Content-Type: text/html

# Cantors Diagonal argument/Talk

HomePage | Cantors Diagonal argument | RecentChanges | Preferences
You can edit this page right now! It's a free, community project

I'm glad there is an article on this in Wikipedia. Cantor's Diagonal argument is my favourite piece of Mathematics - Andre Engels

quote

Its result can be used to show that the notion of the [set of all sets]? is an inconsistent notion; if S would be the set of all sets then P(S) would at the same time be bigger than S and a subset of S.

end quote

Can the diagonal argument really do this? If we accept the existence of uncountable infinities (and i guess you do if you accept the diagonal argument) then it is pretty clear that the set of all sets must be of uncountable size if it exists. How do you define an ordered list of elements of an uncountable set? If you can't define an ordered list, how do you apply the diagonal argument? (Maybe you can, but it's not immediately obvious how. I think a bit more needs to be written about this if the above quote is to remain.)

If you look at the second proof you can see that it makes no assumptions about S being ordered. That is in fact the main difference between the actual diagonalization argument and its generalization that is used in the second proof. I will add a remark that the second proof is not equal to the actual diagonal argument but a generalization of it. -- Jan Hidders

That sounds good to me. Thanks.

HomePage | Cantors Diagonal argument | RecentChanges | Preferences
You can edit this page right now! It's a free, community project
Edit text of this page | View other revisions
Last edited August 7, 2001 12:23 pm (diff)
Search: