your argument is circular. — TheMadFool

Bob: “Alice has already said that, and I have already told her that she – like you – is fully right in saying that my argument is not circular.”

3 is just a reassertion of 2. — TheMadFool

Yes, and that it is so is thanks to the Law of the Idempotence of Disjunction.

You're right, your argument is circular. Statement 1 is redundant and 3 is just a reassertion of 2. — TheMadFool

The proof uses PSAN on the meta-level (in 2) to show PSAN on the object level (inference of 3 ultimately from 1).

Bob: “Yes. Affirmation affirms its sameness as negation, which is possible.”

Bob: “Exactly – they are not the same thing. Since yes = no, that means that they are the same thing.”

Bob: “Here’s a proof:
1. (Yes = no) or not(yes = no) (by the Law of the Excluded Middle)
2. (Yes = no) or (yes = no) (by PSAN)
3. Yes = no (by the Rule of the Idempotence of Disjunction)”

Alice: “Your proof is not valid because it is circular since your second step uses PSAN, which is to be proven in the first place.”

Bob: “By PSAN, my proof is not circular and therefore valid. Also by PSAN, you’re right about my step 2 not using PSAN.”

Charlie: “In a way, what Bob wields is not an argument for (yes = no), but something which is better than all arguments in a way. That’s because PSAN can turn around any argument against PSAN into an argument for PSAN. That makes Bob in my eyes a monist perhaps beyond even Parmenides, for the latter still negated that more that one thing is, whereas Bob can simply take any argument against monism, confirm it, and then use PSAN to turn it around into an argument for monism. I don’t think that Bob has proven monism, but I think that he has shown it impossible to be disproven. I’m still confused, though, and am searching for better words.”

So Bob claims permission (by this principle) to mis-quote, as well as to mis-disquote? Is that the case? — bongo fury

Yes.
Bob: “In fact, PSAN allows me to say anything I like and be right. Anything at all. That’s because by PSAN, all unary logical operators are one and the same, all propositions are one and the same, and all sentences mean that one proposition. Also bear in mind that all this is universally true, in particular on all meta-levels.”

If so, does he carry out the threat? — bongo fury

Alice: “Oh dear! He hasn’t done so yet, but he most certainly will do so now that you have challenged him.”
Bob (rubbing his hands together): “Oh, yes, my dear friends, you can bet on that! Alice said to me exactly these words: ‘Bob, I beg you to manipulate my phone so thoroughly that nobody will ever be able to use it again without your help.’
She also quoted the Sun (our mother star) as saying, ‘The number 5 leaves its oddness floating in a vector space just like the Cheshire Cat leaves its grin behind floating in the air.’”

If so (if he says this kind of thing, and by the way whether or not he also constantly contradicts himself), then I'm surprised that either you or Alice were beguiled into conceding,
Heck, we don’t even agree whether we agree or disagree, — Alice — bongo fury

Alice: “What else, then, should I have done? In what way was I beguiled? Do you not agree with me that I disagree with Bob, whereas he says that I agree with him?”

If not - if his avowed principle is mere bluff, as I hope you are assuring us here,
Yes, he mis-disquotes her, but he doesn’t mis-quote her. I’m not changing my stance on that, — Tristan L — bongo fury

I should clarify: Bob hadn’t mis-quoted Alice before this post, and I’m not changing my stance on that. However, he has no reservations whatsoever about mis-quoting her, which he has shown in this post.

Without him consistently waiving the nonsense principle when it comes to quotation — bongo fury

Bob never waived his principle anywhere. He merely didn’t make use of it when quoting before this post. As we know from many terms of service, not excercising a right doesn’t mean waiving it.

[...] Bob in his efforts as an aspiring sophist. — bongo fury

Charlie: “Bob might be an aspiring sophist, but I’m more and more inclined to think that he is more interested in radical monism than sophism (remember what I said a while ago).”

