I definitely disagree with the meaning that “2 * ” provides, as if you can be “twice” as endless as endlessness, and I definitely disagree with dividing endlessness into arrays of [n] and [n+1] etc. to get there (like you do with “1 x (1+1+1…+1) = infA[2]”).
I can’t currently see any definite issue with “1 x infA = infA”, but I suspect that issues could maybe arise through using the finite “1 x” part to insert finitude into the infinite expression of “infA”.
“1 endlessness” seems strange as it combines finite quantity to a quality that defies finite quantity by definition. So I’m warey of it, but provisionally I’ll allow it depending on what you do with it.
I mean, this is what I’ve been saying for a while now, but I’m glad you now see it as progress.
Which is worse? Disagreeing with almost the entirety of maths professionals, or disagreeing with almost the entirety of physics professionals by denying both relativity and the laws of thermodynamics? I think we’re both way past appeals to authority fallacies here.
Not a distraction, just something to bear in mind when we eventually apply all we’ve learned back with the topic of this thread.
If my logic is right, there’s something wrong with “infW” being smaller than “infR”, and authority doesn’t override that. Logic can though - so let’s get to that. I’m fully aware that there’s something I might be missing that elite mathematicians aren’t - but there needs to be a logical explanation from you that doesn’t contain contradictions, like we’ve had so far. I’d be quite happy to accept one if you have one, once you have it. Repeating anything you’ve already said doesn’t cut it for reasons I’ve explained a great many times by now. I’m looking forward to it
So, unlike everyone else on this board, you do not simply accept ethos argumentation. My compliments.
So now to that proof that apparently surprised many people long ago.
That merely let’s us know that Georg Cantor got the credit for “proving” the idea that infR is bigger than infW.
The name of the method was “Diagnalization”.
Actually, I think there is an easier way to prove it but since belief is ruled by reputation and reputation is not ruled by performance, but politics, it wouldn’t do any good to bother with it.
If you have trouble following that explanation, we can go through it line by line but it is getting late for me, so it will have to be later.
I had no problem following your explanation, don’t worry. Funnily enough I was first familiarised with Diagonalisation through a YouTube channel called “Numberphile” - it’s a decent channel for explaining famous mathematical theorems and the like, if you’re not already familiar with it.
So the primary concern of Diagonalisation is to pair elements between two sets. For example, if the number 2 is in both, you can match them together, or if you have two in one and two squared in the set of square numbers, you can match them together - and the goal is to see if there’s anything left over. If there’s something left over in one and not the other, then that infinity of numbers is said to be bigger than another. Diagonalisation is a way of finding these leftovers.
I think it’s easy enough to dispute that simply by pairing positions of elements in each set. If both sets are infinite, every single position in one will presumably have a corresponding “same” position in another, forever. The intention behind the method of Diagonalisation is to make it appear as though one set is “deeper” than another, like Jakob was aiming after - that there’s gaps left in one set when you match pairs together i.e. that there’s numbers in one set that aren’t in another, therefore it’s “deeper”.
Put another way, for example, the set of the whole numbers “leave out” numbers in the set of the real numbers, and that fact is used to claim that the one with elements left out is “shallower” or smaller - with respect to Cardinality. And yet each set has a 1st element, a 10th element, a 100th, an “nth” forever. They each literally do not have an end, and the difference between how they’re constructed does not change this. This doesn’t mean they have equal or unequal cardinality, just that they both have undefined cardinality by definition of their infinite lengths as sets.
I literally do not give a shit about reputation nor politics when it comes to logic. I am quite happy to discuss any logic you have whoever you are, and whatever it is, and it won’t reflect on you, your political position or inclinations, nor what other people, including myself, thinks of you. If it’s of interest, I want to hear it. If you don’t wish to share, that is fine too.
I really don’t think it can help to quote Wittgenstein. He uses words in such a way as to leave serious doubt as to whether he is being truly logical. I can’t verify one way or another what he truly meant. But I do sense that he is mostly exercising conscious denial without actual sound argument while casting that very accusation on transfinite believers. If we cannot examine very carefully what he meant, we cannot accept what he said as meaningful. In short, spell it out yourself like the rest of us.
