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A brain teaser on the notion of "absolute".

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Does Russell`s paradox suffer the same fallacy?

That would be an elegant way to end a 200-yr mathematical debate - no self references!

I wonder, since computer languages cannot self-reference, why should human languages do that?

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Does Russell`s paradox suffer the same fallacy?

That would be an elegant way to end a 200-yr mathematical debate - no self references!

I wonder, since computer languages cannot self-reference, why should human languages do that?

What's Russell's paradox?

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What's Russell's paradox?

The class of all classes is itself a class, and so it seems to be in itself. The null or empty class, however, must not be a member of itself. However, suppose that we can form a class of all classes (or sets) that, like the null class, are not included in themselves. The paradox arises from asking the question of whether this class is in itself. It is if and only if it is not.

Have fun.

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Does Russell`s paradox suffer the same fallacy?
That's more a case of context dropping. However you want to name the fallacy, it depends on set. Sets do not exist, although there are some things that do exist and that can be called sets (like: the set of dogs, the set of sticks, the set of yellow cars). Under the harmless version of "set", sets are not members of themselves (the set of dogs has all of the dogs as its membership, and it doesn't have any sets as members). Since they're only dealing with word-manipulations, they can define a "set" negatively to to everything that isn't a dog. Hence the paradog. (Leaving out a few steps).

Sets don't exist; but concepts do. What makes the "the set of dogs" fairly harmless is that it's the same as "the meaning of the concept 'dog'" in Objectivist terminology, i.e. the referents (actual mutts). The same doesn't hold of non-sets -- there is no valid concept "non-dog". Russell's paradox depends on such misuse of the term "set". Although, from a strictly fomal POV, what it really shows is that you can construct a contradictory formal system. No surprise there.

The statement "The only absolute is that there are no absolutes" also strikes me as a form of context dropping. An absolute would be a statement that is true of some stated group of existents, e.g. "All dogs are mammals", "all fish are cold-blooded". So if you drop the referents (dogs, fish) and propositional content ("are mammals", "are cold blooded") -- totally drop context -- you end up with the idea of a pure Platonic "absolute", a universal truth saying nothing. But there is no such thing as a universal truth saying nothing, so we can say that there are no absolutes; and furthermore, absolutes absolutely do not exist. So really, "absolute" is a non-referring expression.

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Russell's paradox is usually couched in terms of sets, but the paradox is more fundamental and can be couched without mentioning sets but only mentioning concepts or predicates or properties, or, indeed, just predicate symbols. Though 'concepts', 'predicates', or 'properties' here are not necessarily in an Objectivist sense, which would seem a point in favor of Objectivism except that Objectivism doesn't offer an axiomatization of mathematics.

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Fallacy of self-reference.

You dont get to just make up fallacies. It can be shown that self-reference is dubious in some contexts such when it results in vicious circles (this was the approach Russell tried to take with his theory of types), but theres no reason to say that all self reference is fallacious.

Edited by Hal

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That's more a case of context dropping.
What context was dropped? He started with the axioms of Frege's Grundgesetze and proceded to draw a formal contradiction, with the result that at least one axiom had to be dropped/reformulated.

Sets do not exist
I'm not sure what 'exists' means in this particular context (do numbers exist?), but they were an element of the formalization regardless.

If you dont want sets to be able to have themselves as members, then you're free to include this as an axiom of your set theory. We are dealing the stipulative definition of a new technical term here, so objections based on 'what the word set really means' miss the point.

edit: Grundgesetze not Begriffsschrift

Edited by Hal

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Russell's paradox is usually couched in terms of sets, but the paradox is more fundamental and can be couched without mentioning sets but only mentioning concepts or predicates or properties, or, indeed, just predicate symbols.
Okay: can you present a verion of Russell's paradox that doesn't crucially depend on an invalid concept?

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Okay: can you present a verion of Russell's paradox that doesn't crucially depend on an invalid concept?

1) Click "Post new thread"

2) Thread title: "This thread contains links to all threads on the ObjectivismOnline forums that dont contain links to themselves"

3) Thread body: (A link to the thread itself)

4) Internet explodes

If there were a forum here for 'offtopic/useless posts', I'd actually have posted this :/

Edited by Hal

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What context was dropped?
The empirical content of "set". This is not to say that he didn't show that an existing set of formal stipulations were mutually incompatible. You'll note that the discussion here is about formal ideas that are thought to have some applicability to reality, which is a mistake. Russell's paradox isn't real: it doesn't point to any fact whatsoever about of existence. As a purely internal methodological exercise in mathematics, I have no objection to it, but one shouldn't import the vocabulary of reality into that discussion.

