Defining a transcendental number and countability

• I
BWV
Is there a term for transcendental numbers that cannot be specified by an operation with a finite amount of data?

for example pi or e have various finite definitions and one could generate other transcendental numbers with operations on these.

On the other hand if n= some randomly chosen number on the real number line, there is no real way to express or differentiate that number from some other arbitrarily close transcendental number

The transcendental numbers are uncountable, but is the set of transcendental numbers that can be defined by some finite operation countable?

Homework Helper
Gold Member
I wonder how you define "define". When you say that ##\pi## has a finite definition, I think that you are allowing a wide definition of "define". If that is true, just identifying a transfinite number that has no finite definition might define it. It hurts my head.

BWV
I wonder how you define "define". When you say that ##\pi## has a finite definition, I think that you are allowing a wide definition of "define". If that is true, just identifying a transfinite number that has no finite definition might define it. It hurts my head.
Yes, struggling with the terminology, by define I mean uniquely identify the number - with pi or e one could (mathematically, perhaps not physically) compute any finite sequence of digits within it and differentiate it from some other transcendental number arbitrarily close to it

##\pi## is the ratio of a circle's circumference to its diameter. Is that a definition?

Staff Emeritus
Gold Member
mfb, BWV and FactChecker
BWV
I think you're looking for something like this
https://en.m.wikipedia.org/wiki/Computable_number

A fun fact is that the computable numbers are countable, so almost every real number is not computable.
yes, that is what I was trying to ask, thanks

Homework Helper
I think you're looking for something like this
https://en.m.wikipedia.org/wiki/Computable_number

A fun fact is that the computable numbers are countable, so almost every real number is not computable.
Note that there are numbers that can be defined but cannot be computed. e.g Chaitin's omega.

BWV and PeroK
BWV
So is if the number ‘take pi and replace decimal places 601-610 with my phone number’ is computable, you could generate an infinite amount of arbitrary real numbers with instructions like this - simply manipulating digits of another computable irrational number. But you could not create an uncountable set of these?