Rational numbers == infinitely repeating sequences of digits

[untermensche]

And in theory any finite set can be expressed.

Even a set with more members than there are elementary particles in the observable Universe? A number which is approximately 10⁸⁶.

In theory.

The only constraint is time.

But with infinite numbers to express there is no amount of time in which they can be expressed. By definition they cannot all be expressed. By definition the last digit cannot ever be observed.

 

[lpetrich]

But with infinite numbers to express there is no amount of time in which they can be expressed. By definition they cannot all be expressed. By definition the last digit cannot ever be observed.

untermensche, this is what you seem to believe:  Finitism: "Finitism is a philosophy of mathematics that accepts the existence only of finite mathematical objects. It is best understood in comparison to the mainstream philosophy of mathematics where infinite mathematical objects (e.g., infinite sets) are accepted as legitimate."

 

[untermensche]

I give reasons.

 

[lpetrich]

I will now prove Euler's totient theorem. I will do that using group theory, since that proof is very simple.

The theorem: for relatively prime or coprime positive integer n and nonzero integer a, a^q = 1 mod n, where q = phi, Euler's totient function, the count of all positive integers relatively prime to n.

The proof:

First, show that the positive integers relatively prime to n form a group under multiplication modulo n: Z*.

Then use group theory to prove the theorem.

Z* is the set of all positive integers a that are coprime to n, along with multiplication.

• Associativity? Multiplication is associative. Yes.
• Identity? Multiplication has an identity: 1, and that number is coprime with every positive integer.
• Inverse? That is more complicated, and it uses the finiteness of this putative group's set.

Z*(1) and Z*(2) have {1}, and it is the identity group. The others have at least one non-identity element. Select one of them and take powers k of them. There are at most q = phi distinct values of a^k. So for some k1 and k2, a^(k1) = a^(k2) mod n. This gives us a^(k1)*(a^(k2-k1) - 1) = 0 mod n, and thus, a^(k2-k1) = 1 mod n, since a cannot be 0 and since all powers of a are relatively prime to n.

If m is the smallest positive integer that makes possible a^m = 1 mod n, then a generates a version of cyclic group Z(m). It also means that a^(m-1) is the inverse of a. Thus, every element has an inverse.

Since Z* is a group, every subgroup of it must have a size that evenly divides q. This means that a^q = (a^m)^(q/m) = 1^(q/m) = 1, since q/m is an integer. Since this is true of all a coprime with n, that proves the theorem.

 

[repoman]

seems like we are slightly stuck in the mental framework of thinking that decimal expressions are only valid, sort of an anti fraction bias.

1/7 is a valid and most efficient method of presenting this actual number.

in base 10 it is 0.142857..........

in base 7 (talk about a crazy base) 1/7 would be 0.1 and there are no "8" or "9" symbols.

 

[untermensche]

It all depends on how the word "number" is arbitrarily defined.

If you define "number" as some repeating string of numbers, that can never be fully expressed by definition, there can be no last number, then again by definition you have a number.

But 1/3 is not the same thing as 0.3333......

One is fully expressed and one can never be fully expressed.

They are only equivalent if you just arbitrarily command as a math god, "They are equivalent". If you define them to be equivalent.

But in reality they are never equivalent. Can never be equivalent. There can never be a last digit.

And in the real world one or two decimal places is all that is significant.

 

[lpetrich]

But 1/3 is not the same thing as 0.3333......

One is fully expressed and one can never be fully expressed.

0.3333... can be expressed as "an infinite sequences of 3's after the decimal point". The ability to write them out is irrelevant.

And in the real world one or two decimal places is all that is significant.

What counts as "real world"? untermensche, you have a lot to learn about precision measurement.

 

[untermensche]

But 1/3 is not the same thing as 0.3333......

One is fully expressed and one can never be fully expressed.

0.3333... can be expressed as "an infinite sequences of 3's after the decimal point". The ability to write them out is irrelevant.

It can be relevant or irrelevant.

It all depends on what you arbitrarily decide is relevent.

If you by nothing but arbitrary choice say that not being able to ever express a number is irrelevant it is. By choice not by reason.

All of mathematics is arbitrary choices like this.

What counts as "real world"?

A measurement of something "real".

 

[AdamWho]

I am happy that I stopped arguing about science and math with people who aren't qualified to have an opinion.

