The Absolute Russell Set Exists

I understand this much in simpler terms , normal and abnormal sets are easier to understand without any contradiction between logic and language
Zemelo’s treatment seems most informal for the purposes at hand.

Never the less, it is obvious that primal logic would lead to contradiction. But that such a nuance be so earth shaking to Frege shows , that it is not so easy to shift gears into sub typical nomenclature, from merely one of order as a way of differentiation.

I bring in calculus for the reason that reference to Euclid is as far removed ,do that lest we forget that 3000 years difference in thought can really up the ante.

Are you being serious? Since I gave you a clear and simple explanation of restricted and unrestricted comprehension, a simple thank-you would suffice. Instead, you wrote a post of meaningless gobbledygook. Why? Can’t you accept that I explained something to you that you asked about, and be happy that you learned something?

I have no idea what normal and abnormal sets are. There is no such terminology. There are of course many uses of “normal” in math, such as normal subgroups in group theory, and normal extensions in field theory. But none of these usages have any relation to set theory. There are simply no such things as normal and abnormal sets.

What contradiction between logic and language do you think is involved in my explanation of restricted and unrestricted comprehension? Did you read the Wiki page I linked? Are you trying to learn anything? Or just trying to argue with the most basic of mathematical facts taught to undergraduates?

Really? I find that quite surprising, since it’s Zermelo who did the most in the early days to formalize set theory. In fact Cantor gets the credit and Zermelo did most of the heavy lifting.

Tell me, which particular aspects of Zermelo’s treatment of set theory do you find relevant here?

I’m afraid that I don’t know what primal logic is. Please define it.

Until you define primal logic you’re in no position to make such a claim.

You appear to be speaking in bullshit-ese here. You’re slinging words that have no referents. The above sentence says nothing and means nothing. Are you playing games?

You did? When? Calculus has nothing at all to do with any of this.

Euclid has nothing to do with any of this.

I hope you won’t mind if I’m direct. You’re full of baloney and clearly not engaging in a good faith dialog. Have a nice day.

Well, emotionalism doesen’t enter here and I could refer you to normal and abnormal sets, but briefly, an abnormal set contains all sets including itself and a normal one does not. Primal logic is reducible logic which ends in contradiction, and all others do not.

If philosophy be a matter of naivete, it is as understandable as that, with which Cantor was concerned, yet surely one couldn’t label him ignorant.

If Euclid has nothing to do with this, why did You bring it up?
I see he has everything to do with it& absolutely.

Of course I do not see anything in Your directness, but bravado, and that is perfectly forgivable, as far as I am concerned.

Its like I don’t know who wrote it, Jane Austin, You can make sense without being sensible, or having sensibility. And remember my name , Meno, and You know how he learned.

I chose it somewhere along that line, for an obvious purpose.

And a very good day to you as well.

The link in my previous post should bring you to a proof. If you want to better understand the proof, then please follow the appropriate additional referrals. It’s all there; you just have to click and scroll a bit.

By the absolute Russell set, I do not mean the set of all sets that are not members of themselves; I mean the set of all things that are not members of themselves. As I have proved privately in first-order logic, as I explained in the past debate “The Absolute Russell Set Exists” I proposed at debate.org/debates/The-Absol … -Exists/1/, which I have already indirectly referred to, and as may be common knowledge, the set does not “exist via an easy and standard proof.” I, notwithstanding, present proof to the contrary.

And that contradiction, through ex contradictione quodlibet, leads to no contradiction at all. I brought that up in my original post.

That’s not true. Set theory is consistent both with and without unrestricted comprehension. By the law of non-contradiction, set theory is consistent or inconsistent. If set theory is consistent, then by reiteration, it is consistent. If set theory is inconsistent, then by ex contradictione quodlibet, it is consistent. Therefore by disjunction elimination, set theory is consistent.

Contradictione quidlobet is a shift to a functional approach to verifying a second order logic , grounding a utilitarian-positivism, meaning it tries to overcome the naturalistic fallacy, utilizing a neo-Kantism.

Marbourg school-Cassirer.

It is, as if in Ayer’s behaviorist model, what signifies meaning. through the function of language where usage determines the logical structure , not some intrinsic property.

This structural hierarchy coincides with the epistemological significance of semantic usage, rather then the intractibly logical signifier, of a first order hierarchy , which does cause contradiction. The usage will will synthesize this by reasserting the logical unity between usage , function and meaning.(Typical logical progression. Is de-emphasized, in favor of logically ordered sequences).

This is the ontic, rather then the ontological take on it.

I hate to appear defensive, but a proof predicated on an assumption is only part of the story, and this is an effort to show the fallibility of that assumption.

