A question on ‘the set of everything’.

Quoting Wayfarer

I’d be interested to see if I’m barking up the wrong tree he

Is this the same as:

Russell's paradox
Russell's paradox is one of the most famous of the logical or set-theoretical paradoxes. Also known as the Russell-Zermelo paradox, the paradox arises within naïve set theory by considering the set of all sets that are not members of themselves.

Reply to Tom Storm I think it is, but it’s a slightly different angle on it. Maybe.

Reply to Wayfarer I am kind of fascinated by what paradoxes like this tell us about human conceptualization and descriptive categories. I lack sufficient Wittgensteinian acumen, if that's what's needed, to unravel it properly.

."..there couldn’t be such a set, because it would have to include itself."

Sets can include themselves. E.g. the set of sets with more than one member has more than one member: it therefore includes itself. So the set of all things would include itself jprovided a set is a thing.

I think a problem with the set of 'all things' is to decide what is a thing. Is the letter that I forgot to write a thing? It's something I can refer to and discuss and I can tell untruths about (perhaps I didn't really forget, I deliberately left it unwritten) and therefore also tell truths about. So it's a thing, it can feature in true or untrue statements. But it was never written. It never existed. Is it a member of the set?

Reply to Cuthbert Good question! So maybe the problem, or an additional problem, is that ‘everything’ can’t be defined.

Quoting Cuthbert

So the set of all things would include itself provided a set is a thing.

But if it is ‘a thing’, then no set could ever include it - there would always be ‘everything’, PLUS ‘the set of everything’ which would be separate to the contents of the set, wouldn’t there?

Reply to Tom Storm

I think it’s in the conceptual foundations of maths dept., rather than Wittgenstein as such.

Alain Badiou - who takes set theory to be the best description of ontology that we have - makes a similar point but with an opposite conclusion. That the non-totalizibility of set theory attests not to any deficiency of set theory, but to the impossibility "totalizing" the universe at all. In other words, what you see as a deficiency in the instrument of description (set theory) is read by Badiou to be a positive characteristic of the world itself:

"In [Badiou's] language, the universe does not exist, whereas there are many worlds. [H]is argument for the nonexistence of the All or the universe draws on set-theoretical paradoxes, particularly Russell’s antinomy. Badiou argues as follows: If the All existed, it would have to exist as a member of itself. Otherwise, there would be an all outside of which something else, namely the All, existed. Hence, the All has to be a member of itself. Thus, there is at least one set, which is a member of itself. Nevertheless, there are obviously sets that are not members of themselves.

The set of all bananas is not itself a banana. This entails that the All consists both of sets which are members of themselves and sets which are not members of themselves. Given that the set of all sets that are not members of themselves famously leads into Russell’s paradox, the All cannot exist, because its existence would entail an antinomy". (Markus Gabriel, Transcendental Ontology).

I don't like the idea of treating set theory as ontology at all, so it's a non-starter for me, but I thought this was interesting.

Wayfarer: Yes, I think you are right. https://en.wikipedia.org/wiki/Universal_set

Reply to StreetlightX :up: (Also ...
[url=https://www.amazon.com/Critical existentialism-Nicola-Abbagnano/dp/B0006BVVOM]
Abbagnano[/url]'s non-totalizable "possibility" and Levinas' "infinition" contra "totality", while not strictly mathematical they are logically speculative; Meillassoux's (Badious' student) "not-All" as well. No doubt, SLX, you're familiar with one or more of these references.)

Reply to Wayfarer The continuum hypothesis might be of use.

Quoting Wayfarer

have been watching some documentary material on Georg Cantor and set theory. This gave rise to the following conundrum: I don’t think there could be a ‘set which includes everything’. Why? Because you implicitly then have two things - namely, everything, and also ‘a set which includes everything’.

Remember that a set is not a basket that contains things (ignore this if that's not what you meant)


A set is supposedly an abstract object. Say you have a club that is open to all pygmies. If you're a pygmy, you're a member of the club. A set is like the criteria for being in the club. It's not the pygmies themselves.