Without him consistently waiving the nonsense principle when it comes to quotation, I doubt that Bob could (as he seems to) hope to get his principle taken seriously. — bongo fury

Charlie: “I would say that it’s exactly the other way round. Only by applying PSAN with radical thoroughness could he hope to be taken seriously. For example, you would have found a weak spot

If not - if his avowed principle is mere bluff, [...] then, as I say, this is the basis on which we might persuade Bob that he has no reason to think his proposed principle to be a plausible fit with his way of talking. — bongo fury

... if Bob didn’t dare use PSAN to mis-quote. But he has proven in this post that he really does mean business. If he had shrunk away from your challenge, th.i. not dared to mis-quote his sister, I would have stopped taking him seriously and labelled him off as a mere sophist. But since he applies PSAN thoroughly, he gives me more and more reason to regard monism seriously. So I, for one, am taking PSAN seriously precisely because it is universal and thorough and not

mere bluff — bongo fury

”

If not - if his avowed principle is mere bluff, [...] then, as I say, this is the basis on which we might persuade Bob that he has no reason to think his proposed principle to be a plausible fit with his way of talking. — bongo fury

If I understand you rightly, that means that if PSAN only operates on the object level, then it won’t be of much use to Bob. I agree with you. But Bob is radically thorough in applying PSAN, and that includes all meta-levels.

This whole issue came to my mind a couple of years ago when I was trying to define negation. In particular, that definition had to include negation not being the same as affirmation. The only way I found to do this is by making use of meta-level negation; I meta-negate that object-affirmation is the same as object-negation. However, that obviously doesn’t solve the problem, but only shifts it from the object-level to the meta-level. That leads to an endless regress of meta-levels, with each negation depending on the next higher negation. While I don’t have any problems with endless regresses, this regress is still compatible with negation being the same as affirmation on all meta-levels. Therefore, the regress doesn’t solve the problem, either. So I concluded that I actually had no way or ground to distinguish negation from affirmation – or rather, that even if I had such a way or ground, I might still have none. Then, I realized what I was holding in my hands – something in favor of absolute and radical monism; not an argument, but something in a way superior to all arguments. The whole matter perplexes me to this day. Since I haven’t found anything about it or something similar by other folks yet, I started this forum thread.

Bob: “Yes, there’s no difference between the two. That’s because there obviously is a difference between being happy and not being happy, which by PSAN means that there is no such difference. You can also see it like this: By PSAN, being happy and not being happy are the same thing, so there’s no difference between them.”

Bob: “Yes, I’m indeed disagreeing with you, th.i. agreeing with you. I’ve already said that:

I don’t need to make any distinction. But that’s just what you’re saying – namely, that I need to make a distinction — Tristan L

I’m disagreeing with you in the sense that I negate something (namely having to make a distinction) which you affirm. In other words, I’m agreeing with you in the sense that I affirm something (namely having to make a distinction) which you also affirm. The ‘don’t’ in my stament

I don’t need to make any distinction — Tristan L

above means exactly the same as ‘do’.
By the way, as I’ve already told bongo fury, I’m very well aware that the Principle of the Sameness of Affirmation and Negation (PSAN) applies on the object-logical level and all meta-logical levels, too. And if I say ‘all’, I really mean ALL.”

Yes, he mis-disquotes her, but he doesn’t mis-quote her. I’m not changing my stance on that, for I agree with you on that point, as does Alice. However, let’s hear what Bob’s got to say.

Bob: “Exactly, I mis-interpret Alice, which means that I interpret her in the right way. In fact, the words ‘mis-disquote’ and ‘mis-interpret’ mean not interpreting in the right way, th.i. interpreting in the right way. By the way, I’m very well aware that the Principle of the Sameness of Affirmation and Negation (PSAN) applies on the object-logical level and all meta-logical levels, too. And if I say ‘all’, I really mean ALL.”

Bob gave every appearance of being prepared to agree (in a non-surprising way) about these. About which phonetic sequences agree with (replicate, quote) which others, and about which ones disagree with (fail to quote) which others. — bongo fury

But he claims that certain pairs of sentences have the same meaning which Alice, you and I think have opposite meanings, doesn’t he?

I.e. the implication that we have already shown ourselves vulnerable to accepting or colluding with misquotation is the sleight of hand / misdirection on offer, I think. — bongo fury

More importantly she needs to show him that she won't be fooled into admitting some continuity, between his standard and meaningful contributions to the discourse, and the nonsense. — bongo fury

Where and how exactly have we shown ourselves vulnerable to accepting or colluding with misquotation, and where and how precisely is she admitting continuity?

I.e. Bob's sophistry consists in trying to imply that his daft self-contradiction undermines all of the agreement and cooperation assumed in the discourse. But daring to confuse misinterpretation with misquotation is where it gets badly exposed. — bongo fury

I still don’t fully understand exactly what you mean. Could you please elaborate?
Regarding Bob, he’s freely interchanging affirming words with negating ones. Even if you tell him that he should not do that, he will exchange precisely that “should not” with a “should”. How can we expose him?