So you do not accept the bijection method for establishing cardinality concerning infinite sets? Like I said, you are now contending with the whole of professional mathematics. But they have been wrong before, so let’s see what we can do to come up with agreement.
All arguments are going to say that because one infinite set contains all of the other set’s items plus more, that set is greater, higher cardinality than the other. You say that if more unique items are added to only one of two infinite bijective sets (a new word I have learned), neither set is greater or of higher cardinality than the other. To me, that is an issue of language and the intelligence it allows.
You say that the cardinality for such sets is “undefined”. I would normally disagree, but I think that it would be better to just resolve that issue by giving definition to the undefined (which is exactly what James had done).
Let’s say that under President Trumps new USSPACECOM (Space Command) regulations, it is allowed that private corporations may purchase the space above many regions of Earth (being a capitalist and real estate kind of guy). But in his haste to gain the real estate taxes, a few details are overlooked in the regulations.
Spotting an opportunity, Amazon’s lawyers quickly declare ownership of “all space above” Redmond, WA. Immediately afterward, Microsoft declares ownership of “all space above” Seattle, WA. Trump is immediately thrilled as the IRS begins scooping in the new tax revenue to help build The Wall. And the corporations made a fortune by renting the space back to the US government.
For a while everything was working out great. The economy was booming and everyone (except the Democrats) were happy. But then in their bliss, something happens.
Amazon and Microsoft decided to merge.
Up until that moment, the IRS had a math formula for exactly how much to charge in taxes for “all space above” situations. Their formula had no limit for the distance into space, so they charged by the cubic mile (Americans, you know) and diminished the rate as the distance from Earth extended. To the government’s first surprise, both corporations realized that the tax formula converged such as to yield a calculable and affordable tax rate even for an infinity of space, so naturally, being faithful corporate oligarchs, they declared ownership of the entire infinity of space above their respective cities (a slight increase of the price of their goods easily paid for it all while allowing the rent to establish pure profit).
After the corporations merged, the dutiful tax man dropped by expecting to reap the same booty as always but merely from one accountant rather than from two. Then he got a surprise. The new corporate tax attorney saw that an infinity of space plus another infinity of space was still “just an infinity of space”. So he wrote a check for “an infinity of miles of space”, only half the amount that the IRS expected.
Naturally, the IRS was displeased and demanded that the new merged union had twice the amount of rentable space as each of the separate corporations had alone and so they demanded the same twice amount they had been collecting before the merge. And as always with everything associated with Trump, the lawsuits quickly emerged.
In federal court the issue came down to having to declare definitions for words that indicate a taxable property amount that reflects twice as much as an infinity of space above a city. With such new words, the IRS would be free to simply rewrite the tax code, restore the federal economy, finish The Wall, and save the US from a recession.
So now, seeing how dire the need, what words can we use to indicate a value that is twice as much as an original infinity of cubic miles?
James would have just said something like “2 * infA”, but what word(s) shall we use?
I think the above three lines sum up your point, yes?
You use a good example because 3D volumes extending upwards from the 2D surface of a spheroid diverge as you tend upwards in height to infinity - like a truncated cone with no base, and law requires specific terminology to avoid ambiguity to prevent abuse of any loopholes that would result from any ambiguity. However, the example - and probably any example - can be reduced to whatever the relevant factors are.
Whether land includes the space infinitely above it or not, the infinity doesn’t change for each instance of owned land, and so doesn’t affect any calculation of spatial volume extending above any one or more surface areas.
Example: let V = Ah, where volume, V, is the 3D space above any given plot of land, and A is the surface area of that plot of land and h is the infinite height that extends above this surface area.
The ratio of V₁ to V₂ is the same as the ratio of A₁ to A₂ and is not affected by h in either case, meaning h is not a relevant factor to determining the price of any given plots of land, with or without additions or subtractions from one another. Only if the h values were finitely different would the ratios not be the same.
Therefore, “The new corporate tax attorney” would either write a check for the surface area of both Amazon and Microsoft’s land, modified by any finite factor that represents the infinitude of height above it, or the volume of space above both their combined land areas, or either way separately for each company before adding them together, and it would be the same amount whichever method is used. The infinite height above the plots of land is no different so doesn’t affect anything. For example, assuming both Amazon and Microsoft owned the same land areas as one another, the check for both merged would be twice the amount of just one by itself based on surface area alone. But even without the assumption of this example, the ratio of volumes to land areas owned would be the same in either case and height, h, makes no difference.
