Calculus of Measures: Mapping Natural Numbers to Rationals

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Discussion Overview

The discussion revolves around the mapping of natural numbers to positive rationals within the context of measure theory. Participants explore concepts related to countability, density of rational and irrational numbers, and the implications of these properties on ratios and derivatives of functions derived from such mappings.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions whether it is possible to map natural numbers to positive rationals and derive a function that reveals insights about the measure of irrationals between them.
  • Another participant acknowledges the existence of a mapping between countable sets but argues that such a mapping does not preserve order, rendering the measure between consecutive rationals meaningless.
  • Some participants express confusion about the relative densities of rational and irrational numbers, questioning if the density ratio is fixed across all real intervals.
  • There is a discussion about the nature of cardinality, with one participant asserting that the ratio of countable to uncountable sets leads to confusion regarding the properties of individual elements like 1/2 and π.
  • Participants explore the idea that the infinite nature of the ratio between countable and uncountable sets may imply that the ratio approaches infinity under certain conditions.
  • One participant suggests that the ratio of the densities of irrationals to rationals could be constant across intervals, leading to further speculation about the implications of this ratio being infinite.
  • A clarification is made that algebraic irrationals are countable while transcendental irrationals are uncountable, adding complexity to the discussion of density and cardinality.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and confusion regarding the concepts of countability, density, and the implications of these properties on ratios. There is no consensus on the nature of the ratio between rational and irrational numbers or the implications of their densities.

Contextual Notes

Limitations in understanding the definitions of countability and uncountability are evident, as well as the implications of these definitions on ratios and measures. The discussion remains unresolved regarding the mathematical properties and relationships being explored.

moyo
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i have one question concerning measure theory and this...could we map the natural numbers to the positive rationals...then observe the measure between the rationals being mapped ,as a value giving us a sense of how many irrationals are between them...then generate a function where we take the derivative of this mapping as 1 scaled by the magnitude of the value of the measure between any two rationals...i'm interested , does this dirvative function reveal anything striking
 
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Yes, since both Z and the set of rational numbers are countable, there exist a mapping from one to the other. However, that mapping will not "preserve" order so distance or 'measure' between two consecutive rational numbers is meaningless.
 
I probably haven't grasped these concepts well either , but its interesting,I always thought that since both rationals and irrationals were dense on the reals...then the statement that the rationals are fewer than irrationals was a statement about their relative densities...that rationals are less dense. If this is true ,is the ratio of their densities a fixed number for every possible real interval...?
 
moyo said:
I probably haven't grasped these concepts well either , but its interesting,I always thought that since both rationals and irrationals were dense on the reals...then the statement that the rationals are fewer than irrationals was a statement about their relative densities...that rationals are less dense. If this is true ,is the ratio of their densities a fixed number for every possible real interval...?

The rationals are countable.
That is, we can find a 1-1 function between the rationals and the natural numbers.
The irrationals are not countable.
As far as a ratio is concerned, it's infinite.
That is, there are infinitely (and uncountably) more irrational numbers than there are rational numbers.
 
Thankyou for clearing that up for me.

But that's confusing,

the cardinality of the ratio is uncountably infinite..

if countability and uncountability be the property of a set...then how can a you have a property of a set bet the property of a ratio...

1/2 is not countable...it belongs to a set that is countable...simmilarly pi is not uncountable...it belongs to a set that is uncountable..
so i can't see how you can have a ratio that is one or the other as i have tried to show...because 1/2 also belongs to an uncoutanle set...the reals...so it seems the ratio belongs to a set with uncountable infinite property , but when asked what exactly that ratio we get null , so the null set should be uncountable?

I am really confused
 
moyo said:
Thankyou for clearing that up for me.

But that's confusing,

the cardinality of the ratio is uncountably infinite..

if countability and uncountability be the property of a set...then how can a you have a property of a set bet the property of a ratio...

1/2 is not countable...

Erm... 1/2 is just a single element. How is it not countable?

moyo said:
it belongs to a set that is countable...simmilarly pi is not uncountable...

Erm... again $\pi$ itself is just a single element.
Did you perhaps mean that when written as a decimal number, it has an infinitely long but countable representation?
And it belongs to the the set of the irrationals (and stronger, the set of transcendentals), which is uncountable.
 
I like Serena said:
Erm... 1/2 is just a single element. How is it not countable?
Erm... again $\pi$ itself is just a single element.
Did you perhaps mean that when written as a decimal number, it has an infinitely long but countable representation?
And it belongs to the the set of the irrationals (and stronger, the set of transcendentals), which is uncountable.

so cardinality of a number is how many of them they are?

Then
you had said thsi:

I like Serena said:
As far as a ratio is concerned, it's infinite.
.

So the infinite aspect of the ratio cannot be its cardinality right since it only one ratio

what does it reffer to?
 
Cardinality can only reffer to a set...

Are you simply saying that as we calculate the ratio of their densities , the value of the ratio approaches infinity as the number of iterations in our calculations gets arbitrarily large.
 
So my initial query could be rephrased as,

what is the derivative of this function we are itterating?
 
  • #10
I like Serena said:
As far as a ratio is concerned, it's infinite.

I seem to sense that a ratio can only be infinite in a position where the denominator is zero.or the numerator is infinite and the denomenator is finite...

one canse is undefined and in the other the denominator is infinite if e plug in the cardinals of the irrationals and rationals to calculate the ratio...

Are you saying that we have also "defined" this ratio to be infinite when dividing an uncountable by a countable?

or did we figure it out?
 
  • #11
If we defined it then all the mystery goes away from this dichotomy of countable and uncountable as we could have defined it the opposite way around if we had felt like it...

And if we figured it out ...what was the function used, and what is its derivative?
 
  • #12
It is clear that the ratio has to be constant as the same ratio is present in any interval of the number line.

So if the ratio , irrational uncountable/countable is an uncountable value...

this can only be true when the countable goes to zero or irrational uncountable gets infinite...

since division by zero is undefined
we multiply the irrational uncountable by another infinity
larger than the irrational uncountable for it to have an effective increase. Then the infinite nature of the ratio between the irrationals and the rationals is larger than the cardinality of the uncountable irrationals themselves.is this true?
 
  • #13
One point to mention - the algebraic irrationals are countable; it is the transcendental irrationals that are uncountable.
 

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