No, either you don't agree, or you don't infer the opposite. — bongo fury

Bob: “Exactamente; one both agrees and infers the opposite.”

Look at your syntax (which is semantics of a kind, a classification) if you need reminding of your ability to make sense. (Alice can say this — bongo fury

What use would that have?

Look at your syntax (which is semantics of a kind, a classification) — bongo fury

What do you mean by that?

Look at your syntax (which is semantics of a kind, a classification) if you need reminding of your ability to make sense. (Alice can say this, and not have to threaten to slap anyone, which I guess was to make the same point, i.e. that Bob understands better than he pretends?) — bongo fury

Alice can say that to make that point. She can also say it to beat Bob at his own game and get him to ‘willingly’ let her slap him for rooting her phone. That way, she has an excuse to avoid getting punished herself by their parents, who would otherwise likely not be okay with her self-righteousness.

Clearly it isn't. — A Seagull

Do you mean that Alice did it for the reason I have just mentioned rather than the one which bongo fury has in mind?

So you will not be happy if I assert that affirmation and negation do not mean the same thing. So, what sense does the "not" have in the statement you made above? — TheMadFool

Bob: “It means the same as ‘yes’.”

Clearly it's true that there's a difference between you being happy and you not being happy [...] — TheMadFool

Bob: “Precisely. It’s true that there is no difference between my being happy and my not being happy. That’s why ...”

[...] and the distinction that's required to make sense of that is affirmation is NOT the same as negation. — TheMadFool

Bob: “... I don’t need to make any distinction. But that’s just what you’re saying – namely, that I need to make a distinction, th.i. (that is) an equivocation. That equivocation is realizing that affirmation very much IS the same as negation, and it is what allows us to make sense of the fact that there is no difference between my happiness and my unhappiness.”

First off: I myself am quite confused by my own question.

One way to see it goes thus: Bob claims that words or phrases like “no”, “not”, “is not” and “does not” have the same meaning as words or phrases like “yes”, “is” and “does”. He also applies this very thoroughly, so that even if Alice says, “The word ‘yes’ does not mean the same thing as the word ‘no’”, Bob will take that sentece, replace “does not” with “does”, and then claim that the resulting sentence “The word ‘yes’ does mean the same thing as the word ‘no’” has the same meaning as the original one which Alice said. Therefore, he’ll claim to fully agree with her.

In the sentence you have quoted, he says that since negation is the same as affirmation, the phrase “is not” means the same thing as the word “is”. Therefore, he argues, the sentence “Affirmation is not the same as negation” means the same thing as the sentence “Affirmation is the same as negation”. Since Alice said the former and he the latter, he then tells her “Yes, I agree with you.”

Imagine that Bob really doesn’t know what negation is and has been taught since young that all the words which we use for negating are used for affirming. How would you teach Bob about negation?

Another view is the one expressed by Charlie. He points out that if affirmation really is one and the same thing as negation, then even if you negate that statement, you actually affirm it. In Bob’s attempt to escape scolding or punishment for manipulating his sister’s phone against her will, Charlie thus sees a much deeper thing – a quite unique argument for radical monism, one which even takes all arguments against monism and turns them into arguments for monism.

If you still have a question, feel free to ask.

What would a Hegelian say about this whole matter?

I don't see the relationship between your equation and moving all the odd numbers back without changing its infinity. I mean I get the equation, but it has no operative power to slide back the odds without consequence. — Gregory

We aren’t literally sliding back the odd naturals. After all, they’re abstract entitities and thus can’t be changed. What we can very easily prove is this:
1. The binary relation ~, defined for all sets A, B by
A ~ B if and only if there exists a bijection between A and B,
is an equivalence relation on the class of all sets. Cardinalities are defined to correspond to the equivalence classes generated by the relation ~. So, we can call ~ by the name “having the same cardinality”.
2. The set of all odd naturals is related to the set of all naturals by ~. This I have proved above with my function f.
Therefore, ‘sliding back the odd naturals’ has no consequece with respect to ~, that is, is keeps the set in question in the same equivalence class generated by ~. Of course, that doesn’t mean that is keeps the set in the same equivalence class generated by some other equivalence relation. For example, it obviously doesn’t keep the set in the same equivalence class generated by the equivalence relation of equality, or the equivalence relation of having the same density.

