A set, and everything contained inside it.

A physical substance wherin the mean free path is much less than the characteristic length of the observation. An example would be a fluid.

The cardinality of the set of real numbers and of the power set of the integers. Denoted c.
The question that immedialty springs to mind is: is continuum the second smallest infinite cardinality?

Continuum is the new client for the ever so popular Subspace game. Much has already been written about Subspace and its subculture, that is why I feel obligated to write about this important milestone in the the Subspace history.

Subspace was developed by Virgin Interactive Entertainment a few years ago, and was rather quickly abondoned by the company. Several Subspace fanatics rose and decided to host the game on their personal servers. These servers soon evolved to become the unofficial Subspace universe and began hosting various zones which seemed nothing like the originals developed by VIE.

VIE developed their last version of Subspace (1.34) during '98. As the client was not updated since, cheaters appeared and bugs surfaced. These could not be fixed, as the game's source code was never released, not to mention there was no company to back the game up. And so, after three years, two dedicated Subspace players decided to code a new Subspace client from scratch--Continuum. The client had to be renamed, as Virgin Interactive Entertainment never relinquished the rights to the name `Subspace`.

Continuum, currently in its 0.36 version, is almost an exact copy of the original Subspace cilent. All that sets the two apart are minor alterations and interface bugs, which are quickly vanishing. This is an important step for Subspace because it finally puts all the power into the players' hands. The game can now be updated and improved on a regular basis, finally.

It is important to state that Continuum is not a replacement for Subspace, it is simply a new client, build by Subspace players, for Subspace players.

Power to the people.

In set theory, a continuum is usually defined to be any set for which there exists a one to one map from there onto the open unit interval (0, 1), which is a set that can be shown to be uncountable, and having the cardinality . Many sets that seem to have "more" elements than (0, 1) can actually be shown to have such a one-to-one onto mapping, meaning they all have the same cardinality. For instance, (0, 1) can be one to one mapped onto the whole real line (-∞, +∞) by using the mapping: f:x -> tan(πx - π/2). By applying Bernstein's Theorem, it can be shown that R2 is also a continuum, and by extending the argument, Rn for any positive integer n is also readily seen to be a continuum as well. The argument can even be extended into infinite-dimensional spaces, these can all be shown to be continua.

The power set of any countable infinite set can also be shown to be a continuum. Since all countable infinite sets have a one-to-one onto mapping to the set of natural numbers N, it suffices to show that the power set of N, P(N), is a continuum. It is easy to see that every point in the unit interval (0,1) can be expressed as an infinite sum of the form 2-n1 + 2-n2 + .... The (possibly infinite) set of natural numbers used to produce that sum {n1, n2, ... } is of course an element of P(N). To each element in P(N) there corresponds the point 3-n1 + 3-n2 + ... which is in (0, 1); this point cannot be produced by any other element of P(N). Thus we have created a one to one mapping of P(N) onto (0, 1), showing that it is a continuum, so ℵ = 20.

However, just as the power set of a countably infinite set is not countably infinite but a continuum, the power set of a continuum is not itself a continuum but something even more uncountably infinite than that.

The continuum hypothesis conjectures that there exist no sets intermediate (in the sense of existence of one to one onto maps, or put another way, of cardinality) between a countable infinite set and a continuum, or put another way, ℵ = ℵ1 I believe that it has been shown that this proposition is formally undecidable from the axioms of set theory.

This is an example of the kinds of pitfalls in reasoning you get into when you try to reason about infinity!

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