I think you are talking about finite numbers here rather than infinite numbers. So - actually that's something I looked into as part of my research into finitist maths.
There's an interesting paper here by Van Dantzig called Is 10^10^10 a finite number?
So - that's a number that is too large to express as a row of digits in decimal format in practice. Though - you could write it if you could convert all the stars and galaxies in the observable universe into digits.
Something like 10^10^100 is too large for that though.
So what about numbers you can express in any possible way?
Well you can describe enormously huge numbers using Knuth's up-arrow notation
and so on.
That's just a start.
We could define
A(a,b) = a (a up arrows) b.
Or
A(a,b) = a followed by a (a up arrows) b of those up arrows, then b
etc.
for instance,
But of course, they are all - though huge compared with most calculations we do in our ordinary life - absolutely tiny compared to infinity.
And they are really easy to write out with a few symbols once you've got the trick of them.
So - then - is there a "number too large to write out with our maths"?
Well that gets tricky.
Seems intuitively that surely there must be. But - first, what does "our maths" here mean?
And - does that include proving that the number exists? And if so - what counts as "exists" here?
An example of a very large number we can define uses the "busy beaver" function.
The Busy Beaver Problem
So here, essentially
B(n) = largest number of consecutive 1s that can be printed out by a program with n symbols of computer code.
For instance would be easy to write a short computer program to write out 10^10^10 "1"s.
It's defined using "Turing machines" - this is a simplified programming language, of mathematical interest rather than meant to be of any use for writing computer code.
- but is essentially the same idea if you use any language, e.g. you could define it in c code. E.g. any c program for, e.g. printing out sequence of "1"s - is equivalent to some Turing machine program and vice versa
So - does B(n) really define a number?
For example, this simple program (in pseudo code):
So - does this program go on for ever?
To find out, we need to solve Goldbach's conjecture, and at present nobody knows how to do that.
Goldbach's conjecture
If it stops, then it gives us a finite number of "1"s, so defines a number - which may be absolutely huge, no reason why not really.
If it goes on for ever then it doesn't.
So - the definition of B(n) requires us to solve many unsolved conjectures.
As it turns out, you can prove that we can never find a method of computing B(n) for all n. It's what is called a "non computable" function.
We've found the first few values, but it's believed that probably B(6) will never be calculated.
Philosophers differ here on whether or not the busy beaver function really does define a function.
Intuitionists require all proofs and definitions to be constructive. And the busy beaver function is not.
So e.g. if I define f(10,000) = largest number of "1"s you can print out with a computer program of 10,000 characters
there is no way, even in principle, to find out what that number is and what the computer program is that generates that number. So that's not a constructive definition of a number.
Intuitionists have no problem with the likes of PI. Though we can only write out finitely many digits of PI in practise, we have a procedure we can use to write it out to as many digits as we like in principle. So that's "constructive".
But with the likes of the Goldbach conjecture - there is no procedure we can follow that is guaranteed to either prove the conjecture or find the first counter example.
Most working mathematicians use classical logic. For them, then the B(10,000) is a perfectly acceptable number, all though not computable.
So, in classical logic, then the busy beaver definition does define a function.
And with classical logic you can then go on and define a whole heirarchy of "super busy beaver" functions that are even faster than the busy beaver function.
see
Who Can Name the Bigger Number?
Then in that paper he has this as the challenge:
But - now - let's increase that time out to, say, 1 year. And you can use as much paper as you like.
So, let's define n as the largest finite number a human mathematician could define with a year of work using any papers in the existing published maths literature, with probability of defining it at least 1 in a million?
With two versions - constructive, or classical, depending on your preference.
Does that define a number?
What about a thousand years of work by a project involving thousand mathematicians full time? And reduce probability of success to 1 in a trillion?
Does that define a number?
It's a vague definition of a number, nevertheless - it does seem to define, or at least perhaps, hint, at something - and it is far larger than anything you can define in a few minutes, obviously.
And you can go on and invent increasingly vague natural language definitions of numbers, which nevertheless, seem to make some sense, of a sort.
Intuitively, seems obvious - humans are limited in time and space. There must be some limit to what is the largest number we can define.
