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The field of “useful reals” between rational and real numbers (2019)

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Re: The field of “useful reals” between rational and real numbers (2019)

#111

Post by shawnz »

perl4ever wrote: post #108
shawnz wrote: post #66

Earlier quoted context omitted.

Clearly every single rational number is "useful", plus others that are not rational

So I guess what I lack is an understanding of why that doesn't affect cardinality.

There is a 1-to-1 mapping between the sets, so by definition that means they have the same cardinality.

Let me give a more common example. Consider these two sets: N, the set of non-negative integers, and Z, the set of all integers. Clearly, everything in N is also in Z, and then some. But N and Z still have the same cardinality, because there are 1-to-1 mappings between the two sets. Here is one example of such a mapping:

    N  |  Z
   ---------
    0 ->  0
    1 ->  1
    2 -> -1
    3 ->  2
    4 -> -2
    5 ->  3
    6 -> -3
..etc. The formula for this mapping would be floor(n/2)*(-1)^(n%2). Clearly everything in the left set has exactly one corresponding item in the right set and vice versa, so they must be the same "size", even though the right set contains every item in the left set and then some.

Re: The field of “useful reals” between rational and real numbers (2019)

#112

Post by dchyrdvh »

PureParadigm wrote: post #110
dchyrdvh wrote: post #91

Earlier quoted context omitted.

We don't "come up" with ideas from other ideas using some closed form rules of logic, like in Coq or some Turing machine. Instead, we discover new ideas. There is a world of ideas and the real world. People live in both worlds. When they discover a new idea, often by accident, they label it with a symbol and use it in the real world. Other people can see the same idea and since they can't fully describe it with words…

Even if ideas come from an uncountable set (not convinced yet), there are still only countably many ideas people will ever have. Each time anyone comes up with an idea, I can assign it a new integer.

Im merely trying to drag the concept of separating ideas and reality as two different but very real worlds under the spotlight of everyone's attention. This concept is fundamental and very old. I won't be able to defend this idea with formal proofs.

Re: The field of “useful reals” between rational and real numbers (2019)

#113

Post by dchyrdvh »

lisper wrote: post #97
dchyrdvh wrote: post #92

Earlier quoted context omitted.

So long as every real number exists, has properties and so on. Every such number is a separate idea. They exist, no matter whether we know about them or not.

Ah. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.

We can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.

Re: The field of “useful reals” between rational and real numbers (2019)

#114

Post by lisper »

dchyrdvh wrote: post #113
lisper wrote: post #97

Earlier quoted context omitted.

Ah. Personally, I distinguish between potential ideas and actual ideas. To be an actual idea, it has to reside in someone's brain (or a computer, or some other data-processing system). The reals correspond to the set of potential ideas, but the set of actual ideas is not only countable, but almost certainly finite.

We can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.

Proofs must be finite so there can only be countably many of them.

Re: The field of “useful reals” between rational and real numbers (2019)

#115

Post by dchyrdvh »

lisper wrote: post #114
dchyrdvh wrote: post #113

Earlier quoted context omitted.

We can call it a materialized idea, like an implemented software algorithm. I'm indeed talking about the world of ideas that's not real, i.e. non material. The proof of the Fermat's theorem has always existed, but only recently it's been discovered by Wales.

Proofs must be finite so there can only be countably many of them.

The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.

Re: The field of “useful reals” between rational and real numbers (2019)

#116

Post by lisper »

dchyrdvh wrote: post #115
lisper wrote: post #114

Earlier quoted context omitted.

Proofs must be finite so there can only be countably many of them.

The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.

> The same word can have infinitely many meanings.

But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.

Re: The field of “useful reals” between rational and real numbers (2019)

#117

Post by dchyrdvh »

lisper wrote: post #116
dchyrdvh wrote: post #115

Earlier quoted context omitted.

The same word can have infinitely many meanings. So even if we restrict the length of proofs to 140 chars and restrict the alphabet to Latin, there will be infinitely many proofs there: well just start inventing new meanings for the same words.

> The same word can have infinitely many meanings. But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.

What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.

Re: The field of “useful reals” between rational and real numbers (2019)

#118

Post by lisper »

dchyrdvh wrote: post #117
lisper wrote: post #116

Earlier quoted context omitted.

> The same word can have infinitely many meanings. But only countably many because definitions have to be finite too. The combination of proof + definitions must also be finite, so there can only be countably many of them.

What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.

> What's the definition of "set"?

It is a function from objects onto booleans.

> what's the definition of the implication symbol

The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions.

Next question?

Re: The field of “useful reals” between rational and real numbers (2019)

#119

Post by dchyrdvh »

lisper wrote: post #118
dchyrdvh wrote: post #117

Earlier quoted context omitted.

What's the definition of "set"? Or what's the definition of the implication symbol, i.e. when someone says that something obviously follows from the previous theorems? We don't bother to define a lot of foundational things in math.

> What's the definition of "set"? It is a function from objects onto booleans. > what's the definition of the implication symbol The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions. Next question?

Well, try to define a function without the recursion to sets.

Re: The field of “useful reals” between rational and real numbers (2019)

#120

Post by lisper »

dchyrdvh wrote: post #119
lisper wrote: post #118

Earlier quoted context omitted.

> What's the definition of "set"? It is a function from objects onto booleans. > what's the definition of the implication symbol The implication symbol doesn't have a definition, it's part of a completely different kind of reasoning process. Formal symbolic reasoning is a completely different animal than informal arguments involving words that have definitions. Next question?

Well, try to define a function without the recursion to sets.

Definitional recursion has to bottom out somewhere. (OK, it can also be circular, but I'm guessing you would not find that satisfactory.) Whatever words I use to define "function" you can always turn around and insist that I define those words. It's a never-ending game. It ultimately boils down to the definitions of words like "true" and "false, "same" and "different", whose meanings can only be communicated by way of examples: X and X are the same, X and Y are different.

But none of this has anything to do with the matter at hand. There are a finite number of atoms in the universe. Those atoms can only arrange themselves into a finite number of sentient creatures (or computers), each of which has only a finite brain in which can reside only a finite number of thoughts. So no matter how you slice it, the number of realized ideas in this universe is going to be not only countable but actually finite because there is only a finite amount of time before heat death.

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