there are more sets of things than there are individual things

[michel paul]

"We must therefore accept that there are more sets of numbers than there are individual numbers. And in fact, with one crucial qualification that we shall come back to, this argument can be applied to anything whatsoever: there are more sets of bananas than there are bananas, more sets of stars than there are stars, more sets of points in space than there are points in space, more sets of sets of bananas than there are sets of bananas, and so on. In general – subject to the crucial qualification that I’ve said we’ll come back to – there are always more sets of things of any given kind than there are individual things of that kind. This is Cantor’s theorem."

https://aeon.co/essays/why-some-infinities-are-bigger-than-others

I just happened to discover this article today. It very nicely illuminates a prior thread on this list. 

--

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

[Joseph Austin]

I'm not prepared to admit that there are the same "number" of Evens as Naturals,

in that there is NO (computable) "number" of items in an infinite set,

so comparison by counting or pairing is "meaningless".

So we need a different notion of "size" or magnitude when speaking of "how much" (continuous or infinite concepts) than when speaking of "how many" (discrete entities). 

Which is more: 3 jellybeans or 2 chocolate bunnies?  As any three-year old which he would rather have.

Joe Austin

 

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.
To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+...@googlegroups.com.
To post to this group, send email to mathf...@googlegroups.com.
Visit this group at https://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.

[Maria Droujkova]

Cardinality is not a number at all.

Cheers,
Dr. Maria Droujkova

NaturalMath.com
Make math your own, to make your own math!

 

On Mon, Mar 27, 2017 at 10:08 AM, Joseph Austin <drtec...@gmail.com> wrote:

I'm not prepared to admit that there are the same "number" of Evens as Naturals,

in that there is NO (computable) "number" of items in an infinite set,

so comparison by counting or pairing is "meaningless".

So we need a different notion of "size" or magnitude when speaking of "how much" (continuous or infinite concepts) than when speaking of "how many" (discrete entities). 

Which is more: 3 jellybeans or 2 chocolate bunnies?  As any three-year old which he would rather have.

Joe Austin

On Mar 27, 2017, at 3:18 AM, michel paul <python...@gmail.com> wrote:

"We must therefore accept that there are more sets of numbers than there are individual numbers. And in fact, with one crucial qualification that we shall come back to, this argument can be applied to anything whatsoever: there are more sets of bananas than there are bananas, more sets of stars than there are stars, more sets of points in space than there are points in space, more sets of sets of bananas than there are sets of bananas, and so on. In general – subject to the crucial qualification that I’ve said we’ll come back to – there are always more sets of things of any given kind than there are individual things of that kind. This is Cantor’s theorem."

https://aeon.co/essays/why-some-infinities-are-bigger-than-others

I just happened to discover this article today. It very nicely illuminates a prior thread on this list. 

--

===================================
"What I cannot create, I do not understand."

- Richard Feynman

===================================
"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
===================================

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.

To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+unsubscribe@googlegroups.com.

To post to this group, send email to mathf...@googlegroups.com.
Visit this group at https://groups.google.com/group/mathfuture.
For more options, visit https://groups.google.com/d/optout.

--
You received this message because you are subscribed to the Google Groups "MathFuture" group.

To unsubscribe from this group and stop receiving emails from it, send an email to mathfuture+unsubscribe@googlegroups.com.

[michel paul]

On Mon, Mar 27, 2017 at 7:08 AM, Joseph Austin <drtec...@gmail.com> wrote:

I'm not prepared to admit that there are the same "number" of Evens as Naturals,

​But certainly​

 you

​are ​

prepared to admit that this function is one-to-one: f(x) = 2x  ?

​​

in that there is NO (computable) "number" of items in an infinite set,

so comparison by counting or pairing is "meaningless".

Here is a

​n infinite​

set of ordered pairs

​generated 

by that function: {(1,2), (2,4), (3,6), (4,8), (5,10), ...}

​It is true that without a specific context these ordered pairs don't 'refer' to anything, so yes, we could call the set 'meaningless', but I think it is meaningful to say that, since the function is one-to-one, each domain element corresponds to one and only one range element, and vice versa.

Though not a 'computable' value, the cardinality of the set of ordered pairs equals the cardinality of the set of domain elements as well as the cardinality of the set of range elements.

 

So we need a different notion of "size" or magnitude when speaking of "how much" (continuous or infinite concepts) than when speaking of "how many" (discrete entities). 

​Well, we're in luck. Awhile back a very smart man came up with this insightful way to look at it:​

​"‘By Number we understand not so much a Multitude of Unities, as the abstracted Ratio of any Quantity, to another Quantity of the same kind, which we take for Unity.’​"

- Isaac Newton

​That is beautiful.

--

​ Michel