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 April 23rd, 2018, 10:04 PM #1 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 390 Thanks: 27 Math Focus: Number theory Real #s altered by countable ops? Is there any countable function that affects the continuum of real numbers?
 April 23rd, 2018, 11:01 PM #2 Senior Member   Joined: Aug 2012 Posts: 2,076 Thanks: 593 What's a countable function? ps -- Your title said countable ops as in operations. You could take the real numbers and map all the negative reals to 0 and all the nonnegative reals to 1, thereby disconnecting the continuum, if that's what you mean. Is that what you mean? That's only one op. A function can do pretty much anything. Last edited by Maschke; April 23rd, 2018 at 11:21 PM.
 April 24th, 2018, 12:24 PM #3 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 390 Thanks: 27 Math Focus: Number theory I was looking for an operation, or a countable sequence of them, that could map from countable to uncountable sets (Aleph-null to Aleph-one). This was the "connection" I meant. Such an "operation" would transform a "discontinuum" to a continuum. Is there any operation (or operations) at all which reverse the transform from continuum to countable set, as you mentioned? Aside: can Cantor's diagonal argument contain "spaces," "nulls" or even sets (somewhat like matrices, tensors, etc.) in place of numbers? [Please ignore the term "countable function."]
 April 24th, 2018, 01:25 PM #4 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra The real numbers can be seen as the limits of infinite series of rationals. These are by definition countably infinite sequences of operations on a countably infinite set. Thanks from topsquark and Loren
 April 24th, 2018, 05:19 PM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra You could even extend that to a mapping of natural numbers (including zero) to the reals. The absolute value of the floor function maps the reals to the natural numbers.
 April 24th, 2018, 05:41 PM #6 Senior Member     Joined: Sep 2015 From: USA Posts: 2,163 Thanks: 1135 what it sounds like he's after is a function that maps a discrete set into a continuous one. I don't see how that could be done. Thanks from Loren
April 24th, 2018, 05:59 PM   #7
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Quote:
 Originally Posted by Loren I was looking for an operation, or a countable sequence of them, that could map from countable to uncountable sets (Aleph-null to Aleph-one).
No countable sequence of maps from a countable set could possibly hit every element of an uncountable set. Each map has a countable number of inputs and a countable number of outputs. There can never be more than countably many outputs.

Even if you started with a countable set and then mapped each element to some real number; and then you took each of those countably many real numbers and mapped them to countably many different real numbers; and you iterated that procedure a countable number of times; you still would in the end only hit countably many real numbers. That's because a countable union of countable sets is countable. So at each iteration if you throw all the new targets into a bucket, at the end of countably many iterations you still have only countably many real numbers in your bucket.

I don't know if you consider real numbers as limits of countable sequences of rationals to be a satisfactory answer. It's true that every real number is the limit of a countable sequence of rationals. But that is not the same as saying that every real gets hit. In fact the countable sequence of rationals 3, 3.1, 3.14, 3.141, 3.1415, ... never hits $\pi$. So this idea fails your requirement of mapping the initial countable set to every single element of the uncountable target set. The best you can do is to guarantee that you can get arbitrarily close to each real with a countable sequence of rationals. But you can't hit them all.

And the idea of limits of sequences applies only to the reals or some set with an equivalent topology. The question you asked was about arbitrary sets, one countable and one uncountable. You can NOT compose countably many maps to hit all the uncountably many elements of the target. You can only hit countably many of them.

Last edited by Maschke; April 24th, 2018 at 06:05 PM.

April 24th, 2018, 06:52 PM   #8
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Quote:
 Originally Posted by Maschke It's true that every real number is the limit of a countable sequence of rationals. But that is not the same as saying that every real gets hit. In fact the countable sequence of rationals 3, 3.1, 3.14, 3.141, 3.1415, ... never hits $\pi$.
Shhhh! You don't want zylo to hear you say that!

-Dan

 April 24th, 2018, 10:20 PM #9 Senior Member   Joined: May 2015 From: Arlington, VA Posts: 390 Thanks: 27 Math Focus: Number theory The limit definition of a real number is helpful, near to my approach. To put it simply, what I am after may start with: Domain (Aleph-one) >= Range (Aleph-one) > Domain (Aleph-null) >= Range (Aleph-null) > Domain (Finites) >= Range (Finites).
April 25th, 2018, 11:17 AM   #10
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Quote:
 Originally Posted by Loren Domain (Aleph-one) >= Range (Aleph-one) > Domain (Aleph-null) >= Range (Aleph-null) > Domain (Finites) >= Range (Finites).
That notation is unclear. A function has a domain, not a set or a cardinality.

For example what is the domain of 6? What's the range of the reals?

Last edited by Maschke; April 25th, 2018 at 11:20 AM.

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