BTW, the fact that sets don't include themselves as members is not an axiom, it's a theorem. It stems from the characteristics that define set membership.

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The empirical content of "set".
There is no empirical content of set. I think you're confusing the English language word 'set' with the technical term 'set' within the foundations of mathematics. It's like saying that quantum chromodynamics is flawed because quarks dont really have a colour.

Russell's paradox isn't real
It is real. Its a real contradiction within the formal system in which it was derived.

BTW, the fact that sets don't include themselves as members is not an axiom, it's a theorem. It stems from the characteristics that define set membership.

True.

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1) Click "Post new thread"

2) Thread title: "This thread contains links to all threads on the ObjectivismOnline forums that dont contain links to themselves"

3) Thread body: (A link to the thread itself)

4) Internet explodes

Well, you're wrong. That's just the Liar's Paradox.

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Well, you're wrong. That's just the Liar's Paradox.

What am I wrong about? That would be a version of Russell's paradox that didnt include any invalid concepts. Russell's paradox is very similar to the liar paradox (as is Godel's incompleteness theorem). "Threads containing links" and "sets containing members" are pretty much equivalent. In fact if I changed it to say "this post contains links to all posts that dont contain links to themselves", then you could say that the post is just a set of links.

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Just to clarify, I'm not claiming that Russell's Paradox has any real signficance outside of specific (mainly formal) contexts, nor am I saying that 'there is a contradiction existing in reality!!!' or any nonsense like that. But the RP is a real issue for any theory that involves 'things being subsumed by other things', as it can result in internal contradictions.

Edited by Hal

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What am I wrong about?
The Internet exploding. It's still here, as you can see. In your would-be real world scenario, one of the statements (the thread title) was a lie: that's very much like the "set containing itself". When you can depart from reality (as is possible in the mind), you can describe contradictions and paradoxes -- and nobody, AFAIK, has ever denied that. The point is that when you are speak of reality, or using methods for operating on reality, then you can't have contradictions. As long as you recognise that RP is not about anything real, and is just further evidence (as though any were needed!) that the mind can string together arbitrary sequences of symbols, we seem to agree.

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BTW, the fact that sets don't include themselves as members is not an axiom, it's a theorem. It stems from the characteristics that define set membership.

What characteristics are those, and in what theory? In set theory, membership is a primitive - undefined - except that axioms define by limiting the models of the theory. In any case, that a set is not a member of itself is not ensured by the basic axioms of set theory but is proven from an additional axiom - the axiom of regularity - that was adopted several years after Zermelo's first axiomatization that itself came several years after Russell's paradox became known. The axiom of regularity is adopted pretty much for the purpose of precluding that sets are not members of themselves. (The reason the wider axiom of regualarity is adopted rather than just an axiom that sets are not members of themselves is that the wider formulation is needed to ensure against other unwanted situations, such as x being a member of y and y being a member of x.) Moreover, the relative independence of the axiom of regularity is proven, so that there are relatively consistent set theories in which there are sets that are members of themselves. Moreover, the axiom of regularity is not needed to derive analysis, which is probably the primary mandate of set theory. So the claim that sets are not members of themselves by virtue of basic "characteristics" of set membership is vague and dubious, as well as the claim that sets are not members of themselves by virtue of more basic axioms is terribly misleading since, on the contrary, a special axiom IS needed to ensure that sets are not members of themselves.

The point is that when you are speak of reality, or using methods for operating on reality, then you can't have contradictions. As long as you recognise that RP is not about anything real, and is just further evidence (as though any were needed!) that the mind can string together arbitrary sequences of symbols, we seem to agree.

More fundamentally, any theory of a model is consistent, whether the model is that of a putative reality, fiction, abstraction, or made entirely of counterfactuals. But, unfortunately, axioms rich enough for arithmetic can't be both decidable and those of the theory of the standard model. In this sense, talking about needing a system to be based only on reality quite misses the point. Also, what in Frege's system do you find to be arbitrary, as opposed to another presumably non-arbitrary system? What would that non-arbitrary system be?

