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# The Power Set Proof

A similar proof to the Diagonal proof is the Power Set proof. This is essentially just another version of Georg Cantor’s proof of 1891, (Footnote:
Georg Cantor, ‘*Über eine elemtare Frage de Mannigfaltigketslehre’*, Jahresberich der Deutsch. Math. Vereing. Bd. I, S. pp 75-78 (1891). An English translation of the original can be seen Cantor’s original 1891 proof here.
) and it is usually presented with the same secondary argument that is commonly applied to the Diagonal proof. The Power Set proof involves the notion of subsets. A subset of a set is a set that includes some or all of the elements of a given set. In standard set theory, given a set ** A**, there can be a power set of

**whose elements include every possible subset of the set**

*A***.**

*A*

The Power Set proof states that, given a set ** A** with an infinite number of elements, there cannot be a function that matches each element of the Power Set of

**to each element of the set**

*A***; that is, there is no function that matches every element of the set**

*A***to every subset of the set**

*A***(see also one-to-one correspondence).**

*A*

The usual version of the proof as is commonly used today is as follows:

We start with an initial assumption; the object of the proof is to prove that this assumption cannot be correct. The assumption is that there is a function, which we call ** List(x)**, that maps each element of the set

**to each subset of the set**

*A***.**

*A*

**1**.

We now define a set, that we call the set ** B**, to be the set which includes every element of

**which is matched to a subset that does not contain that actual element itself.**

*A***2**.

It follows that ** B** defines a set, which must either have no elements (and so is the empty set), or have elements which are elements of the set

**.**

*A***3**.

It follows that this set ** B** must be a subset of

**.**

*A***4**.

But it is also the case that the set ** B** must be the set given by the matching function for some element

**n**, that is, that

**.**

*B*=*List*(n)**5**.

Now, since the element **n** of the set ** A** is matched to the set

**, it follows, from the definition of the set**

*B***, that the element**

*B***n**cannot appear in the set

**itself.**

*B***6**.

But this results in a contradiction, since the definition of the set ** B** stipulates that any element of

**which is matched to a subset of**

*A***that does not contain that element must be an element of the set**

*A***.**

*B***7**.

Therefore the original assumption that there can be some matching function ** List(x)** must be false.

And as for the Diagonal proof, this proves that there can be no function that gives a one-to-one correspondence of the elements of a set and the subsets of a set, where the function is in the same language as the definitions of the sets.

And, as for the Diagonal proof, the notion that a ‘Power Set’ must be a set of a ‘bigger’ infinity than the set from which it is derived is a notion that is based on the same unfounded assumptions as occur in the secondary argument of the Diagonal proof.

No-one has ever encountered an infinite set other than by way of some definition. And no-one has ever encountered an infinite subset of any infinite set other than by way of some definition. And every definition must be a definition in some language. Given a definition of an infinite set, we can define various subsets of that set, some of which are finite, others infinite. And we can have a function, ** in a meta-language**, that lists every such definition of a subset. But the Power Set proof does not apply in this case - for the definition of the set

*must be defined in terms of that list - but*

**B***is a definition in a sub-language to that meta-language and the information of the meta-language is inaccessible by the sub-language. For more details, see the Diagonal proof, the secondary argument of the Diagonal proof, and A List with no Diagonal number, and Proof of more Real numbers than Natural numbers.*

**B**

When the Power Set proof is divested of any Platonist assumptions concerning the ‘existence’ of things independently of language, the proof only proves that there cannot be a matching function ** List(x)** that matches up every element of a set to every subset of a set,

**.**

*in the same language as the language being used for the definition of the sets*

For more on Platonism see Platonism, The Myths of Platonism, Platonism’s Logical Blunder, Numbers, chairs and unicorns and the posts Moderate Platonism and Descartes’ Platonism.

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