Logic and Language
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Copyright   James R Meyer    2012 - 2020 https://www.jamesrmeyer.com

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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 A whose elements include every possible subset of the set 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 A 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).

 

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 A to each subset of the set A.

 

1.

We now define a set, that we call the set B, to be the set which includes every element of A which is matched to a subset that does not contain that actual element itself.

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 B, it follows, from the definition of the set B, that the element n cannot appear in the set B itself.

6.

But this results in a contradiction, since the definition of the set B stipulates that any element of A which is matched to a subset of A that does not contain that element must be an element of the set 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 B 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.

 

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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Footnotes:

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Copyright   James R Meyer   2012 - 2020
https://www.jamesrmeyer.com