The Paradox That Broke Set Theory | Hacker News Reader

The Paradox That Broke Set Theory

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12 pointsAnon8422 comments

https://en.m.wikipedia.org/wiki/Russell%27s_paradox

Russell's paradox

Maybe this is in the article which I could not read, but - why is this a paradox? What I see is that Russell tried to define a set, then found out that there is no set which would fulfill his definition. "Let igiveup's number be any prime number which is divisible by four." Why is this not a paradox, while Russell's definition is?

The "paradox" is really a contradiction of the axiom of unrestricted comprehension:

    To every condition there corresponds a set of things meeting the condition.

Unrestricted comprehension allows you to build sets out of whole cloth, and seems reasonable. Of course everything divisible by two is a set, of course every prime number is a set. It's tempting to say that the set is simply defined by a single condition, and if some item fulfills that condition, it gets to be in the set.

Your example would not violate the axiom of unrestricted comprehension. No primes are divisible by 4, so the set of all primes divisible by 4 is the empty set. Russell's paradox does. He chose a condition that depends on the result of creating a set using that condition. "The set of all sets that do not contain themselves" can't contain itself, but it can't not contain itself either -- since either case would imply the opposite must be true.

The result is that unrestricted comprehension gets pared back. Instead of building a set from a condition, all sets must be built from a condition and a pre-existing set. A single infinite set corresponding to the natural numbers is given as an axiom, and all further sets (integers, real numbers, topological spaces, fields, and so on) are built from there.

Thanks, clear.

Though, I think my example was misunderstood. I meant, why is it a problem that there is no set meeting Russell's definition, but not a problem that there is no number meeting my definition. The example would violate an axiom if there was an axiom like:

  To every definition of a certain kind about natural numbers, there is a number which satisfies it.

Which of course would be rather useless, while the unrestricted comprehension axiom is necessary for the rest of the theory.

The reason is (a bit informally), the rules of maths said I could ask for “all the things which satisfy properly X”. So, it would be fine to ask for “all igiveup numbers” — there might be none, one, who knows, but we can ask for that.

Too much of maths asks “get me one (if it exists), or all, things that satisfy X”, that if that can’t be trusted to work, basically everything is suspect.

https://archive.ph/L9a4z

looks like it's not the full article :(

Thank you for this, but infinite captcha for me.

I thought this would be about Skolem's Paradox, which _really_ broke set theory, and which set theorists have done their very best to ignore and handwave away since its introduction.

The Wikipedia article (https://en.wikipedia.org/wiki/Skolem's_paradox) says that 1) it is "not an actual antinomy like Russell's Paradox" (that is, it's an apparent contradiction rather than a real one), and 2) "the result quickly came to be accepted by the mathematical community" (even though Zermelo was highly opposed to it).

So I think your claim is rather misleading.

Oh it's much worse for set theory than an actual antimony. For most of those are mere word-games of the "does not compute" variety, whereas Skolem's Paradox states plainly, and uncontrovertibly, that there's no such thing as an absolutely uncountable infinite set. All sets have countable models, however uncountably vast they may be. It thus completely explodes all of Cantor's work on "countable" and "uncountable" infinities, rendering all infinite sets countable and equal. (As should have been intuitively obvious to begin with.)

And it's ignored because people still find the nesting hierarchies of set theory useful, on occasion, even though what they now mean is very much open to question.

Most of the people who have famously disparaged set theory as a theoretical foundation for mathematics -- like Wittgenstein -- have done so because, after Skolem, the whole thing is quite uncertain.

I think the problem starts earlier. The moment you talk about sets that contain themselves. It’s something akin to infinite, that’s difficult to reason about

Bingo. It's an infinite recursion issue.

You need to explain why that is an issue. A data structure with a pointer to itself is still valid and you can work with it. So math sets have operations people want that aren't valid under those situations.

you have to dereference the pointer to determine truthiness of the value, which then gives you the parent pointer, which you have to dereference the pointer to determine truthiness of the value, which then gives you the parent pointer, which you have to dereference the pointer to determine truthiness of the value, which then gives you the parent pointer, which you have to dereference the pointer to determine truthiness of the value, which then gives you the parent pointer, which you...

'broke' is a weird word here. I never learned math in English, is it usually used in those cases?

Is there a way to read the full article without logging in?

Even worse, I can confirm that logging in is not sufficient, despite what the initial call to action seems to claim.

After logging in the article is still incomplete and the dialog is changed to one prompting the user to "upgrade", meaning payment.

This seems like a dark pattern to me.

Hofstadter talks about such paradoxes in his book -- from what I remember, the paradox stems from self-reference (he also uses the example of "adjectives that describe themselves", aka the Grelling–Nelson paradox)

Self-reference breaks a lot of things, including our arrogance if it's done right

Could you explain how self-reference breaks our arrogance?

If someone were able to properly see themselves and evaluate themselves they are going to be much less likely to be arrogant as they will also consider their flaws

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