Does $1-\frac{1}{2^\omega}$ equal 1? What about $1-\frac{1}{2^{\aleph_0}}$?

Question

(Correct if I'm wrong on any of this.)

Recently, I've been learning about transfinite ordinals and cardinals. For some cases, I understand the difference between ordinals and cardinals, for instance that $$|\{1,2,3,\ldots,\omega\}| = \aleph_0$$ I get that ordinals index a set, while cardinal numbers are the sizes of sets. For some cases, however, I'm still uncertain about which to use.

An equation that would seem to me to work is $$1-\frac{1}{2^\omega} = 1$$

After all, $$\lim_{x\to\infty}{1-\frac{1}{2^x}}=1$$

However, I'm not certain that $\omega$, an ordinal number, is the right infinity to be using. Does this work: $$1-\frac{1}{2^{\aleph_0}}=1$$

Or perhaps, am I making a fallacy and does neither work?

Answer 1

Cardinals measure the number of elements in a set. Since there is no such thing as half an element of a set, there is absolutely no meaning to a reciprocal of a cardinal number. Certainly not an infinite cardinal.

Similarly, ordinal numbers measure the order type of an ordered set. Specifically, a well-ordered set. Since in this case there is no meaning to a reciprocal either, both of the equations $1-\frac1{2^{\aleph_0}}=1$ and $1-\frac1{2^\omega}$ are absolutely meaningless (also, note that cardinal and ordinal arithmetic are very different, $2^\omega=\omega$, whereas $2^{\aleph_0}>\aleph_0$).

It is tempting to say that in the surreal numbers one can find meaning to a reciprocal of an ordinal, but that would be false. Because the surreal numbers form a field, and as such their arithmetic is incompatible with the ordinal arithmetic, and with the cardinal arithmetic as well.

In short, there is no meaning to $\frac1{2^x}$ when $x$ is a cardinal or an ordinal, and there is even less meaning to $\lim_{x\to\infty}\frac1{2^x}$ when $x$ is either an ordinal or a cardinal, not because there is no meaning for limits. There is a meaning of limits in either context of cardinals and ordinals. But because (1) there is no meaning for reciprocals, and (2) there is no meaning for the $\infty$ symbol: do you mean the natural numbers, do you mean the entire ordinals/cardinals? The meaning of $\infty$ is unclear, and just today I was scolded by my colleague for using the $\infty$ symbol in a rather ambiguous set theoretic context.

Answer 2

There are lots of different things in mathematics called "infinity" or "infinite". Infinite cardinals are different from infinite ordinals, and the $+\infty$ encountered in calculus is a different thing from those, and the $\infty$ that is neither $+\infty$ nor $-\infty$ but is approached by going in either direction on the line is slightly different from those (that's the one that is often appropriate when one writes $\lim\limits_{x\to0}\dfrac 1 x=\infty$) and the infinite value of Dirac's delta function is an altogether different thing from all of the above, and the infinite number of Abraham Robinson's non-standard analysis or of Euler's way of doing calculus is different from all of the above, and Knuth's "surreal numbers" (relevant to strategy in games of skill) are different from all of the foregoing. And there are yet others.

Some arithmetic operations are not appropriate for some of these "infinities", just as, for example, one cannot do matrix multiplication when the number of columns on the left fails to match the number of rows on the right.

Accordingly, one does not evaluate things like $\dfrac 6 {\aleph_2}$, etc.