I just learned this : in classical logic the principle of acceptance of contradictory statements is valid, but in relevance logic it is rejected. This backs up the idea that such a logical proof is not totally relevant therefore invalid.

The paradox comes in where two different types of logic are conflated.

It is an interesting mind game, however the claims upon which the assumptions are based have lost their significance.

But thanks for the opportunity !

This post includes a second proof that the absolute Russell set exists. The second proof is a modified and arguably stronger version of the first proof. All conditional statements in the modified proof are strict; that differs from the first proof, in which some conditional statements were material.

Second Proof.

Lemma 2. If a statement is false, then if the statement is true, then some contradiction exists.

Proof of Lemma 2. It is given that a statement is false. Assume the statement is true. Then some contradiction exists. Discharge the assumption. So, by conditional introduction, if the statement is true, then some contradiction exists. This concludes the proof of the lemma.

Postulate 2. If one statement implies it is not true that a second statement, then the first statement does not imply the second statement.

Argument for Postulate 2. Postulate 2 seems intuitively true, despite the fact that Postulate 2 would traditionally be false due to the case in which the first statement is necessarily false, since, as suggested by Property 4 of Theorem 2.1 on page 11 of the March 23, 2014 Edition of Want Theory #6 by me, any conditional statement with a necessarily false hypothesis is true. However, a conditional statement with a necessarily false hypothesis is only vacuously true, and treats the false hypothesis as true, yielding a contradiction that by ex contradictione quodlibet implies and does not imply everything. So, such a conditional statement is true and false simultaneously. Thus, for the sake of Postulate 2, if the first statement is necessarily false, then the conditional statement “if the first statement, then the second statement” is considered false. This concludes the argument for Postulate 2.

Assume the statement “the absolute Russell set exists” is true. By simplification, the absolute Russell set exists. As I’ve proved in my post at http://www.ilovephilosophy.com/viewtopic.php?p=2699066#p2699066, the absolute Russell set both is and is not an element of itself. wtf was critical of that citation in my first proof, so I will explain the linked proof some more here. That proof invokes the fact that the absolute Russell set is an element of itself if and only if it is not an element of itself. That fact is readily derived from a property of the absolute Russell set: a thing is an element of the absolute Russell set if and only if it is not an element of itself. As I had done in the debate I previously proposed and referred to, for more context, I cite pages 432, 433, and 434 of Language, Proof and Logic (1999, 2000, 2002, 2003, 2007, 2008) by Jon Barwise and John Etchemendy, and https://en.wikipedia.org/wiki/Russell’s_paradox#Formal_presentation.

Since the absolute Russell set is and is not an element of itself, some contradiction exists. It follows by ex contradictione quodlibet that no contradiction exists. Discharge the assumption. Thus, by conditional introduction, if the statement “the absolute Russell set exists” is true, then no contradiction exists. So, by Postulate 2, it is not true that if the statement “the absolute Russell set exists” is true, then some contradiction exists. Invoking modus tollens with Lemma 2 and the previous sentence, it is shown that the statement “the absolute Russell set exists” is not false. Thus, the statement is true. Therefore, the absolute Russell set exists. This concludes the second proof.

As to the first proof, although the truth table for material implication may be correct, it is incomplete. Material implication is not truth functional as has been believed. As suggested by my argument for Postulate 2, whenever the hypothesis of a material implication is false, the implication is not just true, but it is also false. That claim is supported by the fact that if a false statement is assumed true, then by ex contradictione quodlibet, all statements are true and false simultaneously. So, a false statement always materially implies and always does not materially imply every statement. For these reasons, the following Postulate 1, which was implicitly invoked in the first proof at the step I explicitly claimed may be questionable, is true.

Postulate 1. If one statement materially implies it is not true that a second statement, then the first statement does not materially imply the second statement.

Note that while the hypothesis and conclusion of Postulate 1 are a material implication and material nonimplication, respectively, Postulate 1 itself is a strict implication.

Bizarre that you’d write all this word salad without bothering to define the “absolute Russell set,” which has no meaning in standard math or logic.

But if you merely mean the class of all sets that are not members of themselves, I already showed you how to define it:

$$R = {x : x \notin x}$$
This is a perfectly well-defined proper class. It’s just not a set.

Proper classes are formally defined in some versions of set theory. In ZFC where they’re not formally defined, they’re used informally, meaning a class that’s “too big” to be a set. The class of all sets, the class of all Abelian groups, the class of all topological spaces, the class of all sets that are not members of themselves, etc. All those are proper classes either in the formal or informal sense.

With your interest in the subject, why don’t you study some basic set theory and logic? You’d find it interesting and fun.