The only way that height would be a relevant factor in determining the volumes of space above plots of land would be if the heights were finitely different - only then would they be a relevant factor in determining pricing.
If you think about it, this is not surprising at all, because infinity isn’t a quantity - it is the exact lack of quantity. It’s a quality of unquantifiable endlessness. Even qualities like how yellow something is can be reduced to quantifiable wavelengths, such that they can affect quantities like pricing, but a quality that exactly opposes quantity? - the infinite can by definition never be reduced to its exact opposite that by definition it lacks. For 2 * infinity or n * infinity, the infinity makes no difference.
This is why I say Cardinality is undefined for infinite sets, because by its very definition, an infinite set defies quantity - whether cardinal, ordinal, or anything. The law need not worry - there is no dire need or any need at all - Trump’s new regulations lack any impact and the Wall remains unfinished. Whether or not Democrats are happy about this, I don’t care either way, I’m just looking at the logic here - irrespective of politics and personal/psychological preferences.
It could make a difference if there was a two or more tier system to distinguish whether you merely own the surface area of the land or the infinite volume of space above it too (perhaps even the finite volume beneath it that converges to a centre point within the planet), but this seems pointless, because volume of space above a plot of land is implied for any erected structure with non-zero height. Perhaps a tier system to distinguish different finite heights above a plot of land, with the last tier being infinite, but “however you finitely slice it” the presence of infinities or not makes no difference - only finites do.
This is why finitely bounding infinities to distinguish them is by definition a category error for the particular example you provided, and the logic extends to general cases by definition too.
I hope this makes sense and lays the topic to rest, but I’m happy to discuss it further if need be.
I read the section you selected - it sounds like Wittgenstein and I would have been in agreement.
Intel had only claimed 500 miles above.
Exxon claimed 1000 miles above.
Amazon claimed infinity above.
So Microsoft claimed infinity above.
Except for Microsoft, they were all paying different tax amounts.
When Microsoft merged with Amazon they individually paid only half as much as before because some idiot taught half of the world that “2 times infinite is still just infinite”.
And yes, you missed the point.
There is twice the amount of space being taxed. The tax schedule is described by a converging formula which allows for an infinity of space to be scheduled. When the corporations merge, they have twice the amount of space together as they had individually. But on the tax form they claim that they own merely “an infinity of space” which yields a tax burden that is the same as either corporation had been paying individually.
How is the IRS going to describe the amount of space the new merged corporation has such that the IRS can tax it for both infinite distances?
seems to me that this all started when you/we were reviewing james’s description of various features of his ‘ontology’ as being ‘infinite’ - i take this to mean not only time and space, but energy. i mentioned that i can’t imagine energy being infinite (with a very brief explanation somewhere back a few pages). it was then that i drew attention to the differences between ‘actual’ and ‘potential’ infinities and tried (half-heartedly) to suggest that a mathematical infinity doesn’t necessarily reference actual quantities or address the question of energy being infinite. in my understanding - and i side with W in the view that mathematics is a convention, not some objective feature of the universe - creating infinite mathematical sets only expresses intension; it is an application of conventional rules which govern our use of mathematical functions, processes and relations. what it isn’t is an extension… not actually an extension. that is to say, we don’t observe infinities anywhere like we do finite sets… but we are only ever approaching them… which is why potential infinity exists in the active application of the rules, but not in product, not actually.
i don’t know if this is making any sense to you, but i can’t imagine how it wouldn’t.
another thing that might be relevent in defending the notion that energy is finite is the fact that despite the number of times a body is divided, the total mass of the collection of pieces combined, never increases. so one can divide infinitely, maybe, but add infinitely? i think here is where mathematicians are fooled into thinking that because they can devise potentially infinite sets, these sets actually represent physical states and things in the universe. so, if the numbers can be added infinitely, the things these numbers represent (bodies with mass) can also be infinite.
that conclusion is certainly not reached a posteriori, so as far as such theory has any empirical efficacy, we’re shit out of luck.
and no, we can’t ‘know’ if energy is finite, either, but a great deal of logical reasoning brings one to that conclusion and is far more parsimonious than the contrary theory; that energy is infinite. i think the idea that it’s infinite is a kind of psychologistic sleight of hand resulting from applied mathematics.
anyway W is saying something incredibly simple. he’s looking at the way we use this concept of infinity and how it is made meaningful… not just ostensibly as in the case of mathematically potential infinities.
