[And here there is a connection between ordinal arithmetic and the arithmetic of the extended semipositives [0, infinity], in that the affinely extended numbers validate the limit-based reasoning of infinity^infinity = infinity in exactly the same way as ordinal arithmetic tells us omega^omega = omega. But the affinely extended values have negatives and fractions and commutativity of addition and commutativity of multiplication and infinity + 1 = 1 + infinity = 2 * infinity = infinity * 2 = infinity and all that, all of which ordinal arithmetic lacks]
As I said, this is just about what definition of exponentiation you use. omega = omega[sub]0[/sub] and aleph_null = aleph[sub]0[/sub] both mean “The first ordinal after the natural numbers”. But aleph[sub]0[/sub] can also mean “The cardinality of the first ordinal after the natural numbers”. [In general, set theorists, following von Neumann, don’t worry so much about distinguishing between ordinals and cardinals, instead identifying cardinals with the smallest ordinals of the corresponding size. Though without the axiom of choice, there are cardinals with no ordinals of corresponding size…]. And this cardinality is also called beth[sub]0[/sub].
With ordinal exponentiation, (aleph[sub]0[/sub])^(aleph[sub]0[/sub]) = 2^aleph[sub]0[/sub] = aleph[sub]0[/sub], because ordinal exponentiation is designed to be continuous in a suitable sense, so that this should be the limit of 0^0, 1^1, 2^2, 3^3, 4^4, …, or the limit of 2^0, 2^1, 2^2, 2^3, 2^4, …, all of which are stilll bounded by aleph[sub]0[/sub].
With cardinal exponentiation, (aleph[sub]0[/sub])^(aleph[sub]0[/sub]) = 2^aleph[sub]0[/sub] = the cardinality of the powerset of the naturals. But this isn’t the definition of aleph[sub]1[/sub], though people often confuse it to be. Rather, the sequence of cardinalities defined via powersets is the beth sequence; 2^aleph[sub]0[/sub] = 2^beth[sub]0[/sub] = beth[sub]1[/sub].
aleph[sub]1[/sub] is by definition the smallest ordinal/cardinal with greater cardinality than aleph[sub]0[/sub]. The assertion that aleph[sub]1[/sub] = beth[sub]1[/sub] is the Continuum Hypothesis.
In general, we have: for a cardinal X, beth(X) is the cardinal corresponding to the powerset of X. For an ordinal/well-orderable cardinal X, aleph(X) is the smallest ordinal/well-orderable cardinal larger in cardinality than X. These give rise to corresponding sequences beth[sub]0[/sub], beth[sub]1[/sub], beth[sub]2[/sub], …, and aleph[sub]0[/sub], aleph[sub]1[/sub], aleph[sub]2[/sub], …, in which successive elements are obtained from the previous element by the beth/aleph operation. [The aleph[sub]0[/sub], aleph[sub]1[/sub], aleph[sub]2[/sub], …, is also called omega[sub]0[/sub], omega[sub]1[/sub], omega[sub]2[/sub], …, particularly when one wants to emphasize its nature as ordinals rather than as cardinals].
Cantor’s theorem tells us that beth(X) >= aleph(X), and thus the same is true of these sequences, but the assertion that these are always equal is the Generalized Continuum Hypothesis.
[Also here’s an aleph: א. And here’s a beth: ב. But typesetting math on SDMB is not really worth the effort…]
Same goes with absolute nothingness. Very hard to visualize. It feels like an opposite counterpart to infinity, even.
Imagine two infinitely long parallel lines. One line is rotated to intersect the other. Where the intersection happens is infinitely far away and happens infinitely fast, i.e. instantaneously.
Whether or not infinities actually exist, and likewise, nothingness as well… remains to be seen. Might have to wait a while…
Even if infinities do not have particular kinds of physical instantiations [e.g., two physical lengths whose ratio is larger than any finite value*], they certainly have other kinds of physical instantiations [the slope of a vertical line; the ratio of my number of sisters to my number of brothers]. Why privilege the one kind of physical instantiation to count as “existence” and not the other? [Answer: I don’t think you should.]
