On Sun, Oct 11, 2020 at 05:47:44PM -0400, Wes Turner wrote:
No, 2 times something is greater than something. Something over something is 1.
Define "something". Define "times" (multiplication). Define "greater than". Define "over" (division).
And while you are at it, don't forget to define what you mean by "infinity". Do you mean potential infinity, actual infinity, Absolute infinity, aleph and beth numbers, omegas, or something else?
I am not being facetious. Getting your definitions right is vital if you wish to avoid error, and to avoid miscommunication. Change the definitions, and you change the meaning of everything said.
(1) In the so-called "real numbers", there is no such thing as infinity. Since there is no such thing as infinity, infinity is not "something" that can be multiplied or divided, or added or subtracted. In the Real number system, there is no coherent way of doing arithmetic on "infinity". "Two times infinity" is meaningless.
In the real numbers, there's no sensible way of doing arithmetic with "infinity" without leading to contradiction.
Informally, infinity in the Real number system is a process that never completes, so doing twice as much doesn't take any longer.
(2) Mathematicians have created at least two extensions to the Real number line which do include at least one infinity. It is possible to construct a coherent system that is not self-contradictory by including either a pair of plus and minus infinity, or just a single unsigned infinity:
But in doing so, we have to give up certain "common sense" properties of finite numbers. For example, with only a single infinity, infinity is both greater than everything, and less than (more negative) than everything. We lose a coherent definition of "greater than".
Even in the extended number lines, two times infinity is just infinity, and infinity divided by infinity is not coherent and cannot be defined in any sensible way.
The IEEE-754 standard, and consequently Python floats, closely models the extended real number line.
(3) In the *cardinal numbers*, there is something called infinity. Or rather, there are an *infinite number* of infinities, starting with the smallest, aleph-0, which represents the cardinality of the integers, i.e. what people usually mean when they think of infinity.
Even in the cardinal numbers, two times infinity (aleph-0) is just aleph-0; however you might be pleased to know that two to the power of aleph-0 is aleph-1.
Arithmetic with infinite cardinal numbers is strange.
(4) In other extensions of the real numbers, such as hyperreal and surreal numbers, we can work with various different kinds of infinities (and infinitesimals).
For example, in the surreal numbers, we can do arithmetic on infinities, and you will be gratified, I am sure, that twice infinity is different from plain old infinity. In the language of the surreals:
2ω = ω + ω ≠ ω
(That's an omega symbol, not ∞.)
Unfortunately, the surreals are very different from the commonsense world of the real numbers we know and love. For starters, they form a tree, not a line. You cannot reach ω by starting at 0 and adding 1 repeatedly. (ω is not the successor of any ordinal number.) Consequently there are other infinite numbers like ω-1 that are less than infinity but cannot be reached by counting upwards from zero but only by counting down from infinity.
And of course, in the surreal numbers, there are an infinity of ever-growing infinities: not just ω+1 and 2ω but ω^2 and ω^ω and so on, all of which are "bigger than infinity".
All very fascinating I am sure, but I don't think that we should be trying to emulate the surreal numbers as part of float.