The set of all logical formulae: Countable or uncountable?

[lpetrich]

That's a geometrical interpretation. I was thinking of an analytic(algebraic) interpretation.

 

[steve_bank]

Ok, it's the ratio between the circumference and diameter of a circle. Still almost infinitely short, compared to an infinite series.

"Pi" is a label. it is not a way of describing that number.

What you might be trying to get at is that pi is a "computable number".

What I'm getting at "pi is a ratio between 2 different lengths, one of which isn't measurable with a non-transcendental, bendy ruler.

If you can't measure a circumference how do you come up with pi? I would have thought it was empirical.

I can think of several ways to measure a circumference. Create a circle and overlay a string, measure the length. You know me, I'm no mathematician just a simple minded practical engineer....

 

[Kharakov]

Ohh, so the idea was that Pi could be described without using an infinite amount of symbols.


Would transcendentals be in the realm of Fuzzy, the not precisely defined logical bear?

 

[Bomb#20]

What I'm getting at "pi is a ratio between 2 different lengths, one of which isn't measurable with a non-transcendental, bendy ruler.

If you can't measure a circumference how do you come up with pi? I would have thought it was empirical.

I can think of several ways to measure a circumference. Create a circle and overlay a string, measure the length. You know me, I'm no mathematician just a simple minded practical engineer....

It's not empirical. There are hundreds of known formulas for pi. Even before computers they were a better way of finding pi than using a tape measure. Here's a smattering...

 

[lpetrich]

Ohh, so the idea was that Pi could be described without using an infinite amount of symbols.

Yes indeed. Like a finite-sized algorithm for calculating that number.

Would transcendentals be in the realm of Fuzzy, the not precisely defined logical bear?

Some transcendental numbers can be specified with a finite number of symbols, like e and pi, but most cannot.

 

[steve_bank]

Which came first, the empirical observation of the ratio or a series expansion?



Math is as much empirical as theoretical. Applied to the real world math is vakidated as science is. For measuring the Egyptians used a rotating wheel not unlike today to measure length. Pi resulted from having to solve real problems, not as an abstraction.

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

 

[untermensche]

Ohh, so the idea was that Pi could be described without using an infinite amount of symbols.

Yes indeed. Like a finite-sized algorithm for calculating that number.

So pi can be calculated?

The whole thing?

Pi is no different from dividing 6 by 9.

It is a ratio that produces an infinite decimal.

There are many such ratio's.

Pi is only interesting because it has use. In itself it is just an infinite decimal from a ratio.

 

[Jokodo]

Ohh, so the idea was that Pi could be described without using an infinite amount of symbols.

Yes indeed. Like a finite-sized algorithm for calculating that number.

So pi can be calculated?

The whole thing?

Pi is no different from dividing 6 by 9.

It is a ratio that produces an infinite decimal.

There are many such ratio's.

Pi is only interesting because it has use. In itself it is just an infinite decimal from a ratio.

pi is very much different from 6/9. 6/9 is a rational numbers that in some bases can be represented as a finite string, e. g. 0.8₁₂ in duodecimal. Pi cannot be represented as a finite string in any integer base, e. g. in base 12 it is 3.184809493B9(...)

 

[untermensche]

6/9 in the decimal system is not anything else in that system.

Changing bases changes the nature of the ratio. You no longer have 6/9 in some other system.

You are comparing apples to oranges.

 

[Bomb#20]

Which came first, the empirical observation of the ratio or a series expansion?

The empirical observation, of course.

Math is as much empirical as theoretical. Applied to the real world math is vakidated as science is. For measuring the Egyptians used a rotating wheel not unlike today to measure length. Pi resulted from having to solve real problems, not as an abstraction.

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

But the estimate of pi you can get from rotating wheels and measurements is 22/7. That served the ancient world just fine for building pyramids and whatnot, but if you tried to build modern precision machinery based on pi=22/7 you'd promptly turn your machines into piles of expensive scrap metal. The abstract non-empirical theory-based series expansions are necessary to solve modern real engineering problems.

