/sci/ - Science & Math

>>15047622
>Ok, so what is this big number you are alluding too
First time I had posted in the thread

>>15047881
In raw predicate logic, if E is any relation, you can quickly prove
[math] \forall x. \big(xEs\leftrightarrow(xEy \land \neg xEx)\big) \to \neg(yEs\lor sEs\lor sEy) [/math]

For any set, if you allow for (even just predicative) comprehension, so that "x not in x" is a valid predicate, then you can define
s := {x in y | x not in x}
and the above logical statement implies
s not in y.

So very moderate forms of comprehension principles have the consequence that the collection of all sets is not consistently a set (because given a candidate y, you can define some s that is definately not in it).

It's fairly easy to defined more collections which can't be sets (e.g. the collection of all ordinals.)

Now with algebraic structure, the issue is that in set theory they are generally defined as "set + function", e.g. a group is any set plus any operation fulfilling some axioms.
But now given any set t, the pair ({t}, id) with id(t)=t is the trivial group.
So given any class y, you can define the class y' which for every x in y holds the group ({x}, id). So there's many collections of trivial groups.
So because there's collections which aren't sets (by predicative comprehension), the "set plus structure" concept is quickly infected by this issue.

Btw. very quickly people had also developed foundations where "x is not in x" is not a legal predicate for comprehension

https://en.wikipedia.org/wiki/Stratification_(mathematics)

but I mean the ZFC deal was fairly quickly settled on and few mathematicians care about foundations