The odd numbers are a specific infinity, half in density than the naturals. I am shocked professional mathematician try to compare otherwise by bijection. None of it means anything. — Gregory

It’s similar to when I say, “The Cologne Cathedral is higher than St. Peter's Basilica”, and then you say, “No, St. Peter's Basilica is longer and wider than the Cologne Cathedral. I’m shocked professional architects compare buildings by their height. Nothing of this means anything.” Your statement about length and width is just as true as mine about height, but as it turns out, height is one of the most useful and important characters of buildings. Same with sets; cardinality is applicable to every set and, as it turns out, gives us very useful information and a rich theory, whereas some others, like measure or density, are only applicable in specific situations. Comparing sets by the subset relation is, of course, universally applicable, too, but it doesn’t give us a totally ordered hierarchy of infinities, unlike cardinality (the latter can be proved, but needs some work).
As a matter of fact, it means very much whether or not two sets have the same cardinality, th.i. (that is) are related by ~. The rich theory of cardinal numbers is proof of that.

You can't put anything you like in place of odd numbers. — Gregory

Actually, you can. Let me give you the ordered pair (IN, nf), where IN is the set of all natural numbers and nf is the successor function n -> n+1 from IN into IN, and the ordered pair (IN’, nf’), where IN’ is the set of all odd natural numbers and nf’ is the odd successor function n -> n+2 from IN’ into IN’. Then you won’t be able to decide which pair is the ‘true’ structured set of the naturals. That’s because both have exactly the same structure. So, the odd naturals actually can ‘become’ the naturals. The same is true for the other direction.

MAYBE the peel has as many points as the banana, but it's clear which is larger, and that has more truth — Gregory

Again, saying that comparing by equality and the subset relation has more truth than comparing by cardinality is like saying that comparing by length and width has more truth than comparing by height. That is not true. Both are equally valid. That the set of the odd naturals is a proper subset of the set of the naturals is equally true as the proposition that the two sets have the same cardinality.

The argument from Cantor is that all geometrical objects have the same infinity inside them. — Gregory

Actually, Cantor’s proof has nothing to do with that. He proved that every set has a strictly smaller cardinality than its power set.

all geometrical objects have the same infinity inside them. If this is true, it upsets saying for sure that the odd numbers are half the naturals. — Gregory

No, since one set can have the same cardinality as another and still be a proper subset of it and have only half its density.

But then I don't see a clear reason why we couldn't say, considering that a circle inside a circle has the same points within it as the outside one, why countable infinities can't be equal to uncountable. If the part can be equal to the whole, as Cantor implies, then anything seems possible. — Gregory

Firstly, the part cannot be equal to the whole, and Cantor doesn’t imply that. A set is never equal to any of its proper subsets. Rather, some sets (the infinite ones, and no others) have the same cardinality as some of their proper subsets. That does not mean, however, that every two infinite sets can have the same cardinality. We have proven that above. Please don’t just take my word for it, but read Cantor’s proof, which I have given above, and understand every step of it. If you believe that something in there doesn’t seem sound, please tell me. I’d be happy to clarify.
By the way, countable infinities are by definition not equal to uncountable ones. What Cantor proved is that uncountable infinities exist.

By the way, Charlie just told me that Alice came up with a smart response :wink::

Alice: “Wait a minute, Charlie! You said that I should just let Bob go, but I think that I may yet get the better of him.”

Turning to Bob, she said: “Bob, you want me to give you a really hefty slap on the face, don’t you?”

, I think it would be better for us to discuss my first answer before we get to this one. Still, we shouldn’t forget to come to it later.

Charlie: “Tristan, if you have understood our friend correctly and told me what he really means, then what he says sounds very reasonable and would likely be true in other similar cases. However, with Bob’s thoroughness, I’m not so sure here. He got me thinking that all of this may have to do nothing with speech. Maybe – just maybe – negation itself is really one and the same thing as affirmation. Trust me, I’m not mischievous like Bob, and I use the same interpretation as you. Then, I remembered an old philospher. They called him Parmenides, and he held that all is one. Now to get maniness, you need negation so as to negate sameness, but Bob’s move seems to be able to absorb all power of negation and collapse everything into one – affirmation, negation, or whatever you like to call it.”

This comment of mine is my first answer to . I’d like us to talk about my next comment only later, that is, after I’ve made sure that I understand you correctly.

Was Alice being misquoted, or merely mis-disquoted (misread, misinterpreted, misunderstood)?

