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Re: Infinity

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Posted by Paul R. Martin on December 15, 2003 01:40:17 UTC

Hi William,

"What about the set of all sets which includes itself?"

Let's back up a little and ask, "What about the set of all sets?" I think "the set of all sets" is an ambiguous concept which cannot be defined in any meaningful way. I think that if somehow you knew what all the sets were, then you would be entitled to form what would be a completely new concept by conceiving of all of those sets in one collection. Otherwise I think you are simply assuming that such a concept exists and you have no basis for that assumption.

But, assuming that you did have a meaningful concept of the set of all sets and then narrowed it down by excluding all those which do not include themselves, you would arrive at Russell's paradox. In my view, 'paradox' in mathematics is a euphemism for nonsense.

So, what about it? It is nonsense.

"This is the universe, is it not?"

Probably not. It all depends on what you mean by 'universe'. It is not the universe of science which is our familiar 4D space-time continuum because this does not contain, for example, concepts like unicorns or the platonic ideals. There are sets of those kinds of concepts which are not part of the universe of science. Similarly, most other notions of 'universe' are not completely inclusive of everything.

"or God?"

Probably not, again. It all depends on what you mean by God. And there are many more different notions of God than there are of the notion of universe. I suspect that if you took a poll, you would find very few believers in God who would consider Him to be a set.

"or infinity?"

No, almost certainly not. Sets which are considered to be infinite do not by any means include all sets, even those of the highest orders of infinity. And there is an error of type here: Infinity is simply the measure of cardinality of a set and is not the set itself.

"or is it finite because it is countable as one set even though it contains all sets."

You may be close here although you have stated your question as a non sequitur. The set of all sets may indeed be finite (I happen to believe it is) but not because it is countable. All finite sets are countable whether they contain all sets or not.

"What is the number of sets if not 8 on its side?"

In my opinion it is a very large finite number.

Warm regards,


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