which map (modal logic) to use based on the territory.

It's a question of how it's implemented. A UBI replacing all other social safety nets would be at least as bad and enable, if not come directly along with, lots of encroachment on those social safety nets. In addition to all other social safety nets, it makes more sense, having some money for food which is not means (or capability, he writes as he remembers the times the UK job centre "fit for work" assessments have been raised as a human rights violation in the UN...) assessed would be nice.

Its introduction is definitely (only) a palliative measure.

I think I disagree. — quickly

For the most part it looks like a difference in emphasis to me.

The question then becomes: which accessibility relations are germane to our universe? — quickly

Which is not something that can be decided by modal logic in full. You pick the axioms that capture whatever sense of modality seems relevant. Hell, you might even want to pick theorems and find axioms that produce them. "Possibly necessary => necessary => true in this world" is very attractive if you want the modal ontological argument to go through. Otherwise - "It is possibly necessary (in the sense of a covering law) that light requires the aether as a transmission medium, therefore it is necessary that light requires the aether as a medium, therefore light requires the aether as a medium", and it does not.

So maybe the accessibility relation can't be an equivalence relation for modalities concerning physical law; but for transcendent a-priori rationally considered possibilities...

It isn't just a matter of modelling, there's a matter of deciding which modalities are meaningful to model and in what context.

But specifying an acceptable answer determines a metaphysics, and therefore circumscribes the set of accessible worlds. — quickly

But this is very backwards, at least in part, if you're using a modal logic to represent metaphysical intuitions, the metaphysical intuitions, the axioms and the theorems all interact; a feeding forward of "acceptable answers" into "metaphysical accounts" makes the appropriate metaphysics for a domain rather arbitrary; or if not arbitrary, we consider the acceptable answers through metaphysical arguments, and at that point we're not just talking about the formal structure of a modal logic either. The extra logical considerations in part determine what logical structure seems appropriate to represent them, so do in part the theorems and axioms of the logic.

Edit: there's also the transition problem I highlighted. The possible world where a ball rolls down a hill instead of being stuck at the top of it is accessible from the one where the ball is at the top of it in the sense of physical possibility; but the sense of physical potential which mechanises this transition between the possible worlds (stuck now, rolling) simultaneously establishes the possibility (because it could happen in some way) but does not care how it happened at all. It emaciates any metaphysics of potential tied to the actual by suspending any consideration of how things work.

Just some thoughts on my frustrations with modal logic and how it relates to any metaphysics of possibility.

That the last step doesn't say anything about what it would mean for the laptop screen to break; to be capable of breaking when struck hard enough. There's a whole metaphysics of potential and actuality which is just elided by making the connection between how the actual world develops and what it may possibly become of precisely the same character.

What allows us to assign accessible worlds to each other in a network of accessibility is a nascent understanding of modality; a physics, a metaphysics, a criterion of logical consistency etc; which is modelled by stipulated properties of the accessibility relation in question. This is the elided metaphysics. A method of presenting an account of (a type of) possibility is much different from determining what an adequate account of (that type of) possibility is; or only a partial picture. A modal logic and its characterising accessibility relation is a map, the domain studied and its metaphysics of development is the territory.

EG: Causal possibilities often have order-like (antisymmetric, transitive) accessibility relations - this just says that cause always precedes effect. But we know that that has issues in some domains. How you'd find out about the issues which change the principles a modal logic works by isn't just about selecting "the right theorems", it's about assessing whether the axioms and theorems (the whole structure) fit the domain. And this is tricky, since:

"Might" has lots of valences, as does "possible", but when we speak of possibility in terms of possible worlds, we implicitly quantify possibility in the discussion over accessibility relations; each of which has a distinct picture of possibility within it. To say "possibility is just saying that the thing happens in some possible world" isn't an account of possibility of the event, what it means for that event to be possible (like what it means for a political party to be electable), it's a way of deferring the question that confuses map and territory; or modelling device and targeted domain.

Edit: another part of this limitation is that possibility and necessity in a modal logic really are statements about graph connectivity; which worlds are connected to which other worlds, whether a path exists and what facts hold in the reachable nodes; not about what it means for one world to transition into another; what it means for something to have a capacity to develop in this or that way. It could be said that what it means for an entity E to have a capacity to develop in way X is for the characteristic properties of development associated with X to be in an accessible world from E, but there's still that question of transition, the development of E in way X is as much about transition properties as connectivity properties of the accessibility graph. So yes, it might happen, but the way matters; such graphs picture state transition with arbitrary means. "How could it happen?" "I don't care because it's possible"

That bolded "another" is the problem here. It doesn't have to be another possible world, just some possible world. It can be this one, or another, and still be some one. — Pfhorrest

"If I punch my laptop screen, it might break"
"What does that might mean?"
"In some possible world"
"That includes this one?"
"Yes, the actual world is a possible world since it is a reflexive accessibility relation"
"Doesn't that mean if I punch my laptop screen, it will break?"
"No, it only might, because we don't know if the world where the laptop screen breaks when you punch it is this world or another world"

Accessibility relations tend to be reflexive.

"If I punch my laptop screen, it might break" doesn't have to mean "There is another possible world in which my laptop screen is broken as a result of my punching it" surely? It could equally mean that it is something which might happen in this one.

Must this be the same kind of possibility as stipulating another possible (counterfactual, developmentally entailed, temporally precedent...)?

Please cite where you got this from? Also, you should edit the giant copypaste so it's more readable.

"If you consistently state policies which are on the left, we have the majority of supporters, it's just a question of getting the message out there" - leftists a month ago.

"Well shit" - leftists now.