And if you limit it to particular types of notation in a simple limited notation system, e.g. to write it out in decimal - is easy to come up with bounds.
But when you permit the full expressiveness of maths and computing, is really hard to come up with any sensible bound on the largest number we could define, or to find a number too large to define.
As you try -it gets more and more vague.
After all if you could define an upper bound to the largest number a human can write out - then since you are human, then you have already defined a larger number in this indirect way.
Still - it does seem intuitively - that we can't possibly be able to define arbitrarily large numbers, there must be some limitation to it. Anything we can define is surely still tiny compared with infinity.
But actually getting a handle on what that limitation is, that's rather elusive. You can just hint at it, that's all.
Both for constructivists who need a construction of a number before they can define it - or classical maths where you can define e.g. B(10,000) and assert it means something although we have no way to compute it by definition - both ways you get this same problem, that surely we must be limited, what numbers we can define, in a practical way, in a practical length of time.
E.g. surely there are numbers too large for any humans or any ETs or any creature to define them, within our observable universe, using on the materials within the universe, and within, say, a billion years of work.
Or indeed, even if you allow unlimited time to work on the problem (as could be possible in some ways of thinking about the universe, e.g. in Freeman Dyson's ideas Page on www.aleph.se), you'd think there must surely be a limit to what even the most ingenious mathematician would be able to express in a formula that can be physically written inside of our universe.
But if so, it seems we can only hint at our limits in an indirect way.
Note, you have to be careful because of the Berry paradox - "The smallest positive integer not definable in under eleven words."
- which I just defined in ten words.
So you might think, it is easy just define U, say, as the smallest number not definable by humans - we don't know what it is, but can still define it.
But - such a simple definition is not going to work because of a Berry type paradox - if that defines a number then it is obviously definable by us, a paradox.
Yet we surely can define enormously large numbers in a clear unambiguous way. And surely there is some limit to what we can do here - but we can only hint at that possibility - any attempt to pin it down becomes yet another way of defining even larger numbers - or else - drops us into paradox.
There's an interesting paper here by Van Dantzig called Is 10^10^10 a finite number?
So - that's a number that is too large to express as a row of digits in decimal format in practice. Though - you could write it if you could convert all the stars and galaxies in the observable universe into digits.
Something like 10^10^100 is too large for that though.
So what about numbers you can express in any possible way?
Well you can describe enormously huge numbers using Knuth's up-arrow notation
That's just a start.
We could define
A(a,b) = a (a up arrows) b.
Or
A(a,b) = a followed by a (a up arrows) b of those up arrows, then b
etc.
for instance,
But of course, they are all - though huge compared with most calculations we do in our ordinary life - absolutely tiny compared to infinity.
And they are really easy to write out with a few symbols once you've got the trick of them.
So - then - is there a "number too large to write out with our maths"?
Well that gets tricky.
Seems intuitively that surely there must be. But - first, what does "our maths" here mean?
And - does that include proving that the number exists? And if so - what counts as "exists" here?
An example of a very large number we can define uses the "busy beaver" function.
The Busy Beaver Problem
So here, essentially
B(n) = largest number of consecutive 1s that can be printed out by a program with n symbols of computer code.
For instance would be easy to write a short computer program to write out 10^10^10 "1"s.
It's defined using "Turing machines" - this is a simplified programming language, of mathematical interest rather than meant to be of any use for writing computer code.
- but is essentially the same idea if you use any language, e.g. you could define it in c code. E.g. any c program for, e.g. printing out sequence of "1"s - is equivalent to some Turing machine program and vice versa
So - does B(n) really define a number?
For example, this simple program (in pseudo code):
Generate the even numbers
2, 4, 6, 8, ...
For each number, see if it is a sum of two primes. If it is, print out a "1". If it isn't, stop.
So - does this program go on for ever?
To find out, we need to solve Goldbach's conjecture, and at present nobody knows how to do that.
Goldbach's conjecture
If it stops, then it gives us a finite number of "1"s, so defines a number - which may be absolutely huge, no reason why not really.
If it goes on for ever then it doesn't.
So - the definition of B(n) requires us to solve many unsolved conjectures.
As it turns out, you can prove that we can never find a method of computing B(n) for all n. It's what is called a "non computable" function.