I do not have to define the absolute Russell set. A property I described it to have should have sufficed for the discussion so far. I now define the absolute Russell set as the set that has a property I have ascribed to it in my previous post; I define the absolute Russell set as the set such that

I’ve already done that. And I am not convinced, despite formal proofs, that neither the absolute Russell set nor the universal set exist.

Just because a set is inconsistent, doesn’t mean it doesn’t exist.

I’ve already showed you how to define the class of all sets (or things, if you prefer) that are not members of themselves:

$$R = {x : x \notin x}$$

As far as your saying you don’t have to define something to show it exists, surely even you can see that’s insane.

I’m not interested in classes; I’m interested in sets. I’m skeptical of the difference between sets and classes. The difference seems rather artificial and unnecessary.

Just because a thing is not defined, doesn’t mean it doesn’t exist.

Are there any sane people on this forum?

Yes, I am a sane person on this forum.

There’s a difference between a definition and a description. All definitions are descriptions, but not all descriptions are definitions. The description I had initially given of the absolute Russell set, while on another web page, exclusive, and sufficient for my purposes, was not regarded by me as a formal definition. I believe I may have regarded so intentionally, because I may have suspected the definition of the absolute Russell set was already given in a printed textbook, the one by Barwise and Etchemendy.

I should not have to define in this thread everything that has already been defined elsewhere. That decreases the value of this thread and increases its unattractiveness to people looking for original thought.

Nonsense. The phrase “absolute Russell set” does not appear on the Internet except as a reference to the standard Russell set.

I truly doubt your text defines such a thing the way you’re using it. Feel free to post a screen shot though, maybe I’ll learn something.

It would not be possible for anyone to do that.

LOL.

I am referring to the standard Russell set. However, the Russell set, according to page 432 of the aforementioned Barwise and Etchemendy (1999, 2000, 2002, 2003, 2007, 2008), is the set of all things that are not elements of themselves, not the set of all sets that are not elements of themselves. The set of all things that are not elements of themselves is a more natural and better set to be called the Russell set than the other set. It’s more comprehensive. I concede I did coin the term the absolute Russell set myself. But my term was readily derived from page 432 of Barwise and Etchemendy (1999, 2000, 2002, 2003, 2007, 2008), where they put the word absolute in parenthesis.

Barwise and Etchemendy (1999, 2000, 2002, 2003, 2007, 2008) do not seem to formally define the absolute Russell set; they only describe and discuss it.

This is where copyright law comes into play. I’m hesitant to copy a copyrighted formal definition or symbolic expression, especially when it may not be common knowledge. I actually, in my past post at viewtopic.php?p=2699066#p2699066, which I previously cited in this thread twice already, did not use the same letter to represent the absolute Russell set as Barwise and Etchemendy (1999, 2000, 2002, 2003, 2007, 2008) do. I was afraid it might be a copyright violation or plagiarism.

I disagree. It could be worse. It could get worse.

I wish to be enlightened and to advance.

Thanks, that only took me a week to drag out of you, after you claimed several times that you found it in a math book.

So what you are calling the “absolute Russell set,” everyone else calls the proper class of all sets (or things, really doesn’t matter) that are not members of themselves. That’s really all there is to all this.

wtf:

I’m not as familiar with class theory as I am with set theory. If you are correct, then my claim is that the absolute Russell proper class exists.

I’d rather keep the discussion focused on sets rather than classes. There’s no need to speak in terms of classes when it’s evident I’m speaking in terms of sets.

Yes. It does. As I’ve noted a couple of times, (R = {x : x \notin x}) defines the Russell class. It’s the class of all things that are not members of themselves. It “exists” in the sense that its definition does not lead to contradiction. Of course here we are talking about formal logical existence, not anything in the real world. I hope we agree on that.

That’s fine. But the claim that (R) is a set leads to a contradiction.

Of course you could say it exists in an inconsistent version of set theory. Nothing wrong with that. As you’ve already pointed out, in an inconsistent system everything is true.

But even here it is not settled, since contradiction and non contradiction are in an Absolute Russell set , self inclusive sets?

The feeling I have is, that redefinition does nothing but reassert the primacy of naive logic. Can that allowance withstand other succeeding arguments? Or, is it other systems of classes , when weighed in, Change the balance ?

Meno_, I read some of your other posts on the forum and I realize that you are only communicating in your own particular style; and not at all trying to annoy me personally. Thus it was wrong for me to attack you personally. I apologize.

I still do not understand most of what you write. But some of your references are interesting as I look them up. I think there is a kernel of interestingness in your exposition, if only I could discern it.

I hope you’ll accept my apology and perhaps try to explain yourself better to a humble math guy like me.