[*: Though there is also an implicit question here of how to determine that a length ratio is larger than every finite bound and not just larger than the largest finite bound tested, and how meaningful a statement like this can be if there is no means of such determination; i.e., how to physically interpret “For all natural numbers…” in an appropriate fashion]
…which is to say, a point arises from the lines’ intersection from 0 to with nothing in between.
From nothing to a finite-something in an infinitely fast instant and infinitely distant.
Makes me think of how the universe may have began in a vague way.
For illustrative purposes in abstract thinking, nothing else. I find these sorts of concepts intriguing as well.
If you’re asking why question whether or not infinities and/or nothingness are realities of the cosmos, well, the current standard model puts limitations on time, space, energy and matter, from the smallest realms to the largest magnitudes.
When we say a singularity is infinitely dense in the center of a black hole, as the math alludes to, is it actually so? Plenty of other examples here between the math, our understandings of the universe, and what actually is the case; whether we ever discover it or not.
Ah, so there’s a lot of repetition of what was learned in math class here, maybe it’s an avoidance reaction because nobody can say what the roots of their maths really is.
I don’t want to know about cardinals and ordinals, I want to know why people are so addictive to maths when it obviously relates to nothing real.
Will maths turn out to be an evolutionary dead end, and one day humanity wil stop doing it and do something less, umm, fictional ? Abstract ?
After all, it evolved into being, why shouldn’t it go extinct ?
Extinct maths that lasted a few thousand years and died out due to it’s own inconsistencies and flaws ?
That’s a new thread.
Yes, math might turn out one day to be an evolutionary dead end. Could happen. Do you have something to offer as to what the alternative might be?
You mentioned inconsistencies and flaws–did you have something particular in mind?
Mathematics does not itself present any arguments that it ought to in any way model anything real. It can’t, since it can’t talk about anything real without first modeling it mathematically, making it a circular argument. But it turns out that, for whatever reason, it really does seem to model plenty of real things. Some mathematicians find this fact eerie, but physicists and other scientists generally just roll with it.
Not particularly, no. Can you imagine the frustration if I did ?
Freaky, eh ? Then again prayers got Columbus to the New World as much as navigation - they thought at the time.
With the alternative being unemployment.
I don’t know, going to chase that one up.
I don’t know, going to chase that one up.
This means you’re not really saying anything, or anything much worth saying. You’re pointing out that something is possible. Well… anything’s possible. Much more importantly: What have you got?
As others have pointed out, mathematics itself makes no claim to be “real.” All mathematical results are conditional – they are ultimately statements of the form “If we assume these things are true then these other things must also be true.” The content, the particular things we study, are those that have proved interesting to mathematicians and to those who try to use math to model the world, originally largely in physics, but today including many other fields, with a large input from computer science. The wonder at the fact that math has been so useful a tool in modeling the world was famously expressed by Wigner as *The Unreasonable Effectiveness of Mathematics in the Natural Sciences. *To repeat what’s been said so often as to be trite, if math weren’t so successful, we wouldn’t be having this conversation via computers (not to mention all the other technology we all use directly or indirectly).
What’s always been the most amazing thing to me, actually, is that someone would be willing to pay me to do it.
Math is continually changing and growing as what we find interesting changes. If math gets replaced by something else, do you know what we’ll call it? Mathematics.
No, but Tegmark does. I don’t know what he’s on about, myself, if anyone could explain I’d appreciate it.
Wigner, yes.
Well, that’s the glass-half-full way of looking at it I suppose. Wait till some bastard uses maths to bankrupt you.
I don’t know what you’re trying to say here, with specific regard to “obviously relates to nothing real”, “evolutionary dead end”, and “died out due to [its] own inconsistencies and flaws ?”
I’m mostly worried about the lawyers handling my publisher’s bankruptcy filing, at the moment. But you said that math “obviously relates to nothing real.” Depending on what you mean by that, it’s obviously true or demonstrably false. In any case, it’s completely separate from the question of the morality of any particular application someone makes of math.
Your posts are all over the place. What point are you trying to make?
I don’t blame the sword.
The 7 foot tall barbarian holding the sword, on the other hand…
I get infinity.
I just don’t let finity’s husband find out.
Exactly! And then you simply define one of those cardinalities to be “three.”