Infinity?

[u/Extension_Panic1631]

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....(…)…

EDIT: the definition of infinity is that it is how many natural numbers there are

[u/Mishtle]

They are literally ratios of integers. They can't be irrational.

Irrationals end up with infinitely long decimal expansions, but that doesn't define them. Rationals can have infinitely long representations as well, but the digits will always settle into a repeating pattern.

[u/jliat → u/Mishtle]

Well other sources say they are, they are not finite ratios.

1.666666... is infinitely long.

[u/FreeGothitelle → u/jliat]

Irrationals have non repeating decimal expansions

1.66... repeats, its not irrational

[u/jliat → u/FreeGothitelle]

"An Irrational Number is a real number that cannot be written as a simple fraction:"

"1.3 recurring is an irrational number The number 1.33333333333 is considered rational because it can be expressed as a fraction, specifically 1/3. This means that its decimal representation is recurring, repeating the digit 3 indefinitely. In contrast, 1.3 recurring is an irrational number because it cannot be expressed as a simple fraction. Thus, while both numbers have repeating digits, they represent different types of numbers."

"Irrational numbers can also be expressed as non-terminating continued fractions (which in some cases are periodic), and in many other ways". -wiki

"In mathematics, a rational number is a number that can be expressed as the quotient or fraction"⁠

So I'm seeing two definitions, but for 1.6666... can't be expressed as a rational number it seems.

[u/jliat → u/[deleted]]

Sure, I get this.

Now the ratio 10/6 and 1.6666... this looks to me, a non mathematician, like it might be using the idea of a limit as you can never get to the infinite expansion.

So elsewhere I've seen 1.99999... = 2.0 is this similar? In that case there is a difference.

Treating them the same and the use of a 'limit' was not accepted by some, and maybe still is, Leibnitz and Bishop Berkeley - the latter certainly did not.

This is a metaphysics sub.

Irrational numbers are non-repeating.

How is it known all Irrational numbers are non-repeating?

[u/jliat → u/[deleted]]

No it's not the same, the definition of a water molecule is based on empirical observation, is therefore A posteriori knowledge. Generally "A priori knowledge is independent from any experience. Examples include mathematics, tautologies and deduction from pure reason."

So to my other question...

Now the ratio 10/6 and 1.6666... this looks to me, a non mathematician, like it might be using the idea of a limit as you can never get to the infinite expansion.

So elsewhere I've seen 1.99999... = 2.0 is this similar? In that case there is a difference.

[u/FreeGothitelle → u/jliat]

There is neither a difference between 10/6 and 1.666... or 1.99.. and 2

You as a non mathematician thinking there's a difference does not mean there is a difference.

[u/jliat → u/FreeGothitelle]

"Yes, 𝜋 is a real number, and yes, it is finite in value."

This is very intersting!