Your point is that you have 1 lot of infinite space added to 1 other lot of infinite space, and so in pricing 1 * infS + 1 * infS, one ought to equate that to 2 * infS so as to avoid paying the same price for 2 lots together as you do for 1 by itself, yes?
You can refer to me as an idiot and make accusations all you like, but it’s only serving to entrench yourself in your current position and making it harder to see the sense in a position that shows the flaws in your position.
The following conflation is being made in your argument:
We have different variables at work here: the infinite space that both Amazon and Microsoft own, and the respective finite prices or finite taxation of these infinite spaces.
In order to calculate a finite amount of tax due on these infinite spaces, you offer “a converging formula” to convert the infinite space into a finite price.
It is required for the infinite space to be converted in such a way to a finite value in order to deal with it quantitatively, and to then convert it to a finite price, in order to extract a finite tax from it. Without it, infinite space added to infinite space is still nothing more than infinite space, which can’t be mapped to anything but an infinite price and infinite tax - and that is no use, and the price and tax would be just as infinite as the space, whether combined with other spaces, prices and taxes or not.
So what you’re doing is referring to the addition of two infinite spaces in order to argue about two finite prices, to conclude that we need to have 1infS + 1infS equal to 2infS, when the only thing we need is to convert infA into a finite so that 1price + 1price can equal 2price.
The only way to conclude as you do is to conflate the finite price with the infinite space that it quantifies, which is reliant on the conversion of the infinite to the finite in order for you to do so.
Your argument is fine for Intel and Exxon, who claim finite miles above their plots of land in your example - no conversions and conflations would be required and it’s nice and simple. You can use the conversion to standardise the whole taxation calculation between finites and infinites, sure, but that doesn’t conflate the two, it only allows you to treat them both as though they were finite for the purposes of price and taxation.
But none of this conversion and/or conflation is needed in the first place, because you can just treat the surface area of the plots of land as the only relevant factors and get the same desired finite results without tying yourself up with infinity contradictions and paradoxes - as I explained in my last post. Obviously I was jumping too far ahead for you with that as I keep doing, but the case is open and shut in this post alone. Though I’ll happily consider new scenarios that you come up with for as long as you need me to.
I read your post and addressed it a few pages ago, though it probably got lost. I explain how both logically and empirically energy can’t be infinite.
I haven’t referred to you as an idiot.
Yet.
And I am not making an argument either.
How many times do people have to repeat themselves to you before you can understand what they say?
I asked a question. I asked for a word or words which specify how much space a corporation owns.
The IRS cannot accept another form for each plot.
How much do they own?
That is false.
James did it simply by defining one term, “infA = [1+1+1+…+1]”
That would allow the IRS to handle the situation perfectly. But you don’t like that term, so pick one of your own. So far, you are saying that what the corporation owns cannot be described except by long sentences.
I don’t remember who, but I remember someone in academia pointing out that without the ability to concisely describe complex situations, intellectual development is lost. That would explain a lot and is why the word “dumb” refers to both people who cannot speak as well as people who cannot think. Language matters.
You say that such can’t be done. That is absurd. You could just declare that:
Becor = an infinity of space.
Then easily say that the corporation owns two becors.
Then I guess the “idiot”, in your words, who “taught half of the world that 2 times infinite is still just infinite”, which is what I’m teaching, was an idiot for other reasons than for his teachings…
The evasion of explicit commitment through technical amiguity in the implied… well, whatever you say, man.
And you’re not making an argument?
Well you are, but you’re currently more concerned with trying to keep me firmly within the confines of a particular train of thought in order to later bring it back to your argument, in the way you need me to, in order to accept your argument. You get very annoyed when I poke holes in it and show you a better train - I’m almost tempted to just agree with everything you say just to get you back to the argument, but the whole problem is in the flawed way you’re getting to it - leaving us both in this frustrating purgatory. You want things to go a very specific way to validate your argument, I want to show you the problem with your very specific way => you accuse me of evasion and distraction and complain about having to repeat yourself when all I’m doing is trying to save you the trouble.