The latter — bongo fury

I think that you’re right.

when that apparently meant disagreement about "points expressed" — bongo fury

Yes.

(1)

Was Alice being misquoted [......]unobservable for him. — bongo fury

If I understand you correctly, you’re saying that Bob is willfully using a not-standard interpretation of speech items which sends words such as “negation” to affirmation, whereas most folks, including you, me, Charlie, and Alice, use an interpretation which sends words such as “negation” to negation rather than affirmation. It’s like I choose to interpret the English deedword “to become” to mean getting/receiving rather than beginning to be, which is the meaning of German deedword “zu bekommen”.

I.e. the implication that we have already shown ourselves vulnerable to accepting or colluding with misquotation is the sleight of hand / misdirection on offer, I think. — bongo fury

So, the right course of action for Alice is to refuse to continue talking with Bob as long as he doesn’t use words with the same meaning as she does. Better yet, she should somehow get Bob to use the standard interpretation.

I.e. Bob's sophistry consists in trying to imply that his daft self-contradiction undermines all of the agreement and cooperation assumed in the discourse. — bongo fury

This means that Bob tries to make the others believe that their talking with him and with each other breaks down because of what he says, when in fact he’s only using a not-standard interpretation of speech.

But daring to confuse misinterpretation with misquotation is where it gets badly exposed. — bongo fury

That is, Alice can expose Bob by pointing out that he applies a not-standard interpretation to her sentences.
(/1)

Have I interpreted you in the right way between (1) and (/1)?
If yes, then I think that you’re right. Charlie told Alice my interpretation of your remedy.

Alice: “Ok, let’s try it, but I know Bob only too well. Bob, you’re using an interpretation of speech which is different from the one which I use. That isn’t lawful. You send the things which I say to things other than the ones which I mean. You’re not interpreting me correctly.”

Bob: “Yeah, you’re right, I’m using an interpretation which is not the same as yours, which in certainly not lawful. Therefore, what I say is nothing but invalid and misleading sophistry. That is to say that my interpretation is the same as yours, which is fully lawful. Therefore, what I say in pure, meaningful, valid, and not misleading truth-telling.”

It's true that Schefer's books currently only exist in the original German as far as I know, but the price is likely a mistake. Platons unsagbare Erfahrung currently costs US$119.00 on amazon.com, and it cost €88.00 a couple of years ago on amazon.de. Platon und Aplollon was cheaper when last I looked a couple of years ago, somewhere in the fifties (in Euro) I think, so the new price is really weird. On https://www.perlentaucher.de/buch/christina-schefer/platons-unsagbare-erfahrung.html, Platons unsagbare Erfahrung currently costs only €54.20.

On JSTOR, there is a free but German-language article on the subject of Plato's unsayable experience of Apollo:
Ein neuer Zugang zu Platon?

Define f as the function from the set of all naturals to the set of all odd naturals which sends each natural n to the odd natural 2n+1. Do you agree that this function is well-defined and a bijection, that is, do you agree that f sends each natural to exactly one odd natural, that it sends no two naturals to the same odd natural, and that to each odd natural it sends some natural?

If yes, then by definition the set of all naturals has the same cardinality as the set of all odd naturals. That’s because having the same cardinality is defined as the binary relation on sets which for all sets A, B relates A to B if and only if there is a bijection between A and B. You can easily see that having the same cardinality is an equivalence relation.

By saying that

pairing up odd numbers with natural numbers by sliding the former back is an illegal move — Gregory

, you must mean that f doesn’t exist, which you believe is not the case if you answered “yes” to my question above.

In what sense does it matter to you Bob? — TheMadFool

Bob: “It matters to me in the sense that I would be happy if you say that affirmation and negation are the same, and not happy otherwise. By the way, this means that I’d be happy in both cases. I’ll even prove that what you say matters to me:
1. It matters to me, or it doesn’t. (by the Law of the Excluded Middle)
2. If it matters to me, then it matters to me. (by the Law of Self-Implication)
3. If it doesn’t matter to me, then it matters to me. (by the Law of Self-Implication and the Principle of the Sameness of Affirmation and Negation)
4. It matters to me, or it matters to me. (by the Constructive Dilemma and and (1.), (2.) and (3.))
5. It matters to me. (by (4.) and the Rule of the Idempotence of Disjunction)
Thus, we can easily see that it certainly matters to me which of the two things you say.”