We've found the first few values, but it's believed that probably B(6) will never be calculated.
Philosophers differ here on whether or not the busy beaver function really does define a function.
Intuitionists require all proofs and definitions to be constructive. And the busy beaver function is not.
So e.g. if I define f(10,000) = largest number of "1"s you can print out with a computer program of 10,000 characters
there is no way, even in principle, to find out what that number is and what the computer program is that generates that number. So that's not a constructive definition of a number.
Intuitionists have no problem with the likes of PI. Though we can only write out finitely many digits of PI in practise, we have a procedure we can use to write it out to as many digits as we like in principle. So that's "constructive".
But with the likes of the Goldbach conjecture - there is no procedure we can follow that is guaranteed to either prove the conjecture or find the first counter example.
Most working mathematicians use classical logic. For them, then the B(10,000) is a perfectly acceptable number, all though not computable.
So, in classical logic, then the busy beaver definition does define a function.
And with classical logic you can then go on and define a whole heirarchy of "super busy beaver" functions that are even faster than the busy beaver function.
see
Who Can Name the Bigger Number?
Then in that paper he has this as the challenge:
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
But - now - let's increase that time out to, say, 1 year. And you can use as much paper as you like.
So, let's define n as the largest finite number a human mathematician could define with a year of work using any papers in the existing published maths literature, with probability of defining it at least 1 in a million?
With two versions - constructive, or classical, depending on your preference.
Does that define a number?
What about a thousand years of work by a project involving thousand mathematicians full time? And reduce probability of success to 1 in a trillion?
Does that define a number?
It's a vague definition of a number, nevertheless - it does seem to define, or at least perhaps, hint, at something - and it is far larger than anything you can define in a few minutes, obviously.
And you can go on and invent increasingly vague natural language definitions of numbers, which nevertheless, seem to make some sense, of a sort.
Intuitively, seems obvious - humans are limited in time and space. There must be some limit to what is the largest number we can define.
And if you limit it to particular types of notation in a simple limited notation system, e.g. to write it out in decimal - is easy to come up with bounds.
But when you permit the full expressiveness of maths and computing, is really hard to come up with any sensible bound on the largest number we could define, or to find a number too large to define.
As you try -it gets more and more vague.
After all if you could define an upper bound to the largest number a human can write out - then since you are human, then you have already defined a larger number in this indirect way.
Still - it does seem intuitively - that we can't possibly be able to define arbitrarily large numbers, there must be some limitation to it. Anything we can define is surely still tiny compared with infinity.
But actually getting a handle on what that limitation is, that's rather elusive. You can just hint at it, that's all.
Both for constructivists who need a construction of a number before they can define it - or classical maths where you can define e.g. B(10,000) and assert it means something although we have no way to compute it by definition - both ways you get this same problem, that surely we must be limited, what numbers we can define, in a practical way, in a practical length of time.
E.g. surely there are numbers too large for any humans or any ETs or any creature to define them, within our observable universe, using on the materials within the universe, and within, say, a billion years of work.
Or indeed, even if you allow unlimited time to work on the problem (as could be possible in some ways of thinking about the universe, e.g. in Freeman Dyson's ideas Page on www.aleph.se), you'd think there must surely be a limit to what even the most ingenious mathematician would be able to express in a formula that can be physically written inside of our universe.
But if so, it seems we can only hint at our limits in an indirect way.
Note, you have to be careful because of the Berry paradox - "The smallest positive integer not definable in under eleven words."
- which I just defined in ten words.
So you might think, it is easy just define U, say, as the smallest number not definable by humans - we don't know what it is, but can still define it.
But - such a simple definition is not going to work because of a Berry type paradox - if that defines a number then it is obviously definable by us, a paradox.
Yet we surely can define enormously large numbers in a clear unambiguous way. And surely there is some limit to what we can do here - but we can only hint at that possibility - any attempt to pin it down becomes yet another way of defining even larger numbers - or else - drops us into paradox.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
4.8m answer views110.4k this month
Top Writer2017, 2016, and 2015
Published WriterHuffPost, Slate, and 4 moreHuffPost, Slate, Forbes, Newsweek, Business Insider, and Science 2.0