But whatever, again, with the meta stuff. To the content, right?
What is the difference between “becor” and “infA”? If I rejected infA, why would I accept becor when it’s defined the same way? What is this quest to find a better vocabulary for a flawed concept, when the whole problem is not the word we’re using but the fact that what it means leads to paradoxes and contradictions?
Fine, for the sake of the IRS, let’s call “how much space a corporation owns” 1 becor instead of infA or an infinity of space, whatever.
Anything to give an infinity the semblance of finitude, right? We’re back to “1 infinity” forcing quantity into the notion of anti-quantity like I already explained, but who’s complaining about repeating themselves, right?
Whoever is saying “the inability to concisely describe complex situations is the loss of intellectual development” is just reiterating Occam’s Razor.
All I’m saying is (¬f) ≡ ¬(f), where f is finitude.
Oh look, the corporation now owns “2 becors” therefore 2 infinities are twice as many as the single infinity “1 becor” that they each owned previously!
I feel like I’m playing peekaboo with a child… - I’ve seen where your train of thought wanted to end up decades ago, I hope you appreciate I’m only playing along to diminish the intermittent tantrums you play out when I’m not playing your game “right” (how you want me to). I’m “distracting”, “evading”, “missing the point” when I point out the flaws in each stage and thus the conclusion, and when I suggest a better way that doesn’t encounter contradictions.
So you’ve proved that 2 finite-ified concepts of infinity “becors” (a contradiction) are twice as much as 1 in the same way that it’s already been proven that for finites 1+1=2, I’m so proud of you!
So back to the discussion, let’s examine the issues in treating the infinite as finite. Do you want to do philosophy or are you happy with kindergarten?
James never said “2 infinities are twice as many as the single infinity”.
There are practical and rational reasons dictating the words used in maths. The word infinity had one specific intended and limited meaning. As James explained (and you can check out the truth of it) the word was simply not defined sufficiently to be used in arithmetic operations. When you try to use its originally intended definition in an arithmetic operation, you cause confusion because important details of the word are missing.
New words are made because old words were insufficient.
James made a new word “infA” specifically for the purpose of giving sufficient detail to allow arithmetic operations, such as counting, adding, multiplying, and so on to be used on practical concerns about infinite issues and qualities. The most obvious of his examples is when there are two or more infinite lines or dimensions in space and you want to know how many variations of field strength can exist.
And you want to know that so the question posed by this thread can be rationally answered.
You cannot answer that question without doing something very similar to what James did with his infA = [1+1+1+…+1]
They say that need is the mother of invention. James had a pregnant idea. He answered a question that could not have been logically answered without the detailed definition that he provided. Your words would never have allowed you to answer that question.
You are stuck with “we don’t know but my opinion is…” because your words don’t allow you to figure it out.
James defined the (by definition and derivation) IN-finite i.e. undefinable.
Accolades and glory to the man who dared to so boldly contradict himself…
Please reach out and touch back down to earth by acknowledging that (¬f) ≡ ¬(f). One thing is not its definitional negation, and treating it as its negation isn’t exactly creative or new…
Fuck it, maybe I’m wrong - e.g. complex numbers are a direct affront to logic and yet can produce some astounding and useful mathematics as a consequence of accepting their usage - maybe other contradictions can enjoy the same fate.
I just wish this mickey mouse bullshit could end if you find yourself unable to concede to obviousness.
Is logic “just my opinion”? We can discuss this if you like?
It’s super obvious to figure out what you’re trying to do, I’m not unable to figure out what you mean by any stretch of the imagination, it’s like the most basic thinking that I feel bad to humour you as much as I already am doing.
He contradicted nothing. Although he agreed with what are now the most famously brilliant mathematicians on the planet.
As I implied at the very beginning of this, YOU have problems with the rational use of words (as it seems Wittgenstein had) and that is what is leading you into these conflicts and connundrums with the better mathematicians.
It has been your misuse of the words while trying to construct logic. Of course you are going to eventually be wrong.