Still, a failure of the left, just a different flavour from usual. Poor marketing decisions.

This. You don't learn to improvise first. Seeing the relevant holes in things requires first knowing what is relevant to them. Learning what's relevant to what is learning; when you know what's relevant to what you can ask better questions, when you ask better questions you can learn better.

Don't you know that people are poor because of the welfare state? If companies don't have to employ people, they won't. Governments create artificial scarcity in labour markets through the administration of welfare, and since they're less efficient at allocating human resources than markets they always run at a loss. Moreover, if companies didn't have recourse to the state, there would be no regulatory capture! And wealth would be much more liquid, flowing free like business. As it stands, the state is a bastion of corruption and creates shortages where really there are none.

(it's so easy... so easy...)

But, whether reasonably or not, I baulk at the effort of posting such work here only to have it flooded with a series of banal one-liners barely related to the subject... or that God did it... or some other variation on the ever-popular delusion that because a thing seems that way to someone it must therefore be the case. — Isaac

You can ignore or report the posts. Also, maybe it's a personal thing, but I gain a lot from going through something, even if I'm making loads of mistakes.

How about you try making a thread like that? Pick something that interests you, start doing exegesis for the general reader. Set some requirements for discussion in the OP that keep the thread on the track you want.

Try to publish on academia.edu or arxiv.org depending on the content type. You're more likely to be able to upload philosophy things on academia.edu than otherwise. arxiv.org has limited peer review and generally requires technical/scientific content.

If we want to keep threads like that going it would be useful to know why they stopped wouldn't it? — Isaac

In my experience we lose the energy to continue it. It takes a lot of effort to write exegesis and discuss it in spare time. I keep coming back to a similar exegetical thread on Das Kapital, sitting down to write and reading previous notes, but it's a lot of stuff to do to do right - in spare time it's hard. In my experience everyone wants high quality and in depth content, but it's rare that we have the energy or time to produce lots of it.

I don't think failing to complete such ambitious projects (or similarly in depth discussions in threads) is a bad thing though, I've certainly learned a lot from starting and participating in them over the years. We have an outstanding discussion on conceptual schemes and Friston's paper to progress, after all.

There's been a good Philosophical Investigations reading group within the last while. We never finished the book, though.

:up: Looks good to me. I've never even thought of using numerical methods in proofs! That's very cool.



Think it's curly braces. $\{\}$ They need \ in front of them in math mode.

It can be easily shown that the series is convergent by Cauchy's criterion (yes, I just looked up the name - hey, I am three decades out of practice, you guys should be doing this :)) — SophistiCat

Hmmm. The series goes complex for some $k_i$, if you meant the interval of rationals $(0,1)$, I guess you mean $k_i$ is either $0$ or $1$ for all $k_i$. But that makes sense!

The series can converge to any real number in the interval [0, 1]. — SophistiCat

So the procedure says:

"For all input sequence k, the series will converge to a real number in [0,1]"

Now the questions are:

"For all r in the reals, there exists an infinite sequence of inputs such that the series will converge to r?" (this would show the procedure is surjective)

Let's assume that's true. Then the surjection is from infinite binary sequences to infinite binary sequences... Which is unsurprising. It's a surjection from the unit interval of reals to the unit interval of reals, rather than from the unit interval of rationals to the unit interval of reals.

Here..!

The proof works if you understand it, what you've suggested is complete nonsence with regards to the sum you posted — Umonsarmon

(1) The sequence contains irrationals. The infinite sum remains rational.
(2) The sequence can consist only of rationals. The infinite sum can be irrational.
(3) The sequence can consist only of rationals, it can be strictly increasing or decreasing, but not converge in the rationals. (see 2)
(4) The sequence can consist only of irrationals, it can be strictly increasing or decreasing, but converge to a rational.

You don't have the sequence/series facts down.

Now if the sum is a rational number then at no point in the sequence 1/4+1/8 etc can the sum be an irrational number otherwise the sum of the whole sequence will be irrational. — Umonsarmon

No.

1 + root(2) + 1/2 - root(2) + 1/4 + 1/8 ...

Also, if your sequence must output a rational, it can't output an irrational.

If we have the sequence 1/4 + 1/8 + 1/16 etc etc no matter how many numbers I remove from that list the product will be rational. — Umonsarmon

$e^x -1 = \sum_{i=1}^{\infty}\frac{x^i}{i!}$

No.

With this procedure you can produce rationals arbitrarily close to irrationals but not irrationals. If you want the procedure to be infinite, you're no longer dealing with rationals (since the series doesn't converge in the rationals, it only converges in the reals).

I don't know the derivation for that one, just looked on the wiki page for series expansion of root 2. How about:

$e^x = \sum_{i=0}^{\infty}\frac{x^i}{i!}$

sub in $x=1$.

You can do that one yourself. Derive the Taylor series of $e^x$, show the radius of convergence includes $x=1$ (it's actually infinite, the function is analytic). Sub in $x=1$. All the finite sums are rational. The infinite sum is the irrational $e$.

If we cut through all of the banter it all boils down to whether you believe that a rational number + or - a rational number equals a rational number. — Umonsarmon

Finite sequences don't automatically have the same properties as infinite sequences. It doesn't boil down to that at all.

No matter how many times I perform that operation it will always result in a rational number. — Umonsarmon

$\sum_{i=0}^{N}(-1)^{i}\frac{(2i-3)!!}{2i!!}$

where $x!!$ is all the even numbers up to $x$ multiplied together. Is always rational for any $N$.

$\sum_{i=0}^{\infty}(-1)^{i}\frac{(2i-3)!!}{2k!!}$

Is $\sqrt{2}$