# Cantors "Paradox" of the greatest cardinal number

#### Hartlw

This addresses the “proof” of Cantor’s paradox in Suppes (Axiomatic Set Theory), pg 5.

Let S1 be the set of all sets EXCEPT S1 with cardinality n.
Let S2 be the set of all subsets of S1 EXCEPT S2 with cardinality p.

Then every member of S1 is a member of S2 and every member of S2 is a member of S1. Therefore n=p and Cantors “paradox” is not a paradox.

In all fairness to Suppes, he made the mistake of using intelligible language.

#### Shakarri

I am confused, is n the cardinality of every set in S1 or is n the cardinality of S1? I read it as the first meaning but it looks like you interpreted it as the second.

Not every member of S2 is a member of S1.
Suppose S1={T1,T2}
The the subsets of S1 are {T1}, {T2}, {T1,T2}, {} where {} is the empty set.
{T1,T2} is not a member of S1 and nor is {}

2 people

#### Hartlw

n and p are clearly defined.

Your premise is incorrect. As stated previously:
Let S1 be the set of ALL sets EXCEPT S1 with cardinality n.
Let S2 be the set of ALL subsets of S1 EXCEPT S2 with cardinality p.

If x is a member of S1 it is a subset of S1 and hence is a member of S2.
If x is a menber of S2 it is a member of S1 because S1 contains all sets (except S1)
Therefore S1=S2 and n=p.

S1

#### Shakarri

n and p are clearly defined.
No they are not. Saying so and changing your underlining does not help. I pointed out how the wording of your post was confusing, why don't you just reword it.

If x is a member of S2 it is a member of S1 because S1 contains all sets (except S1)
If S1 contains all sets it must be infinite in size. Equating n to p would be equating 2 infinities which is a risky business and can cause paradoxes.

1 person

#### Hartlw

I am relying on Suppes, Dover edition.

"...we consider the cardinal number n of the set of all sets.".... "But we may also consider the set of all subsets of S, and its cardinal number p." pg 7.

He uses Russels "set of all sets," which of course leads to a paradox. I do not do so so there is no paradox.

Last edited:

#### Hartlw

Shakkari: I suspect Cantor's "paradox" was meant to show that an infinite set doesn't have a cardinality p. He proves this by showing the assumption it does leads to a paradox. All I do is show the assumption it does does not lead to a paradox so the proof is invalid.

So what is your proof that the set S of all sets except S has no cardinality?

Last edited:

#### zzephod

Shakkari: I suspect Cantor's "paradox" was meant to show that an infinite set doesn't have a cardinality p. He proves this by showing the assumption it does leads to a paradox. All I do is show the assumption it does does not lead to a paradox so the proof is invalid.

So what is your proof that the set S of all sets except S has no cardinality?
Deleted, decided to teach my pig to sing instead.
.

Last edited:
1 person