Maybe?

http://forumserver.twoplustwo.com/47...99-1-agree-83/

tbf I had it wrong till the youtubes

As masque de Z said that is the Continuum hypothesis. It is possable to assume either true or false.

Which you choose would depend on how your are applying this model of transfinite arithmatic. Otherwise they are just two consistent while mutually inconsistent theories.

But then the next class, I tell them that I line up my sheep in the following order: 2, 6, 10, 14, ..., 4, 8, 12, 16, ...

I have now arranged my sheep so that it looks like I have infinitely many left over by the time you run out of your sheep. In fact, you could let your sheep through in the normal order, and I'll have enough left for you to go through all your sheep again.

This wasn't directed at me was it?

The reason why you want to deny greater cardinalities is the same reason you were struggling with Godels theorem fwiw.

If there are greater cardinalities - and I understand that this is a contentious topic - then how might these different types of infinity be described to a child (or her dad)? Are we going beyond the real line to find such things?

I guess I'm looking for concise conceptual descriptions of different types of infinities, even if they are not mathematically precise - like 'infinitely complex' or 'infinite in magntitude'.

Enrique, I'll try my best to work through your post and thanks for taking the time to lay this out.

Its not a contentious topic. People are free to deny the existence of ANY infinite set and plenty of good mathematicians rejected essentially the existence of the continuum at first but nobody that understands could possible accept infinite sets, the continuum and then somehow reject greater cardinalities.

That might be tough to impossible. All the examples I can think of/find on the internet are not going to be suitable for a child, unless they can just accept the power set of the reals as being larger.

Potentially relevant video from Numberphile just posted today.

https://www.youtube.com/watch?v=BBp0bEczCNg

thanks! great video.

does any math (or related field) require numbers that are greater than all real numbers (N) to solve a problem? if N* contains k that is greater than all N, there was some mention that 1/k has a useful quantum application, but the video didn't elaborate.

@dessin - does the power set of reals contain k? or is k also > the power set of reals? or can you just do this an infinite number of times where there is always a k' > previous largest set. and if so, is this useful and isn't the whole thing just that very first type of infinity, now more properly defined as really really big rather than just big?

actually reading through the start of the thread Aaron/dereds posted/mentioned, there was one post that did make me think.

what does 1/3 + 2/3 (i.e. 0.99999999~) x 2 = ?

is it 1.99999999~ as claimed or more accurately 1.99999999~8 ?

isn't there some argument for the latter and if so, have we created a number set with a new property where something different can happen after infinite repeats, therefore at least creating the possibility that there's a set that allows some cardinality in-between rational and real numbers.

so glad I can ask these questions.

hi masque de Z,

can you elaborate more on this please

We do not use 0.999999... for 1 to avoid having different representation for the same number ie 1 here.


Also the powerset (expressed as 2^A ) can be used to extend cardinality to higher levels of infinity.

https://en.wikipedia.org/wiki/Power_set

And if the cardinality of natural numbers is and the cardinality of the real numbers is c then it can be shown that c=


And my ultimate gift on this topic and other related to you is this link hahaha.

http://mathematicslibrary.blogspot.c...uction-to.html
get the link from it of the djvu format file

http://db.tt/hKoZ8ufU


(if you do not have djvu you should install it its an alternative to pdf to read ebooks that actually uses much less memory per file. See https://en.wikipedia.org/wiki/DjVu and follow the links to the official site ie "http://djvu"+".sourceforge"+".net/" (ignore the + signs and quotes when rejoining the quoted parts to form the full link address) (the 2+2 editor censors this link for some reason that is funny as this is a safe legitimate site) that has the program to download and install)


Then go to pages 36 to 43 to read all the things that relate to this thread and which i think ultimately can be explained to a good student even if they are not near 16,17,18 yet.

(sorry no pdf for this but i have the original book so if it becomes impossible to do any of this i may have to scan certain pages, but djvu is a must have for reading tons of scientific ebooks available out there that often in quality are better than pdf. Dont be afraid to install djvu, i have it in all my computers for many years now.)

See these links first to see what i mean really.

https://en.wikipedia.org/wiki/Finitism

https://en.wikipedia.org/wiki/Ultrafinitism


(essentially what is the meaning of a number of 10^10^10^10^10^10 etc in terms of its digits if its representation requires more objects than the entire universe has available if the universe is finite?)

I see infinity as a concept non realizable anywhere in nature and our brain (when doing math) is just nature also so we deal with infinity in mathematics without it being realized in anything. I do not think infinity is needed to derive any of the usual math used in sciences. We can always avoid this and see it as limits and progressively smaller errors etc. Basically although infinity is used in calculus etc all the time you do not really need to go that way if you want to avoid the concept and imagine only very big numbers up to a point that is consistent with what can be represented in a physical system that is not itself ...infinite.

I do not believe the number 2^(1/2) exists for example anywhere in nature. Same with Pi etc.

do you have a PDF version of George Simmons article?

It depends on how precise you want things to be.

I'd start with the fact (speaking to an adult) that if you have N objects, you have more than N ways of making subsets. In fact, there are 2^N ways you can make such subsets (each object is in or out, and that's two options for each of the N elements), and that's always more than what you started with.

To a kid, I'd be explicit. If you have 2 objects {a,b}, then you can make 4 different collections: Take none, take just a, take just b, take them both. And 4 is greater than 2. You can do the same with 3 objects and have fun counting through all 8 combinations (and maybe do a little combinatorics on the side for fun). It turns out that this pattern of getting bigger keeps happening even if there are infinitely many objects. You can climb to higher powers of infinity by taking different combinations of them.

An even looser way to make the connection is to think about all the points on the real line, and then think about all the different curves you can make on the plane. There are lots and lots of squiggles you can draw on the plane, and it turns out that this is more than the number of dots on the real line.

It may or may not help depending on the level of rigor you require, but something like that was my first exposure to the different infinities. I don't remember what book it was in, but I remember the squiggles.

Ok i found an online version that i am not sure of the legality of the site i will link but of course it allows you to at least read all the relevant pages . The djvu file is much better quality though.

http://www.scribd.com/doc/109795930/...-Modern#scribd


(You do not need to sign up or anything to see it, just scroll the page down all the way to the suggested book pages)


Maybe i can post here; (maybe save these pages in case they are not valid links in the future)

Infinity here is as real as any other mathematical concept. I.E as real as the uses the models can be put to.

My father's masters thesis was The Arithmetic Of Transfinite numbers so I obviously don't really "deny" greater cardinalities.

And I barely know Gödel's theorem except it creates a barber paradox.

And yet, at a certain level you do.

You asked the key question with such precision that I assume you knew the answer but wanted someone else to answer.

Fair enough.....I'd recommend learning Godels Theorem. I know your generation typically never learned how to code so learning Godel numbering would help sharpen your thinking quite a bit i imagine.