Okay: can you present a verion of Russell's paradox that doesn't crucially depend on an invalid concept?

Any theory that produces a contradiction must be invalid. My original point stands: the source of invalidity is more fundamental than just the notion of a set.

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What characteristics are those, and in what theory? In set theory, membership is a primitive - undefined - except that axioms define by limiting the models of the theory.
We're speaking of Objectivist epistemology, and as I said, the closest analog is the concept. Concepts are defined in terms of essential similarities (and omitting nonessential differences), so the members of the "set" would have some defining characteristic (such as color, size, use, whatever...). No concept identifies both actual dogs and the concept "dog" (they have no similarities). It is irrelevant that set theory allows arbitrary collections to be called "a set" -- as I pointed out, the concept "set" refers to reality only in a limited set of cases, and it excludes the type of set that's required for RP to be a "paradox".
Also, what in Frege's system do you find to be arbitrary, as opposed to another presumably non-arbitrary system? What would that non-arbitrary system be?
I make it a practice not to make claims about Frege because I don't understand him. However I am quite certain that Frege is not the only one who allows or even encourages arbitrary claims. Here's an example: AxEy(P(y)->Q(x)). That's an arbitrary string of symbols. The terms have no real referents -- they are arbitrary.

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Russell's paradox is not tied to Objectivist epistemology nor is set theory. So, of course, you're free to stipulate whatever you want to be the "characteristics" of sets or anything else unrelated to Russell's paradox or set theory. But your explanation does not even give a hint of a basis for mathematics.

Concepts are defined in terms of essential similarities (and omitting nonessential differences), so the members of the "set" would have some defining characteristic (such as color, size, use, whatever...)

The defining characteristic of the Russell set is the characteristic of not being a member of oneself. Since membership is fundamental, there can't be any objection to using it as a basis for characterization (membership and identity are THE bases of characterization in set theory). But what about identity? Surely, you don't propose that x = x be disallowed? So, self-reference is not disallowed either. So, your Objectivist explication of the paradox is of no help.

Since you've mentioned concepts, do you have a concept of concepts? Is 'beauty' (the concept, not the word) a concept? Is 'intangibility' a concept? Is 'abstractness' a concept? Is 'concept' a concept?

However I am quite certain that Frege is not the only one who allows or even encourages arbitrary claims. Here's an example: AxEy(P(y)->Q(x)). That's an arbitrary string of symbols. The terms have no real referents -- they are arbitrary.

You are "quite certain"? It seems you don't understand the difference between a mathematical claim and recognition that a formula is simply well formed, so your example is nonsense as an example.

And while you take exception to what you call the 'arbitrariness' of symbolic logic, my question remains standing: what non-arbitrary system for mathematics do you or any Objectivist propose? (By the way, though model theory came after Frege, symbolic formulations are given reference through structures, and Frege has a quite sophisticated theory of meaning.}

It's interesting that mathematics of the last hundred years has proceeded without discovered contradiction and has been vital for such things as the creation of digital computers, space travel and a myriad of everyday practical applications, but without any help from Objectivism. An Objectivist may decry that mathematics of the last hundred years bases itself on "the arbitrary", but mathematics seems to work quite well, and to the benefit of Objectivists too, without the slightest recognition even that it should right itself to Objectivist epistemology. For that matter, what IS an Objectivist basis for mathematics? I've looked for an answer to this question, but so far have only found an Objectivist "blank out".

Also, the discussion of the meaning of 'essential' in ITOE is hardly satisfactory, since it relies upon terms that are no more defined than 'essential' and stipulates that comparisons of properties be performed, but gives no method for these comparisons nor basis for evaluation of them other than a vague decree. Unless you give a precise definition of 'essential', then your claims as to the essentiality and non-essentiality of properties are arbitrary indeed.

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For that matter, what IS an Objectivist basis for mathematics? I've looked for an answer to this question, but so far have only found an Objectivist "blank out".
It's discussed briefly in ITOE: I know it's de rigeur to avoid actually reading what Rand said, but still I do suggest that book in particular. The only thing that you really need to pay attention to is that arbitrary assertions cannot be asserted as true (or false), and derive a valid conclusion. There are a few conceptual corrections, at least for abstractologists who make those kinds of mistakes, that sometimes need correcting. For example, it's false that there exist an infinite set of integers. It's true that the method for creating an integer is unbounded. Basic enumeration is a low-level ordinary method concept ariseing from seeing 1 dog, 2 dogs, 3 dogs, 2 cats, 3 cats etc. Numbers themselves only except as method-concepts. Triangles and circles, OTOH, are first order abstractions from actual objects.