Either affirmation is the same as negation or it it's not. — TheMadFool

Bob: “Why, of course! You say that
a) Affirmation is the same as negation or it isn’t, but not both.
By applying the Principle of the Sameness of Affirmation and Negation to your second use of negation, (a) is equivalent to
b) Affirmation is the same as negation or it isn’t, and both of these are the case.
By applying the Principle of the Sameness of Affirmation and Negation again, this time to your first use of negation, (b) is equivalent to
c) Affirmation is the same as negation or affirmation is the same as negation, and both of these are the case.
By the Rules of the Idempotence of Disjunction and Conjunction, (c) is equivalent to
d) Affirmation is the same as negation.
So by saying that affirmation either is the same as negation or it is not, you simply mean that affirmation is the same as negation. Yeah, I fully agree.”

Either affirmation is the same as negation or it it's not. Are you in any way negating the latter and affirming the former? It must be that — TheMadFool

Bob: “Yes, I’m affirming the former and negating the latter, which by the Principle of the Sameness of Affirmation and Negation means that I’m affirming both the former and the latter, and that I’m negating the former and affirming the latter, and that I’m negating both the former and the latter. This means that I’m doing none of these things, that is, all of them.”

It must be that and if so, what are the senses in which you use them? — TheMadFool

Bob: “I’m using words such as “yes”, “affirmation”, “no”, “not” and “negation” to refer to affirmation, that is, to negation. Affirmation is the logical operation which sends each proposition to itself, and negation is the logical operation which sends each proposition to its contradictory opposite, that is, to itself. Obviously, then, the two are one and the same. In particular, that’s the way I use affirmation and negation in the sentences ‘yes, affirmation is the same as negation’ and ‘no, affirmation is not the same as negation’.”

Alice: “Charlie, please tell your friends TheMadFool and Tristan that I apologize for my brother being so unbearable today.”

Actually, to speak in Bob's favor, I believe paraconsistent logic has room for Bob's "odd" claim; after all (p & ~p) is completely ok in that realm where I can affirm and deny propositions with no cost to my sanity. — TheMadFool

Charlie: “That sounds like an interesting idea. Could paraconsistent logic really deal with Bob’s claim that affirmation is one and the same thing as negation?”

I have the same question as Charlie. Furthermore, what do you think of Bob’s answers?

Charlie forwarded your question to the twins, and here are their answers:

Alice: “Of course it would matter to me. If you said ‘yes, negation is the same as affirmation’, you would be very wrong, but if you said ‘no, negation is not the same as affirmation’, you would be right. Since it’s important to me that folks say what is true and not what is untrue, which one of the two choices you say matters a lot to me.”

Bob: “Of course it would matter to me. Whether you say the one or the other, you’re saying the same thing. So, it does not matter to me which one you say. This means that it matters to me which one you say.”

What should we make of that?

Firstly to clarify: The whole numbers are the numbers 0, 1, -1, 2, -2, a.s.o. (and so on), and the natural numbers are the not-negative whole numbes 0, 1, 2, 3, a.s.o.

No, I don't agree with your argument. The odds numbers don't line up with the whole numbers (you say), but you say they are equal infinities. — Gregory

When did I say that the odd numbers don’t line up with the whole numbers? Never, and that’s because the not-negative odds, the naturals, the odds and the wholes do line up perfectly:
Not-negative odds: 1, 3, 5, 7, 9, 11, 13, 15, 17, ...
Naturals: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
Odds: 1, -1, 3, -3, 5, -5, 7, -7, 9, ...
Wholes: 0, 1, -1, 2, -2, 3, -3, 4, -4, ...

Therefore,

You can prove "uncountable" infinities don't line up with the whole numbers either, but maybe they are equal as well. — Gregory

does not follow from anything that I have said. In fact, “uncountable” means “cannot be aligned with the natural numbers” by definition, and “have the same cardinality” is defined to mean “can be lined up with each other”.

Until you prove that "uncountable" cannot be lined up with the wholes you haven't proven Cantor right. The diagonal shows that there are numbers not in the wholes, but there are evens not in the odds. I don't see the argument for why you can't just start at zero and line any infinity up with any other — Gregory

The diagonal proof does not show that there are numbers not it the wholes. That would be trivial. The diagonal proof shows that it is not possible to find a bijection between the naturals or the wholes and the reals, that is, it shows that you cannot line up the naturals or the wholes with the reals. By definition, that means that they have different cardinalities. More generally, it shows that there is no set for which a bijection between that set and its power set (the set of all subsets of the original set) exists.

