infinities in maths or is it

other infinities?

I placed a question in stack-overflow re maths and infinities. They put the question on hold. So copied bellow the question and initial replies.

In case they delete the content..


The following question was flagged as possibly Off-Topic. Am trying to re-phrase in a more mathematical oriented manner:

The question is, in a sense, of whether there are some mathematics that indeed go into parameters of infinities other than size.

Cantor, as far as I gathered did a lot of work - in mathematics, to do with size of infinities. I think its interesting that these infinities might be only mathematical.

While attempting to research for information about mathematical perceptions and descriptions of infinities, other than size, I could not find Any answers. It seems odd to me, perhaps via a certain ignorance about maths, that there was no data about mathematics of infinities that deal with possible numerals that aren't to do with sizes of infinities.

Hence the original question had examples of elements that seemed like might have more to do with numbers, rather than logic or philosophy. (e.g. nothing to do with questions of endlessness, beginning of infinities, etc.)

Indeed, it seems that even if the questions ultimately produce replies to the tune of: negative, there are no such mathematical infinities - we are still within the realms of maths. The interest is indeed specific to mathematics. Whether or not there are such infinities expressed in some sort of mathematics a person like myself, with a relatively limited interaction with maths, might have not heard of.

However, I also think that there were a few things missed in the 1st questions set, hence the following contains slight alterations:

    Do infinities have various times, or speeds? (As far as I know, speed and time are calculable, hence seemed that might have some maths to do with infinities - do they not?)

    Can infinities be differentiated by rhythms? (As far as I gather,rhythms can be expressed in maths.)

    Perhaps there are other elements that offer theories of different kinds of infinities? (e.g. colour? kind of topography perhaps?) Here perhaps colour and topography are misleading and apologies! 1st I meant to write "topology". 2nd, regarding colours, I thought of gradual colour variations, say from blue to purple and so on. These seem a bit like the lines that may be used to illustrate infinitesimals. Hence the question is, in a sense, whether as far as mathematics goes, there can be an infinite colour gradients between, for example, blue and purple.

Another element that I neglected to ask about is Volume. If there are infinite sixes, are there infinite volumes?

Hopefully these clarifications make the question a bit better.

Again, have tried all manner of search engines and search quarries - however the links I get with the terms are to do with fanciful doctrines, and mystical perceptions. Some philosophical enquires and many musical titles and lyrics. My interest here is indeed on the mathematics of infinities rather than these other, possible insights, to do with infinities.

Many thanks for the replies so far. They are indeed illuminating! Is it possible to + everyone??

Cheers!

(..am keeping the 1st question bellow in case it does interest some people..)

I understand that Cantor came to realize there are a fair few infinities via learning they have different sizes.

Could not find information about other possible aspects to do with infinities. Do infinities have various times, or speeds? Can infinities be differentiated by rhythms? Perhaps there are other elements that offer theories of different kinds of infinities? (e.g. colour? kind of topography perhaps?)

Am really asking here from a certain point of classlessness mixed with wonders. If these are wrong questions - will be good to know :)

Cheers and many thanks in advance!
set-theory infinity
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put on hold as off-topic by Andrés Caicedo, Tim Raczkowski, S.Panja-1729, 6005, Normal Human 8 hours ago

This question appears to be off-topic for this site. While what’s on- and off-topic is not always intuitive, you can learn more about it by reading the help center. The users who voted to close gave this specific reason:

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If this question can be reworded to fit the rules in the help center, please edit your question.

	
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This is like asking whether numbers have different colors,etc. Not literally meaningful. – user254665 14 hours ago
	
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I think you've misunderstood something about "infinities" somewhere along the line.

What Cantor realized is that it is possible to have two infinite sets that don't have the same cardinality, which is a very specific notion of "size". An "infinity" is not a real-world object that has particular properties such as color or shape or anything else. Rather, "an infinity" is a sort of "size" label that we give to every member of a particular family of sets that have the same cardinality. There is nothing more to it than that abstract concept.
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answered 14 hours ago
Cameron Buie
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Mathematicians don't use any of the words you use to describe infinite sets (the things Cantor described). Writers and artists are welcome to find, forge, and render thematic connections, of course. You might start looking at the math side of things by going Wikipedia-diving, starting at Set Theory, Ordinal Number, and Cardinal Number. You can also look at what other people have done for inspiration.
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answered 13 hours ago
Mike Haskel
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As Cameron Buie indicated, the "size" aspect of various orders of (cardinal) infinity is a very specific kind. Two "infinities" are of different orders if they cannot be placed into a one-to-one correspondence with each other. One must be very careful about this: It's not enough to say that there's a mapping that takes one infinity into a strict subset of the other; there has to be absolutely no way you can map each one, one-to-one, onto the other. Just because you don't happen to think of such a "bijection" doesn't mean that one doesn't exist.

Thus, for instance, the even natural numbers 2,4,6,8,…2,4,6,8,… have the same cardinality as all the natural numbers 1,2,3,4,…1,2,3,4,…, even though the naïve identity mapping takes the even natural numbers into a strict subset of the natural numbers. There is a straightforward mapping k↔2kk↔2k that is a bijection, as required.

All that being said, there is something that could be said to be a different distinction between "infinities" other than cardinality: One can also distinguish between cardinal infinities ℵ0,ℵ1,ℵ2,…ℵ0,ℵ1,ℵ2,… and ordinal infinities ω,ω+1,ω+2,…,ω⋅2,…,ω2,…ω,ω+1,ω+2,…,ω⋅2,…,ω2,… The latter sequence, in addition to conveying "size" (all the ones I listed have cardinality ℵ0ℵ0, but there are lots more), also convey a structure of ordering, hence the name.

To give you an idea of what the notion of an ordinal comprises that a cardinal does not, consider the ordering 2,1,4,3,6,5,8,7,…2,1,4,3,6,5,8,7,…—that is, the even natural numbers alternating with the odd natural numbers, with the even naturals coming "first" in each pair. We can envision that as an ordinary list of numbers running in two columns—

1357⋮2468⋮
12345678⋮⋮

—except that we read each row from right to left, rather than left to right. This ordering can be associated with the ordinal 2⋅ω2⋅ω, which is just ωω. That's not surprising, since we can easily come up with a bijection that maps between the kkth element of the usual arrangement of the natural numbers (which are associated with the ordinal ωω) and the kkth element of this ordering.

On the other hand, consider the ordering 1,3,5,7,…,2,4,6,8,…1,3,5,7,…,2,4,6,8,…—in other words, all the odd naturals in order, "followed" by all the even naturals in order. Such an arrangement of the naturals does not have "more" numbers than the usual arrangement, so their cardinalities are the same, but there is no mapping between the kkth element of this arrangement and the kkth element of the usual arrangement. This is akin to reading the numbers down the left-hand odd column, then down the right-hand even column, and this ordering has ordinality ω⋅2ω⋅2, which is not the same as ωω. We have to read this ordering in two "strands", as it were.

Although it is easy to get wrapped up in the apparent whimsy of infinities, one must, however, be careful in thinking about them. For example, I've talked about them rather loosely and informally above, to convey some intuition about them, at the cost of some rigor. Questions that appear to muse absent-mindedly about infinity are liable to draw some fire.
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answered 13 hours ago
Brian Tung

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