 

[Jokodo]

6/9 in the decimal system is not anything else in that system.

Changing bases changes the nature of the ratio. You no longer have 6/9 in some other system.

So, dividing a physical object in 9 equal parts and summing over 6 of them results in a different quantity when you weigh the result in a decimal or duodecimal system of weights?

You are comparing apples to oranges.

Google "rational number" and don't come back before you've understood the definition.

 

[untermensche]

6/9 in the decimal system is not anything else in that system.

Changing bases changes the nature of the ratio. You no longer have 6/9 in some other system.

So, dividing a physical object in 9 equal parts and summing over 6 of them results in a different quantity when you weigh the result in a decimal or duodecimal system of weights?

You are comparing apples to oranges.

Google "rational number" and don't come back before you've understood the definition.

No object can be divided into 9 equal parts. You can only get 9 parts that are very close.

I am talking about a specific ratio in base 10.

It is not like anything else.

You can only get approximate equivalencies if you look at other bases since in base 10 it has no final value.

 

[Jokodo]

So, dividing a physical object in 9 equal parts and summing over 6 of them results in a different quantity when you weigh the result in a decimal or duodecimal system of weights?



Google "rational number" and don't come back before you've understood the definition.

No object can be divided into 9 equal parts. You can only get 9 parts that are very close.

A lot of objects can. Any pure crystal consisting of only the same molecule over again can be divided into 9 equal parts in exactly those cases where its number of molecules is an integer multiple of nine.

Every distance and every timespan can be divided into 9 equal parts, unless you can provide evidence that time and space are quantized (in which case you should be collecting your Nobel Prize instead of debating us dimwits).

I am talking about a specific ratio in base 10.

You're literally confusing the map for the countryside. Base ten is a language to talk about numbers, not a system of numbers.

It is not like anything else.

You can only get approximate equivalencies if you look at other bases since in base 10 it has no final value.

Sure it does. The "final value" is 2/3. Some languages, like duodecimal, express this value in a string that ends in a repeating sequence of 0s which is left out by convnetion; others, like decimal or binary, express it as a string ending in a repeat sequence that includes non-0-digits (6 for decimal, (10) for binary).

The value itself is every bit as finite as that of 1/5 (which, incidentally, has a non-zero repeat sequence in both binary and duodecimal).

You know, like people who talk about 'Ifriqiya and people who talk about Africa are referring to the same landmass. Like "Australia" having 10 letters against "Asia"'s 4 doesn't make Australia the bigger continent.

 

[untermensche]

A lot of objects can. Any pure crystal consisting of only the same molecule over again can be divided into 9 equal parts in exactly those cases where its number of molecules is an integer multiple of nine.

Isotope

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

Every distance and every timespan can be divided into 9 equal parts, unless you can provide evidence that time and space are quantized (in which case you should be collecting your Nobel Prize instead of debating us dimwits).

I agree with the dimwit part.

Distances and time spans can't be divided at all.

The measurement of a distance can be divided with pencil and paper but the distance can't be divided.

And a span of time can't be divided either. All that can be divided is some less than perfect measurement of a time span. And again divided with pencil and paper.

Abstractly divided.

I am talking about a specific ratio in base 10.

You're literally confusing the map for the countryside. Base ten is a language to talk about numbers, not a system of numbers.

The language metaphor is crude but I'll accept it.

I am talking about a specific word in a specific language. 6/9

You are trying to tell me that Schmetterling is the same word as butterfly.

Sure it does. The "final value" is 2/3.

I've told you this before. 2/3 is an operation not a final value. 2 is a value and 3 is a value.

 

[Jokodo]

Thank you very much, I know what an isotope is. Not all elements have multiple stable isotopes.

And, yes, you "told" me before that 2/3 is not a number. You were as wrong then as you are now. 2/3 is a perfectly valid way to refer to a number in the language of fractional notation. It's the same number that is referred to by 0.8 in duodecimal or 0.(6)* in decimal.

