Steve3007 wrote: ↑June 8th, 2021, 11:50 am
Terrapin Station wrote:I didn't catch what the thread title promised: what's the "overlooked" part of Russell's paradox?
Thomyum2 wrote:Didn't Kurt Gödel resolve this dispute about 90 years ago?
Hopefully philosopher19 will tell us more explicitly what he thinks he's added.
There is something that I've noticed as a result of the discussion I was having with RJG with regards how I can the overlooked part of Russell's paradox better.
Suppose you have a house with 3 rooms. Each room contains three lists. Of the three lists that each room contains, one of these lists is a member of itself. So there are at least three lists in this house that are members of themselves (because they reference/list themselves).
So now I form a new list: The list of all lists in this house that are members of themselves
There are definitely three items on this list. Does this list include itself as an item? In other words, should this list list itself?
You would think that this is entirely optional, but it isn't. As in you'd think, if we choose for this list not to list itself, then no contradictions occur, and if we choose for this list to list itself, then again, no contradictions occur. The former is not contradictory, whereas the latter is. Here's why:
You cannot take something as an ALL, and then add to it whilst keeping the context the same.
First you established ALL the self-listing lists in the house (of which there were 3), then you made a list of ALL the self-listing lists in the house (of which there were 3). So the ALL had already been exhausted. It is contradictory for this list to list itself because the list will not longer semantically qualify as ALL the self-listing lists in the house. You cannot have ALL plus one.
Now suppose you found a secret room in this house that also contained a list that lists itself. You'd just amend your list of all self-listing lists in this house
to include this new self-listing list. But in doing so, you have acknowledged that your previous list was in fact inaccurate, precisely because it was not a list of ALL the self-listing lists in the house because it did not include the list in the secret room.
You can even form a new self-listing list in your house
(like a list of all lists in this house
), and then form a new list of ALL self-listing lists in this house
to include this
new list. What you cannot do, is form this new
list, and then add itself to it as that would amount to ALL plus one. ALL plus one
is semantically not the same as ALL
A list of all self-listing lists in this house can never list itself because it would amount to ALL plus one (which is not the same as ALL plus nothing). This is why you can never have a set of ALL sets that are members of themselves that is itself a member of itself.