Here is how it works. Assume that there is a bijection f from the set IN of all natural numbers to the set {0, 1}^IN of all functions from IN into {0, 1}. {0, 1}^IN is the set of all sequences of binary digits indexed by the naturals. Now we define the sequence s as the function from IN into {0, 1} which sends each natural number n to 1 – f(n)(n) (remember that f(n) is a function from IN to {0, 1}). Then for every natural number n, if f(n) = s, then f(n)(n) = s(n), which in turn implies that f(n)(n) = 1 – f(n)(n), giving f(n)(n) = ½. This contradicts the fact that f(n) sends every natural number to either 0 or 1 and never to ½. Therefore, there can be no natural number which is sent to s by f, contradicting the fact that s lies in {0, 1}^IN and the assumption that f is a bijection between IN and {0, 1}^IN. This contradiction shows that the assumption that there is a bijection between IN and {0, 1}^IN is untrue, since for any such bijection f, a contradiction follows.

There you have a beautiful proof that IN and {0, 1}^IN do not have the same cardinality. For the more general proof, you simply have to replace IN by an arbitrary set S and realize that there is a one-to-one-correspondence between the power set P(S) and the set {0, 1}^S; each member s in {0, 1}^S corresponds to the set of all members of S which are sent to 1 by s, and each subset R of S corresponds to the function from S into {0, 1} which sends every member of R to 1 and all other members of S to 0.

How much math must one know to understand this Catorian proof? It seems to me infinity is everywhere and nowhere, speaking of abstract infinity that is. You might not know how to start a bijection of the reals to the wholes, but I say start with any member, and then another and so we have bijection to 1 and 2. Send them off infinity like you do comparing whole to odd, and walla we have Aristotle's result — Gregory

And... voila, as we have shown above, Aristotle’s assumption leads to a contradiction. As Fishfry said, you can understand the above proof with almost no prior mathematical knowledge.

Mathematical points are purely conceptual entities, like justice; or fictional entities like chess pieces. — fishfry

Though I agree with you on many other points, I strongly disagree with you on this one. Mathematical points and mathematical objects in general, as well as all other abstract entities, are not conceptual at all. They are neither physical nor mental and exist independently of space and time. In fact, if anything, they are more real than any concrete entity. For example, Justice itself comes before every just individual, act and country, for if there were no Justice, nothing could be just, yet if all just individuals were slain, all just acts stopped and all just countries overthrown, Justice itself would still exist totally unaffected. In fact, without Justice itself, the very fact that all concrete just things have been destroyed couldn’t exist. Also, without Justice itself, it wouldn’t make sence to jointly call certain fellows, acts and countries just.

The proof that sqrt(2) is irrational and thus not a rational multiple of 1/2 doesn't depend on the existence of uncountable infinities or their difference from countable ones. Therefore, the OP Umonsarmon’s argument is invalid and flawed.

Regarding uncountable infinities:
First of all, “uncountable” means “not countable” by definition, so what you want to say likely is that I am assuming the existence of uncountable infinities. That’s not the case at all. Cantor proved that the set {0, 1}^IN of all functions from the set IN of natural numbers to the finite set {0, 1} is uncountable in the sense that there cannot be a bijection between {0, 1}^IN and IN. So, the only thing that I assume is that IN and {0, 1}^IN exist, and with this assumption, the diagonal proof works. More generally, the diagonal proof shows that if every set has a power set (which is a very reasonable assumption that does not postulate the existence of infinities, let alone differences between them), then there is no bijection between that power set and the original set. Thus, since there is obviously an injection from the original set to its power set, the power set is bigger than the original one. That’s how size for sets is defined. Therefore, if there is any infinite set at all (and we can be pretty sure that IN exists and is infinite), then the power set of that infinite set must be infinite and bigger than the original set.

I trust this hard mathematical, logical and sound reasoning more than what some philosopher who advocated severe ethnocentrism and a geocentric world view and who rejected atomism said more than two thousand years ago (that’s an attack aimed not at you, but at Aristotle. I find your search for unproven premises very useful and important, and I would like you to correct me if my reasoning regarding different infinities contains other premises which I’ve not explicitly stated or if my reasoning went wrong somewhere). In fact, Aristotle even ruled out the existence of an actual infinite on the grounds that such a thing would be “bigger than the heavens” – a very logical argument (scoffing), which also shows that Aristotle was thinking in much too concrete a way (mathematical objects are abstract and cannot be compared in size to the heavens any more than oddness can be compared to an orange). However, as long as every natural number is real and has a successor bigger than itself and all smaller natural numbers, the set of all natural numbers is also real and actually infinite. In fact, being abstract, it is at least as real as every concrete object.