At first glance it doesn't seem problematic to have a set of everything. In fact the word "everything" in a certain context is doing just what we want that set to do: it's picking out everything. There are no things that are not members.

The problem comes when we start thinking of the set of all sets that are members of themselves. And then R, the set of all sets which are not members of themselves: that's Russel's paradox.

Quoting 180 Proof

o doubt, SLX, you're familiar with one or more of these reference.)

Indeedy. I haven't heard of Abbagnano before tho. Good?

Quoting Cuthbert

I think a problem ...is to decide what is a thing.

I think you've just summarized every possible philosophy.

Reply to StreetlightX Thanks. Very close to what I was getting at.

Reply to 180 Proof :up:

By the way, this question was prompted by a 2007 BBC Documentary called Dangerous Knowledge. It's about four great and controversial mathematicians - Cantor, Boltzmann, Godel and Turing - all of whom died by suicide ([s]Cantor[/s] Godel by refusing to eat and basically starving to death.) It's a bit sensationalist at times but worth watching. Features cameos with some current maths greats, including Chaitin and Penrose.

Quoting StreetlightX

Alain Badiou - who takes set theory to be the best description of ontology that we have - makes a similar point but with an opposite conclusion.

Why is it the best description of ontology?

Reply to StreetlightX Interesting to say the least.

Is the problem that "things" and "concepts" are being lumped together? The number of things is finite, and the set "universe" contains all of them. But there is no cap on the number of concepts that one can come up with. Concepts are second order wrt things, and cannot be treated the same way.

Reply to hypericin But there are 'sets of natural numbers' and 'set of prime numbers' and so on. So are they concepts? Or are they real? (Thorny question, I know.)

Quoting Wayfarer

a 2007 BBC Documentary called Dangerous Knowledge. It's about four great and controversial mathematicians - Cantor, Boltzmann, Godel and Turing - all of whom died by suicide

What's controversial about Godel and Turing?

What source does that film provide for its claim that Cantor died by suicide?

Quoting Wayfarer

So if ‘everything’ includes ‘every possible set’ - which it must do, otherwise it would be incomplete - then there couldn’t be such a set, because it would have to include itself.

Self-inclusion is not in itself paradoxical.

However, three ways to derive a contradiction from a claim that there exists a set whose members are all and only the sets are Russell's paradox, Cantor's paradox, and the Burali-Forti paradox.

Reply to TonesInDeepFreeze Good question! Now that you ask me, I find that:

Cantor retired in 1913, living in poverty and suffering from malnourishment during World War I.[33] The public celebration of his 70th birthday was canceled because of the war. In June 1917, he entered a sanatorium for the last time and continually wrote to his wife asking to be allowed to go home. Georg Cantor had a fatal heart attack on January 6, 1918, in the sanatorium where he had spent the last year of his life.[18]

I'll re-visit that section of the video, in case I mis-stated it.

suffering from malnourishment during World War I.

That doesn't say that he died by self-imposed starvation.

Quoting Wayfarer

Cantor, Boltzmann, Godel and Turing

Cantor died of a heart attack. Boltzmann was a physicist. Turing was most likely killed by the Brits because he was blackmailable and knew too many secrets. So some say. The Beeb ain't what it used to be. Also FWIW sets can contain themselves.

https://en.wikipedia.org/wiki/Non-well-founded_set_theory

https://plato.stanford.edu/entries/nonwellfounded-set-theory/

Reply to Wayfarer Same. However you treat numbers, numbers are one thing, sets of numbers are another. You can't treat these the same either (even the degenerate sets {1}, {2}, {3} are not the same as 1, 2, 3)

Reply to Shawn idk. He just kinda says it is, and draws out the implications of that. At least as far as I know.

Although I fudged the description somewhat - Badiou really says mathematics is ontology, and what philosophy does is 'meta-ontology': is explicates what is implicit in the math, in a way that even mathematicians are not able to necessarily do. This is not a Pythagorean thesis that being is mathematical, but that math is the langauge in which being is best spoken of.

