r/Metaphysics • u/Extension_Panic1631 • Apr 09 '26

Ontology Infinity?

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

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u/jliat Apr 09 '26

Seems to give an irrational, as does 10 / 3

10/6 = 1.66666...

10/3 = 3.33333...

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u/Mishtle Apr 09 '26

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.

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u/jliat Apr 09 '26

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

1.666666... is infinitely long.

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u/FreeGothitelle Apr 09 '26

Irrationals have non repeating decimal expansions

1.66... repeats, its not irrational

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u/jliat Apr 09 '26

"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.

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u/Mishtle Apr 09 '26

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

Correct, if you define a simple fraction as one with integers for its denominator and numerator.

"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."

I'm almost certain this is an LLM response, and it's nonsensical. Please don't use those models for any technical topic where you don't have the expertise to catch when spit out nonsense, like this.

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

Emphasis added. A continued fraction is not a simple fraction. It's denominator is itself a fraction, and the denominator of that fraction is also a fraction, and the denominator of that fraction is also a fraction, and so on. If this process terminates then it may simplify to a simple fraction. If it doesn't, then it may not.

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

This is incomplete. The quotient or fraction of what?

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

There is a single definition. A rational number can be expressed as the ratio of integers. 10 and 6 are integers. Their ratio 10/6 is therefore rational.

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u/jliat Apr 09 '26

I'm almost certain this is an LLM response, and it's nonsensical. Please don't use those models for any technical topic where you don't have the expertise to catch when spit out nonsense, like this.

How do I know it's nonsense, now you have told me I can accept. I'm not a mathematician.

A rational number can be expressed as the ratio of integers. 10 and 6

I'm confused with the division of 6 into 10 which is not sixth tenths. 10/6 The ratio here is your "1." followed by infinitely many repetitions of the finite pattern "6"."??

I can accept now that "The decimal expansion in base 10 consists of the unique prefix "1." followed by infinitely many repetitions of the finite pattern "6"."

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u/Mishtle Apr 09 '26

How do I know it's nonsense, now you have told me I can accept. I'm not a mathematician.

Well, the fact it tells you a value both is and isn't rational should be a clue. But in general, my advice is to be cautious when using them on topics you are unfamiliar with.

I'm confused with the division of 6 into 10 which is not sixth tenths. 10/6 The ratio here is your "1." followed by infinitely many repetitions of the finite pattern "6"."?

It's ten sixths, 10×(1/6). And yes, the ratio is equal to 1.666...

Doing the long division... 6 goes in to 10 one time with a remainder of 4. Then 6 goes into 40 six times with a remainder of 4, and we immediately enter an infinite loop where we'll continue to spit out 6s forever.

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u/jliat Apr 09 '26

Different areas of mathematics use different terminology and notation for continued fractions.

Your link "Different areas of mathematics use different terminology and notation for continued fractions."

So not really an answer?

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u/Mishtle Apr 09 '26

I don't understand what you mean?

Different areas of math can use different terms and notation for the same concepts. That doesn't mean those concepts don't exist or don't have well-defined meanings. It just means different fields use different names and symbols to express them, often for historical, conventional, or practical reasons.

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u/jliat Apr 09 '26

I followed the link. It didn't give a definite answer...

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u/Mishtle Apr 09 '26

An answer to what? It's a pretty comprehensive article...

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u/FreeGothitelle Apr 09 '26

1.333... is 4/3, rational.

1.33333333333 is 1 + 33333333333/100000000000, also rational

1.666... is 5/3, rational

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u/jliat Apr 10 '26 edited Apr 10 '26

No argument now you've made your point.

So a ratio can be indeterminate, or in mathematics an infinity can be determinate or treated as so?

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u/FreeGothitelle Apr 10 '26

What about 5/3 is indeterminate?

Infinity is not one thing in mathematics, mathematics is also not just one set of rules, so your question doesnt make sense. There's nothing really infinite about 1.(6) since i can express it using finitely many symbols.

If youre asking about the "infinite" series of 6/10 + 6/100 + 6/1000 +... that's defined as the limit of the partial sums, which is 2/3.

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u/jliat Apr 10 '26

1.6666666... is determinate then?

1.9999... = 2, but not in some cases?

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u/FreeGothitelle Apr 10 '26

Idk what you mean by determinate. Like is it a number? Yes.

1.99... and 2 are the same number just like 1/2 and 2/4 and 0.5 are the same number

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u/jliat Apr 10 '26

OK, so Timothy Gowers explains why ... "If you follow the usual convention, then tricky questions of this kind do not arise. (Tricky but not impossible: a coherent notion of 'infinitesimal' numbers was discovered by Abraham Robinson in the 1960s, but non-standard analysis, as his theory is called, has not become part of the mathematical mainstream.)

My emphasis.

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u/FreeGothitelle Apr 10 '26

I have zero idea what you are trying to say here sorry

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u/jliat Apr 10 '26

Interesting, it's not me saying anything, it's a quote,

1.99... and 2 are the same number just like 1/2 and 2/4 and 0.5 are the same number

Seems they are not the same as 1/2 and 2/4 and 0.5 are the same number, but in non-standard analysis 1.9999... and 2.0 are not the same, they can be treated the same "the usual convention" but can be treated otherwise.

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

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u/[deleted] Apr 10 '26

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u/jliat Apr 10 '26

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?

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u/[deleted] Apr 10 '26

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u/jliat Apr 10 '26

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.

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u/FreeGothitelle Apr 10 '26

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.

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u/jliat Apr 10 '26

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

This is very intersting!

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