I think the problem stems from the fact that you don't understand basic terms such as "concept" and "arbitrary". The other problem seems to stem from your misunderstanding of epistemology. Epistemology pertains to cognition, and psychology is its empirical anchor to reality. You seem to be looking for a Platonic definition of e.g. "essential". The word means exactly what it means in ordinary language. The only sensible complaint that I can imagine about the idea of an "essential" is that you don't know a priori what is essential vs. non-essential. That would be because this is an empirical question, and depends on the concept.

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It's discussed briefly in ITOE: I know it's de rigeur to avoid actually reading what Rand said, but still I do suggest that book in particular.

I've read the book. I'm familiar with the discussion in that book of of 'essential'. That's why I mentioned that discussion in that book.

The only thing that you really need to pay attention to is that arbitrary assertions cannot be asserted as true (or false), and derive a valid conclusion.

Objectivism and the notion of abritrariness depend on the notion of essentialness. It's question begging to say that some things are arbitrary since they don't meet the test of essentiality, but then evade giving a definition of 'essential' by saying that all we need to worry about is avoiding the arbitrary.

I think the problem stems from the fact that you don't understand basic terms such as "concept" and "arbitrary"

No, the problem is that 'essential' is not adequately defined.

The other problem seems to stem from your misunderstanding of epistemology. Epistemology pertains to cognition, and psychology is its empirical anchor to reality. You seem to be looking for a Platonic definition of e.g. "essential". The word means exactly what it means in ordinary language. The only sensible complaint that I can imagine about the idea of an "essential" is that you don't know a priori what is essential vs. non-essential. That would be because this is an empirical question, and depends on the concept.

First, please, don't, in effect, put words in my mouth that I'm looking for a "Platonist" definition. I asked for nothing of the kind, whatever you may mean by "Platonist" definition. Also, I didn't opine as to what Objectivist epistemology is. As to ordinary language, I do have a common sense, everyday understanding of the idea of essentiality, but we were talking about mathematics, philosophy, and philosophy of mathematics. An everyday sense of 'essential' is inadequate for these. Moreover, Rand took it upon herself to give more than an every day explanation. And, I believe you that the Objectivist notion of 'essential' is not a priori, or shouldn't be. But that essentiality is determined empirically does not answer the question of just what this empirical determination is or how one is to evaluate these determinations, unless we are to be satisfied with the circularity you've so far provided. Edited by LauricAcid

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Objectivism and the notion of abritrariness depend on the notion of essentialness. It's question begging to say that some things are arbitrary since they don't meet the test of essentiality, but then evade giving a definition of 'essential' by saying that all we need to worry about is avoiding the arbitrary.
Hang on Nelly, what makes you think that the arbitrary is the non-essential? An arbitrary statement is one that relates to no known fact. An essential characteristic is one which is not omissable in identifying the extension of a concept. Can you connect the dots for me: I have no idea how you got this connection between the arbitrary and an essential characteristic.
No, the problem is that 'essential' is not adequately defined.
An essential characteristic is one which is not omissable in identifying the extension of a concept. Hmm... I'm being redundant.
First, please, don't, in effect, put words in my mouth that I'm looking for a "Platonist" definition.
Sure, I'll reciprocate when you do.
As to ordinary language,  I do have a common sense, everyday understanding of the idea of essentiality, but we were talking about mathematics, philosophy, and philosophy of mathematics. An everyday sense of 'essential' is inadequate for these.
Mebbe, but I've not only given you an adequate definition for our purposes, I've also told you that the concept "essential" does not impinge on "arbitrary" in any useful way, w.r.t. the issue at hand. What I'd like is your proof that the Objectivist concept "arbitrary" is incomprehensible without a definition of "essential", and that the concept "essential" is incomprehensible without a definition of "arbitrary".

If it helps you to understand Objectivist epistemology, "concept" is closer to being axiomatic than either "essential" or "arbitrary".

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