Quoting TonesInDeepFreeze

Self-inclusion is not in itself paradoxical.

:up:

Reply to TonesInDeepFreeze Sorry, in my original post, I mis-named Godel as 'Cantor', typing in haste. I have corrected that. Apologies.

Godel died from malnourishment after refusing to eat, some say because he believed people were trying to poison him.

Cantor died from a heart attack in a sanatorium. However I think it's fair to say that he was treated appallingly by his peers, is it not?

Quoting hypericin

However you treat numbers, numbers are one thing, sets of numbers are another.

Interesting, then, that set theory is regarded as the basis for number theory, no?

Reply to hypericin

In set theory, numbers are sets.

0 = the empty set
1 = {0}
2 = {0 1}
etc.

This is not a claim that numbers are "really" sets (whatever "really" might mean as pertains to abstract objects), but rather that they are treated definitionally that way in set theory.

Quoting Wayfarer

set theory is regarded as the basis for number theory, no?

Set theory is one way to axiomatize mathematics.

Quoting TonesInDeepFreeze

(whatever "really" might mean as pertains to abstract objects

Abstract names for abstract objects, has it’s limits. Paradox, is where abstract meets logic and neither may win! What is REAL in an abstract world is the present moment, because it’s the only thing that is here, right now!

Reply to Wayfarer I recall someone else starting a thread in the same vein:

1. Assume U = the set that contains everything
2. If there's a that U contains everything, U must contain itself

Line 2 might need some explaining. Suppose E = everything. U = {E}. But then U can't be {E} because we can think of {{E}, E} that's a better match for U. However, {{E}, E} ain't it either because there's {{{E}, E}, {E}, E} [nested sets]...the process reiterates ad infinitum and, I suppose, ad nauseum.

There's also the matter of power sets. Suppose P(A) is the power set of set A, I believe a proven theorem in set theory is that the P(A) > A.

1. There's a U that's the set of everything [assume for reductio ad absurdum]
2. P(U) > U [The theorem I spoke of above]
3 . If U is the set of everything then, any set A is such that A < U
4. Any set A is such that A < U [from 1, 3 modus ponens]
5. If any set A is such that A < U then, P(U) < U
6. P(U) < A [from 4, 5 modus ponens]
7. P(U) > A and P(U) < A [2, 6 conjunction, also a contradiction]
Ergo,
8. There's no U that's the set of everything [1 to 7 reductio absurdum]

Quoting TheMadFool

E = everything

In set theory, 'everything' doesn't name a thing. Rather, 'everything' is used for quantification.

(1)

Suppose ExAy yex. ("There exists an x such that every y is a member of x")

Let Ay yeU.

So UeU.

'UeU' is not a contradiction (self membership is consistent with ZFC-regularity).

(2) Cantor's paradox

Suppose ExAy yex.

Let Ay yeU.

So PU is a subset U. ("The power set of U is a subset of U")

So Ef f is an injection from PU into U.

So Ef f is a surjection from U onto PU.

Previously proved theorem: Ax ~Ef f is a surjection from x onto Px.

So ~Ef f is a surjection from U onto PU.

So ~EAy yex.

Quoting TonesInDeepFreeze

In set theory, 'everything' doesn't name a thing. Rather, 'everything' is used for quantification.

(1)

Suppose ExAy yex. ("There exists an x such that every y is a member of x")

Let Ay yeU.

So UeU.

'UeU' is not a contradiction (self membership is consistent with ZFC-regularity).

(2) Cantor's paradox

Suppose ExAy yex.

Let Ay yeU.

So PU is a subset U. ("The power set of U is a subset of U")

So Ef f is an injection from PU into U.

So Ef f is a surjection from U onto PU.

Previously proved theorem: Ax ~Ef f is a surjection from x onto Px.

So ~Ef f is a surjection from U onto PU.

So ~EAy yex.

Above my paygrade fellow forum member. Thanks for the effort. Much appreciated! When I do get round to studying set theory in earnest, I might just be able to have a intelligent conversation with those like you who know their stuff.