 

[untermensche]

So what are you going to divide into 9 identical pieces?

And how?

2/3 is a way to refer to one number divided by another.

You want to pretend it is a number so you can pretend it is the same thing as the result of the division.

It's like saying the wood and the wood chipper are the same thing as the wood chips.

 

[Jokodo]

https://en.wikipedia.org/wiki/Fraction_(mathematics)

Glad I could help, and good luck learning something new.

 

[Speakpigeon]

Both of them are indeed "infinitely long".

There are several examples that I can choose, but I'll chose the exponential function:
\( \exp x = \sum_{n=0}^\infty \frac{x^n}{n!} \)
exp(1) = e, the base of the natural logarithms. It is known to be transcendental.

You know, I can express Pi as Pi, and that's only 2 symbols. It's almost infinitely short. Not quite, but almost.

That's a good point, a valid point in my view, and one I made on another forum. If Pi is taken as the unit then its representation becomes "1", so obviously no longer a transcendental. However, this only concerns physics because physics care about units. Maths doesn't care about units as far as I understand it. So the question of the existence of something like a Pi quantity of something physical is meaningless because you can always change units and make the Pi number disappear as a quantity of something physical. However, the ratio of a circle's circumference to its radius still involves Pi. A ratio is not quite a quantity, though, but it seems real enough. I would say it's a property of the real world. And changing units won't make that transcendental Pi disappear.

So the question that needs to be answered here, and I admit I'm stuck in this respect, is whether it's possible to make Pi disappear in maths, too. Changing units won't do. Is there something else? Is there something like representing numbers in base Pi, for example? In this case, Pi's representation in base Pi would be "10" and its transcendental nature would disappear.

Still, I would assume that other numbers, like 1 and 23867 for instance, expressed in base Pi, would now get a transcendental representation. So the problem still wouldn't go away.

Any view on that?
EB

 

[Speakpigeon]

Ok, it's the ratio between the circumference and diameter of a circle. Still almost infinitely short, compared to an infinite series.

"Pi" is a label. it is not a way of describing that number.

What you might be trying to get at is that pi is a "computable number". It is one that can be computed to arbitrary accuracy by running a Turing machine some suitable finite number of steps. In other words, using a finite-sized algorithm whose size stays fixed as one goes to higher and higher accuracy, even as it is run for more and more steps.

Computable numbers include all algebraic numbers and most familiar transcendental numbers, like e and pi. Chaitin's family of constants are non-computable, since they are associated with whether or not a Turing machine will halt, and there is no Turing machine that can decide that in the general case.

 Definable real number mentions "Definability in models of ZFC" (set theory with the Zermelo-Fraenkel axioms and the axiom of choice).

A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (see Kunen 1980, p. 153). This notion cannot be expressed as a formula in the language of set theory.

So I will call such numbers "definable numbers". These numbers include Chaitin's family of constants.

All these sets of numbers are countable, with cardinality aleph-0:

• Positive integers
• Nonnegative integers
• Integers
• Rational numbers
• Algebraic real numbers
• Computable real numbers
• Definable real numbers

Each set contains the sets above it in this list.

There are infinitely more real numbers than any of these kinds of numbers, meaning that nearly all real numbers are forever beyond our grasp. We know that they exist, but we cannot specify any of them with a finite-sized specification.

Yes, but the idea I think is that we can specify Pi uniquely through a finite expression "the ratio between the circumference and diameter of a circle", as Kharakov put it. This in turn provide the rationale for characterising Pi as "computable", though never as actually computed. So, I guess the idea is that maybe we could do something similar with non-computable numbers...
EB

 

[Speakpigeon]

And showing people their philosophical errors is important.

Sadly something you've never ever managed to achieve, in one zillion of posts. Making a convincing argument is clearly beyond your linguistic capabilities. It's probably also beyond your mental capabilities. What is telling is that you don't seem to realise the futility of your "efforts" in this respect. You're reduced to making noises. At least you get the attention of other posters. Is that gratifying enough for you? Seems so.
EB