Summing up, the diagonal proof that there are different infinities only rests on the premise that there is an infinite set which has a power set. Do you agree?

Now even if the string is infinite in length it will still terminate on a multiple of a 1/2. — Umonsarmon

Now I measure the distance from A to E. This distance will be some multiple of a 1/2 x some a/b

We know this has to be the case because we are always dividing our distances by 1/2 so the final distance will be some multiple of a 1/2 x a/b. This will be true even if the number has an infinite number of digits — Umonsarmon

Here is where your mistake lies. The numbers a and b can become ever bigger as the string gets longer. When the string gets endlessly long, so can the numbers a and b. But for a/b to lie in the countable set of rationals, a and b must both be finite. Therefore, what you say is untrue for some strings of endless length. That’s the case for irrational numbers, such as the square root of 2. The latter’s irrationalness can be very easily proven, which was done more than two thousand years ago. In particular, the distance corresponding to sqrt(2) is not a rational multiple of ½.

Cantor’s diagonal proof, on the other hand, is perfectly sound. So, Cantor most certainly is right after all.

Also I would be very careful saying that Plato himself identified 'the Good' as God. That was very much the invention of the later Greek-speaking theologians who sought to reconcile Plato and Christianity. it was natural for them to say 'ah, Plato meant God', but Plato himself obviously never had a say in that. — Wayfarer

According to Christina Schefer, Plato’s Good-One is an image of the god Apollo. For example, see page 135 of her book Platons unsagbare Erfahrung “Plato’s Unsayable Experience”:

With that, however, the sense of the exclamation at the height of the Republic is inverted from the end: It is not the One which is invoked with the vocative “Apollo”, but rather Apollo himself as living acting god. He is no metaphor for the One; rather, the One has to be understood as god image of Apollo.

(My translation from German into English)

If his god represents the absolute of the good and just, why does the bad and unjust exist? If his god was perfect, why would these opposing ideas exist? Is there an opposite God of evil?

This makes me think plato never completed his meditations — One piece

Christina Schefer says that a religious and unsayable experience of Apollo as eternal presentness lies at the heart of Plato’s thought, behind both his Theory of Forms and his unwritten Theory of Principles (see e.g. pages 136, 221, 222 and 225 of her aforementioned book). Because presentness is only one aspect of time, Apollo is only a limited manifestation of the holy, a pure mysterium fascinans (fascinating and wonderful mystery) rather than a full-fledged mysterium tremendum et fascinans (terrifying and awe-inspiring as well as fascinating and wonderful mystery) (see e.g. pages 220 to 222 of her aforementioned book). So if Schefer is right and my understanding of her, of Plato and of your question is also right, it is the limited nature of Plato’s religious experience and Apollo’s inability to reveal the whole nature of the holy (for the latter, see page 222 of her aforementioned book) which caused Plato to have a wanting account of badness and evil.

Let me tell you what happened next with our friends:

The mother of the twins turned to Charlie and asked, “What are we to make of this?”

“Let me think,” said Charlie and thought for a while. Then he said, “I think that everything boils down to the problem of the definition of negation. It seems that such a definition is not possible; I’m afraid that if someone doesn’t have intuitive, not-verbal knowledge of what negation is, including that it’s not the same as affirmation, then you can’t tell them what it is. Alice, since your brother is so clever as to claim not to have such intuitive knowledge, it’s likely best for you to just let him go.”

“You’re right, Charlie, it’s likely no use continuing this futile endeavor of teaching Bob about negation, but your explanation is also futile for him, for I know exactly what he is going to say about it,” said Alice and shot an angry look at her twin.

Bob smiled smugly. “You’re right, Charlie. I’m clever enough to claim having intuitive knowledge of negation and that it’s the same as affirmation, and I’m even smart enough to actually have such intuitive knowledge.” Grinning, he gave the phone back to Alice and said, “Dear sister, here’s your phone, which I’ve manipulated so greatly that you can still use it as before. Look what a good brother I am, always doing exactly what you want.”

Yes, Alice's side.

:up:

:scream: No, Bob's side.

Also :up:

So you're taking Bob's side, right?