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All of us, maybe in our first incursions on pure mathematics or going further in logic or set theory, need to face the rare and "pretty easy" to understand Russell paradox, but maybe the further implication of this encounter is that there is not a set such that contains the sets that do not be log to it selves . But for it importance i have thought that this paradox should imply further meditations. I mean i am not going seriously in logic yet, but i need to know why this paradox is so important for mathematicians, or may be if it is an actually easy to understand first encourage with math or set theory.

When i first read the paradox and the proof i understand it as a demonstration of impossibility of the existence of a biggest class or the biggest set, besides the normal implication that all of us know, so i don´t understand it as well as i should understand it, and now i don't know if out of logic, basic analysis and basic topology the Russell paradox is that important, are there any interpretations in real life, applied math, physics of RP.

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If I remember correctly, modern math now doesn't allow one to define a set by mere describing. They now iterate each of the sets step by step. See Von Neumann universe. So that sets like the one in Russell Paradox simply doesn't exist in this universe of sets. – Voldemort Sep 22 '12 at 3:51
up vote 4 down vote accepted

I think the main effect of the discovery of Russell's paradox on mathematicians (outside the field of logic) was to make the axiomatic foundations of mathematics slightly more complicated than was previously hoped. Since most mathematicians do not argue directly from the axioms of set theory anyway, this has a minimal effect on mathematical practice. The main effect is that definitions of the form $\{x : P(x)\}$ must be replaced with definitions of the form $\{x \in Y: P(x)\}$ where $Y$ is a set large enough to contain all the $x$'s you want. It is usually not hard to show that such a set $Y$ exists.

I do not think that Russell's paradox shows up in real life, applied math, or physics, although it would be pretty neat if it did and I hope to see another answer proving me wrong on this point. At the core of Russell's paradox is the much older "liar paradox" which shows up in several other places in mathematics: notably in the halting problem, which arguably does have practical implications.

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Oh, speaking of liar paradox, Gödel's incompleteness theorem, one of the greatest theorems in 20th century, is based on "liar paradox". – Voldemort Sep 22 '12 at 5:48
@Voldemort Quite right. I was going to mention that, but then I'd have to admit that it didn't have any practical applications either :) – Trevor Wilson Sep 22 '12 at 5:50
Gödel's incompleteness theorem, or its theoretical computer science equivalent, halting problem, points out the ultimate limitation of all the modern computers (which all based on Turing machine,) for example, for any computer that has some instruction set, there exists a mathematical problem that can never be solved by it. – Voldemort Sep 22 '12 at 6:02

I was told that at the time many mathematicians didn't really care about Russell's paradox. They believed that most sets you would encounter in everyday mathematics are too small to be pathological like that.

Much like nowadays many mathematicians don't really pay attention to large cardinal research for the same reason (and then there are those guys who just use them regardless).

The main implication from Russell's paradox is that not every definable collection makes a set. This is quite a shock to the naive approach taken at the end of the 19th century.

There are two major results (both, I believe, are due to von Neumann). The first is the development of the concept of a class, namely a definable collection. This meant that we can talk about nonexistent collections as long as we can describe them by a formula. This allowed in some sense "virtual sets" like the Russell class, or other paradoxes, to virtually exists as classes. We could refer to them, manipulate them, but they are not objects in the semantical universe (and so they don't really exist).

The second result is the idea of describing the universe as an increasing union of sets, i.e. the von Neumann hierarchy. This allows us to avoid the paradox in the sense that whenever we want to describe some collection, we can limit ourselves to a certain point in the hierarchy and see how the collection looks like at that point. In fact a definable collection is a class if and only if it is bounded in the hierarchy.

Outside of set theory, the Russell paradox doesn't have much impact, or rather we cannot see it nowadays when ZFC (a set theory in which the paradox is resolved) is the de facto meta-theory of mathematics. However you can see remnants of the paradox in every self-referential definition, like the category of all categories.

As for your last question, Russell's paradox (as most set theory) has little (or virtually no) relation to the physical world, as much as we know. Applied mathematics, everything physicists do, and real world-expressible shenanigans are usually very very small in terms of sets (I mean most are bounded below the power set of the real line).

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" Applied mathematics, everything physicists do... is bounded below the power set of the real line." > The physical world is finite, eg: computers are Finite State Machines, not Turing machines; Lie algebras to simplify PDEs can be manipulated in Mathematica (a FSM); Dirac estimated the ratio of the mass of hte universe to the proton at ~10^80; the most energetic cosmic rays detected @ ~10^20 eV; the universe has a finite radius and age, etc. – alancalvitti Dec 20 '12 at 9:59
You do know that physicists nowadays use infinite Hilbert spaces, right? And all that shebang about the time and space continuum being essentially $\mathbb R^4$? Your comment makes no sense. – Asaf Karagila Dec 20 '12 at 10:07
Now you're talking about mathematical representations of the physical world whereas before you were talking about the physical world. What doesn't make sense? – alancalvitti Dec 20 '12 at 10:09
Your comment here in the first place. It is a fact that applied mathematics and physics use "continuous approximations" of the real world. No one thinks about Newtonian mechanics in terms of discrete mathematics, it wouldn't be wrong but it would be particularly dumb as it would require a lot more effort. My point was that all those "approximations" were small enough that no one using them would ever encounter the Russell paradox there. – Asaf Karagila Dec 20 '12 at 10:14
Your answer re Russell paradox is fine. But when you say "No one thinks about Newtonian mechanics in terms of discrete mathematics" - that's nonsense, have you ever programmed a robot arm or try to get robots to walk? The Newtonian (and Hamiltonian) dynamics are represented and controlled with finite state machines and digital control, which are discrete. – alancalvitti Dec 20 '12 at 10:16

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