Am I thinking about this correctly?Russell's paradox shows that the Russell Set (R), the set of all sets which are not members of themselves, yields a contradiction. If R is not a member of itself, then R is a member of itself. If R is a member of itself, then R is not a member of itself. It loops back and forth forever like a recursion error. Does that not simply prove that the Russell Set doesn't exist? Shouldn't this just mean that we can know for certain that the contrary of the Russell Set is true? In other words, we can be certain that all sets which are not members of themselves are false, and thus all sets are members of themselves. The way I see it, this shows that all sets contain themselves as a means of self reference and definition. How could any concept be coherent if that concept made no reference to itself? Take for example, the set of all things that are blue. Is it not obvious that all things that are blue must be contained in the set of all things that are blue? How could you have a set that violates its own definition? Now take the reverse of the Russell Set, the set of all sets that contain themselves. Does the set of all sets contain itself? If it does, then it does. If it doesn't, then it doesn't. This reverse Russell Set is the most basic truism about sets, but it's fundamentally vacuously true, it says nothing at all. It seems to be the definition of definition. For a set to be defined, it must contain itself. If a set contains itself, it is defined. Otherwise, it is undefined. So, it looks like the Russell Set is simply the set of all undefined objects, and because they are undefined they glitch out when the rules of logic are applied to them, as these rules are fundamentally concerned with defined objects, things that actually exist. Language goes beyond those rules because all statements are inherently false. They can be mapped onto reality and used for computing results, but all statements are vacuous in nature.
>>16824747https://en.wikipedia.org/wiki/Constructive_set_theory
>>16825004what does that have to do with russel's paradox specifically
>>16825006you couldn't ctrl+f "russell"?
>The most famous paradoxThe grandfather paradox?
>>16824747RP links the most beautiful mathematical tools of our time: Set Theory, Decidability, Compatibility, and The Distribution Of The Primes into a single moment of Knotness.
>>16825004Constructive set theory (and mathematics) doesn't work, it implies infinite regression. If you can only form subsets of a set using formulas whose quantifiers range over previously given sets and not over all sets, then you need a base set that it all ties back to, so you need a power set for constructive set theory which it doesn't like.
>>16825728I don't follow why you tie forming subsets of given sets back to the wish to have the power set.If we're fine with the empty set, singletons and pairs, we can certainly start forming sets. We can use (predicative) seperation to filer from such sets - but of course we don't get anyhting new with that that we couldn't also form with basic set operations as you find them in Python or whatever.Now to get interesting predicates (e.g. number theoretic ones) to be used via Seperation from also more interesting sets, you need N in set theory. That's a "good" base set, upsurping the simple Neumann ordinal sets into a set term itself. If you don't got Infinity, you're just doing naive finite set theory (which is fine, but formally weak.)So yeah Infinity is commonly used. So you got plenty of sets to quantify over. Evenmoreso if you postulate function space. That still doesn't need powersets.Having adopted lets of sets, and maybe even the big N, I don't know where you spot an "infinite regression". I can only imagine this is a more conceptual critique of axiomatic set theory as a whole?The justification Münchausen-type trilemma of course indeed exists in set theory - you need to speak of a collection of variable symbols to formalize "collection". But that's not specific to constructivism. If you only mean such a philosophical ugliness is not resolved by constructivism, then you're right, but that also doesn't tie in to the Powerset axiom.
I want to add that impredicativity and the regress associated with it, is/was top of mind in some constructivists thinking - but it is exactly the restriction to not allow quantifications over classes (such as just all sets) which avoids this.Allowing unbounded quantifiers, you get a regress, and the most straight-forward justification for such impredicativity is a target model. Then the language can be recursive and just must fit the model. So again, staying on the syntatic level, it is exactly steps like only using Predicative Separation which avoids such "regressive" defintions.
>>16824747So the act of grouping all things blue , has a color and its blue?..- dumb schitzo anon.
>>16824747You really can't think of a set that doesn't contain itself?Just take any set of numbers, because a set of numbers is not a number. Otherwise, now we are running into the problem that elements are not elements.What operation extends numbers to sets?If 2 is a successor(1) then set is what?(x)