 

[untermensche]

I am happy that I stopped arguing about science and math with people who aren't qualified to have an opinion.

What exactly qualifies somebody to point out that a repeating digit that can in theory never end is very different from a series of digits that can possibly end?

Forget it.

You are not qualified to speak on such matters.

 

[AdamWho]

I am happy that I stopped arguing about science and math with people who aren't qualified to have an opinion.

What exactly qualifies somebody to point out that a repeating digit that can in theory never end is very different from a series of digits that can possibly end?

Passing Algebra 2

 

[untermensche]

It's a single topic.

All that is necessary is understanding the difference between something defined such that it can never be expressed and something defined to be able to be expressed.

 

[Jokodo]

It all depends on how the word "number" is arbitrarily defined.

If you define "number" as some repeating string of numbers, that can never be fully expressed by definition, there can be no last number, then again by definition you have a number.

But 1/3 is not the same thing as 0.3333......

One is fully expressed and one can never be fully expressed.

They are only equivalent if you just arbitrarily command as a math god, "They are equivalent". If you define them to be equivalent.

But in reality they are never equivalent. Can never be equivalent. There can never be a last digit.

And in the real world one or two decimal places is all that is significant.

you failed the Turing test.

 

[untermensche]

It all depends on how the word "number" is arbitrarily defined.

If you define "number" as some repeating string of numbers, that can never be fully expressed by definition, there can be no last number, then again by definition you have a number.

But 1/3 is not the same thing as 0.3333......

One is fully expressed and one can never be fully expressed.

They are only equivalent if you just arbitrarily command as a math god, "They are equivalent". If you define them to be equivalent.

But in reality they are never equivalent. Can never be equivalent. There can never be a last digit.

And in the real world one or two decimal places is all that is significant.

you failed the Turing test.

I bet you are one of those who don't understand how useless Turing said the so-called "Turing test" is.

Turing understood you could do a lot with simple principles of computing.

Wolfram showed you could create incredible complexity with simple rules of computing.

It doesn't mean the brain uses either.

But all you have done is show you don't understand what I said because it is all self evident.

Humans created the concept of a number and all the rules of mathematics.

It is only the human mind that arbitrarily decides what the rules are.

Human minds play around with the concepts they invented and every now and then a rare human adds something but only because he was exposed to what others had done.

 

[bilby]

It's fascinating and educational to observe how someone can be an apparently functional member of human society, while both being very obviously and demonstrably wrong about an incontrovertible fact, and totally incapable of changing their position - or even of stopping themselves from vehemently repeating their error.

The human capacity for counterfactual beliefs is truly astonishing.

This thread (or any of myriad similar ones) should be required reading for anyone foolish enough to utter the phrase 'but nobody could possibly believe that'.

 

[untermensche]

The above post is full of juvenile fury and devoid of any meaningful content.

It could be said of anything at any time.

And always be as empty.

Serious minds can see that.

 

[Jokodo]

The above post is full of juvenile fury and devoid of any meaningful content.

It could be said of anything at any time.

And always be as empty.

Serious minds can see that.

Radio traffic announcement: "There's a wrong-way driver on A 38, be careful."
untermensche: "A wrong-way driver my ass - there must be hundreds!"

 

[untermensche]

Yeah.

What a shame you can't demonstrate there is one thing I am mistaken about.

When I say: Humans created the concept of a number and all the rules of mathematics.

That is a challenge, and it isn't dismissed with absolutely nothing.

Nothing about mathematics is self evident.

Another challenge.

 

[lpetrich]

This thread (or any of myriad similar ones) should be required reading for anyone foolish enough to utter the phrase 'but nobody could possibly believe that'.

untermensche claiming that mathematics is arbitrary? Or that there is no such thing as an infinite sequence?

 

[Jokodo]

Yeah.

What a shame you can't demonstrate there is one thing I am mistaken about.

When I say: Humans created the concept of a number and all the rules of mathematics.

That is a challenge, and it isn't dismissed with absolutely nothing.

Nothing about mathematics is self evident.

Another challenge.

I was mostly amused by the use of "serious minds" as a first person singular pronoun.

Also, what challenge? All you're doing is demonstrating yet again that you can't tell the map from the landscape, quite literally. Talking of strings used to represent numbers as if they were the numbers is akin to saying mountains don't exist because you've checked the map and it's flat.